trigonometry problem solving with right triangles

Chapter 4.2: Right Triangle Trigonometry

Using right triangle trigonometry to solve applied problems, using trigonometric functions.

In previous examples, we evaluated the sine and cosine in triangles where we knew all three sides. But the real power of right-triangle trigonometry emerges when we look at triangles in which we know an angle but do not know all the sides.

How To: Given a right triangle, the length of one side, and the measure of one acute angle, find the remaining sides.

• For each side, select the trigonometric function that has the unknown side as either the numerator or the denominator. The known side will in turn be the denominator or the numerator.
• Write an equation setting the function value of the known angle equal to the ratio of the corresponding sides.
• Using the value of the trigonometric function and the known side length, solve for the missing side length.

Example 5: Finding Missing Side Lengths Using Trigonometric Ratios

Find the unknown sides of the triangle in Figure 11.

We know the angle and the opposite side, so we can use the tangent to find the adjacent side.

We rearrange to solve for [latex]a[/latex].

We can use the sine to find the hypotenuse.

Again, we rearrange to solve for [latex]c[/latex].

A right triangle has one angle of [latex]\frac{\pi }{3}[/latex] and a hypotenuse of 20. Find the unknown sides and angle of the triangle.

Right-triangle trigonometry has many practical applications. For example, the ability to compute the lengths of sides of a triangle makes it possible to find the height of a tall object without climbing to the top or having to extend a tape measure along its height. We do so by measuring a distance from the base of the object to a point on the ground some distance away, where we can look up to the top of the tall object at an angle. The angle of elevation of an object above an observer relative to the observer is the angle between the horizontal and the line from the object to the observer’s eye. The right triangle this position creates has sides that represent the unknown height, the measured distance from the base, and the angled line of sight from the ground to the top of the object. Knowing the measured distance to the base of the object and the angle of the line of sight, we can use trigonometric functions to calculate the unknown height. Similarly, we can form a triangle from the top of a tall object by looking downward. The angle of depression of an object below an observer relative to the observer is the angle between the horizontal and the line from the object to the observer’s eye.

How To: Given a tall object, measure its height indirectly.

• Make a sketch of the problem situation to keep track of known and unknown information.
• Lay out a measured distance from the base of the object to a point where the top of the object is clearly visible.
• At the other end of the measured distance, look up to the top of the object. Measure the angle the line of sight makes with the horizontal.
• Write an equation relating the unknown height, the measured distance, and the tangent of the angle of the line of sight.
• Solve the equation for the unknown height.

Example 6: Measuring a Distance Indirectly

To find the height of a tree, a person walks to a point 30 feet from the base of the tree. She measures an angle of [latex]57^\circ [/latex] between a line of sight to the top of the tree and the ground, as shown in Figure 13. Find the height of the tree.

We know that the angle of elevation is [latex]57^\circ [/latex] and the adjacent side is 30 ft long. The opposite side is the unknown height.

The trigonometric function relating the side opposite to an angle and the side adjacent to the angle is the tangent. So we will state our information in terms of the tangent of [latex]57^\circ [/latex], letting [latex]h[/latex] be the unknown height.

The tree is approximately 46 feet tall.

How long a ladder is needed to reach a windowsill 50 feet above the ground if the ladder rests against the building making an angle of [latex]\frac{5\pi }{12}[/latex] with the ground? Round to the nearest foot.

• Precalculus. Authored by : OpenStax College. Provided by : OpenStax. Located at : http://cnx.org/contents/[email protected]:1/Preface . License : CC BY: Attribution
• Example: Determine the Length of a Side of a Right Triangle Using a Trig Equation. Authored by : Mathispower4u. Located at : https://youtu.be/8jU2R3BuR5E . License : All Rights Reserved . License Terms : Standard YouTube License

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Key Questions

In Triangle #ABC# with the right angle at #C# , let #a# , #b# , and #c# be the opposite, the adjacent, and the hypotenuse of #angle A# . Then, we have

#sin A=a/c Rightarrow "m"angle A=sin^{-1}(a/c)#

#sin B=b/c Rightarrow "m"angle B=sin^{-1}(b/c)#

I hope that this was helpful.

Right triangles are a fundamental concept in geometry, and understanding how to solve them is crucial in various fields, including engineering, physics, and architecture. Trigonometry provides the tools to tackle these problems effectively. This guide will walk you through the process of solving right triangles using trigonometric functions and the Pythagorean Theorem.

Understanding Right Triangles

A right triangle is a triangle with one angle measuring 90 degrees. The side opposite the right angle is called the hypotenuse, which is always the longest side. The other two sides are called legs.

Trigonometric functions, such as sine (sin), cosine (cos), and tangent (tan), relate the angles of a right triangle to the ratios of its sides. These functions are defined as follows:

• Sine (sin): Opposite side / Hypotenuse
• Cosine (cos): Adjacent side / Hypotenuse
• Tangent (tan): Opposite side / Adjacent side

Solving Right Triangles: Different Scenarios

There are two primary scenarios when solving right triangles:

1. Given One Acute Angle and One Side

If you know one acute angle (other than the right angle) and the length of one side, you can find all the missing parts of the triangle using trigonometric functions.

Suppose you have a right triangle with one angle measuring 30 degrees and the hypotenuse measuring 10 cm. You can find the lengths of the legs using sin and cos:

• Find the opposite side: sin(30°) = Opposite side / 10 cm. Solving for the opposite side, we get: Opposite side = sin(30°) * 10 cm = 5 cm.
• Find the adjacent side: cos(30°) = Adjacent side / 10 cm. Solving for the adjacent side, we get: Adjacent side = cos(30°) * 10 cm ≈ 8.66 cm.

2. Given Two Sides

If you know the lengths of two sides, you can use the Pythagorean Theorem and trigonometric functions to find the missing side and angles.

Consider a right triangle with one leg measuring 6 cm and the hypotenuse measuring 10 cm. You can find the other leg and the angles using the following steps:

• Find the other leg: Using the Pythagorean Theorem (a² + b² = c²), where a and b are the legs and c is the hypotenuse, we get: 6² + b² = 10². Solving for b, we get: b = √(10² – 6²) = √64 = 8 cm.
• Find the angles: Using the trigonometric functions, we can find the angles. For example, to find the angle opposite the 6 cm side, we can use sin: sin(θ) = 6 cm / 10 cm. Solving for θ, we get: θ = sin⁻¹(6/10) ≈ 36.87°.

Key Points to Remember

• Always remember the definitions of sine, cosine, and tangent.
• The Pythagorean Theorem is essential for solving right triangles when two sides are known.
• Use a calculator to find the trigonometric values and inverse trigonometric functions.

Solving right triangles using trigonometry is a fundamental skill with wide applications. By understanding the relationships between angles and sides, you can effectively solve for missing parts of right triangles in various practical situations. Practice applying the concepts and formulas to build your proficiency in this essential area of mathematics.

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Chapter 2: Trigonometric Ratios

Exercises: 2.3 Solving Right Triangles

Suggested Homework Problems

Exercises Homework 2.3

Exercise group.

For Problems 1–4, solve the triangle. Round answers to hundredths.

For Problems 5–10,

• Sketch the right triangle described.
• Solve the triangle.

[latex]A = 42{^o}, c = 26[/latex]

[latex]B = 28{^o}, c = 6.8[/latex]

[latex]B = 33{^o}, a = 300[/latex]

[latex]B = 79{^o}, a = 116[/latex]

[latex]A = 12{^o}, a = 4[/latex]

[latex]A = 50{^o}, a = 10[/latex]

For Problems 11–16,

• Without doing the calculations, list the steps you would use to solve the triangle.

[latex]B = 53.7{^o}, b = 8.2[/latex]

[latex]B = 80{^o}, a = 250[/latex]

[latex]A = 25{^o}, b = 40[/latex]

[latex]A = 15{^o}, c = 62[/latex]

[latex]A = 64.5{^o}, c = 24[/latex]

[latex]B = 44{^o}, b = 0.6[/latex]

For Problems 17–22, find the labeled angle. Round your answer to tenths of a degree.

For Problems 23–28, evaluate the expression and sketch a right triangle to illustrate.

[latex]\sin^{-1} 0.2[/latex]

[latex]\cos^{-1} 0.8[/latex]

[latex]\tan^{-1} 1.5[/latex]

[latex]\tan^{-1} 2.5[/latex]

[latex]\cos^{-1} 0.2839[/latex]

[latex]\sin^{-1} 0.4127[/latex]

For Problems 29–32, write two different equations for the statement.

The cosine of [latex]15 {^o}[/latex] is [latex]0.9659\text{.}[/latex]

The sine of [latex]70 {^o}[/latex] is [latex]0.9397\text{.}[/latex]

The angle whose tangent is [latex]3.1445[/latex] is [latex]65 {^o}\text{.}[/latex]

The angle whose cosine is [latex]0.0872[/latex] is [latex]85 {^o}\text{.}[/latex]

Evaluate the expressions and explain what each means. [latex]\begin{equation*} \sin^{-1} (0.6), (\sin 6{^o})^{-1} \end{equation*}[/latex]

Evaluate the expressions and explain what each means. [latex]\begin{equation*} \cos^{-1} (0.36), (\cos 36{^o})^{-1} \end{equation*}[/latex]

For Problems 35–38,

• Sketch a right triangle that illustrates the situation. Label your sketch with the given information.
• Choose the appropriate trig ratio and write an equation, then solve the problem.

The gondola cable for the ski lift at Snowy Peak is [latex]2458[/latex] yards long and climbs [latex]1860[/latex] feet. What angle with the horizontal does the cable make?

The Leaning Tower of Pisa is [latex]55[/latex] meters in length. An object dropped from the top of the tower lands [latex]4.8[/latex] meters from the base of the tower. At what angle from the horizontal does the tower lean?

A mining company locates a vein of minerals at a depth of [latex]32[/latex] meters. However, there is a layer of granite directly above the minerals, so they decide to drill at an angle, starting [latex]10[/latex] meters from their original location. At what angle from the horizontal should they drill?

The birdhouse in Carolyn’s front yard is [latex]12[/latex] feet tall, and its shadow at [latex]4[/latex] pm is [latex]15[/latex] feet [latex]4[/latex] inches long. What is the angle of elevation of the sun at [latex]4[/latex] pm?

For Problems 39–42,

[latex]a = 18, b = 26[/latex]

[latex]a = 35, b = 27[/latex]

[latex]b = 10.6 , c = 19.2[/latex]

[latex]a = 88, c = 132[/latex]

For Problems 43–48,

• Make a sketch that illustrates the situation. Label your sketch with the given information.
• Write an equation and solve the problem.

The Mayan pyramid of El Castillo at Chichén Itzá in Mexico has [latex]91[/latex] steps. Each step is 26 cm high and 30 cm deep.

• What angle does the side of the pyramid make with the horizontal?
• What is the distance up the face of the pyramid, from base to top platform?

An airplane begins its descent when its altitude is 10 kilometers. The angle of descent should be [latex]3{^o}[/latex] from horizontal.

• How far from the airport (measured along the ground) should the airplane begin its descent?
• How far will the airplane travel on its descent to the airport?

A communications satellite is in a low earth orbit (LOE) at an altitude of 400 km. From the satellite, the angle of depression to earth’s horizon is [latex]19.728{^o}\text{.}[/latex] Use this information to calculate the radius of the earth.

The first Ferris wheel was built for the [latex]1893[/latex] Chicago World’s Fair. It had a diameter of [latex]250[/latex] feet, and the boarding platform, at the base of the wheel, was [latex]14[/latex] feet above the ground. If you boarded the wheel and rotated through an angle of [latex]50{^o}\text{,}[/latex] what would be your height above the ground?

To find the distance across a ravine, Delbert takes some measurements from a small airplane. When he is a short distance from the ravine at an altitude of [latex]500[/latex] feet, he finds that the angle of depression to the near side of the ravine is [latex]56{^o}\text{,}[/latex] and the angle of depression to the far side is [latex]32{^o}\text{.}[/latex] What is the width of the ravine? (Hint: First find the horizontal distance from Delbert to the near side of the ravine.)

The window in Francine’s office is [latex]4[/latex] feet wide and [latex]5[/latex] feet tall. The bottom of the window is 3 feet from the floor. When the sun is at an angle of elevation of [latex]64{^o}\text{,}[/latex] what is the area of the sunny spot on the floor?

Which of the following numbers are equal to [latex]\cos 45{^o}\text{?}[/latex]

• [latex]\displaystyle \dfrac{\sqrt{2}}{2}[/latex]
• [latex]\displaystyle \dfrac{1}{\sqrt{2}}[/latex]
• [latex]\displaystyle \dfrac{2}{\sqrt{2}}[/latex]
• [latex]\displaystyle \sqrt{2}[/latex]

Which of the following numbers are equal to [latex]\tan 30{^o}\text{?}[/latex]

• [latex]\displaystyle \sqrt{3}[/latex]
• [latex]\displaystyle \dfrac{1}{\sqrt{3}}[/latex]
• [latex]\displaystyle \dfrac{\sqrt{3}}{3}[/latex]
• [latex]\displaystyle \dfrac{3}{\sqrt{3}}[/latex]

Which of the following numbers are equal to [latex]\tan 60{^o}\text{?}[/latex]

Which of the following numbers are equal to [latex]\sin 60{^o}\text{?}[/latex]

• [latex]\displaystyle \dfrac{3}{\sqrt{2}}[/latex]
• [latex]\displaystyle \dfrac{\sqrt{3}}{2}[/latex]
• [latex]\displaystyle \dfrac{\sqrt{2}}{3}[/latex]
• [latex]\displaystyle \dfrac{2}{\sqrt{3}}[/latex]

For Problems 53–58, choose all values from the list below that are exactly equal to, or decimal approximations for, the given trig ratio. (Try not to use a calculator!)

[latex]\cos 30{^o}[/latex]

[latex]\sin 45{^o}[/latex]

[latex]\tan 30{^o}[/latex]

[latex]\cos 60{^o}[/latex]

[latex]\sin 90{^o}[/latex]

[latex]\cos 0{^o}[/latex]

Fill in the table from memory with exact values. Do you notice any patterns that might help you memorize the values?

Fill in the table from memory with decimal approximations to four places.

For Problems 61 and 62, compare the given value with the trig ratios of the special angles to answer the questions. Try not to use a calculator.

Is the acute angle larger or smaller than [latex]45{^o}\text{?}[/latex]

• [latex]\displaystyle \sin \alpha = 0.7[/latex]
• [latex]\displaystyle \tan \beta = 1.2[/latex]
• [latex]\displaystyle \cos \gamma = 0.65[/latex]

Is the acute angle larger or smaller than [latex]60{^o}\text{?}[/latex]

• [latex]\displaystyle \cos \theta = 0.75[/latex]
• [latex]\displaystyle \tan \phi = 1.5[/latex]
• [latex]\displaystyle \sin \psi = 0.72[/latex]

For Problems 63–72, solve the triangle. Give your answers as exact values.

• Find the perimeter of a regular hexagon if the apothegm is [latex]8[/latex] cm long. (The apothegm is the segment from the center of the hexagon and perpendicular to one of its sides.)
• Find the area of the hexagon.

Triangle [latex]ABC[/latex] is equilateral, and its angle bisectors meet at point [latex]P\text{.}[/latex] The sides of [latex]\triangle ABC[/latex] are 6 inches long. Find the length of [latex]AP\text{.}[/latex]

Find an exact value for the area of each triangle.

Find an exact value for the perimeter of each parallelogram.

• Find the area of the outer square.
• Find the dimensions and the area of the inner square.
• What is the ratio of the area of the outer square to the area of the inner square?
• Find the area of the inner square.
• Find the dimensions and the area of the outer square.

Trigonometry Copyright © 2024 by Bimal Kunwor; Donna Densmore; Jared Eusea; and Yi Zhen. All Rights Reserved.

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Right Triangle Trigonometry

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Problems & Videos

Solve the following right triangles, \(\textbf{1)}\) find the missing sides and angles. show answer the answer is \(m\angle d=48^{\circ},\,\,\, e\approx 10.8,\,\,\, f\approx 16.1\), \(\textbf{2)}\) find the missing sides and angles. show answer the answer is \(m\angle a\approx55^{\circ},\,\,\, a\approx 8.2,\,\,\, c\approx 5.74\) show work \(\text{solve for side } c\) \(\,\,\,\,\,\,\sin{\theta}=\frac{opposite}{hypotenuse}\) \(\,\,\,\,\,\,\sin{35^{\circ}}=\frac{c}{10}\) \(\,\,\,\,\,\,c=10\cdot\sin{35^{\circ}}\) \(\,\,\,\,\,\,c\approx 5.74\) \(\text{solve for side a}\) \(\,\,\,\,\,\,a^2+c^2=b^2\) \(\,\,\,\,\,\,a^2+(5.74)^2=(10)^2\) \(\,\,\,\,\,\,a^2+32.95=100\) \(\,\,\,\,\,\,a^2=67.05\) \(\,\,\,\,\,\,a=\sqrt{67.05}\) \(\,\,\,\,\,\,a\approx 8.2\) \(\text{solve for }m\angle a\) \(\,\,\,\,\,\,m\angle a + m\angle b + m\angle c = 180^{\circ} \) \(\,\,\,\,\,\,m\angle a + 90^{\circ} + 35^{\circ} = 180^{\circ} \) \(\,\,\,\,\,\,m\angle a + 125^{\circ} = 180^{\circ} \) \(\,\,\,\,\,\,m\angle a = 55^{\circ} \), \(\textbf{3)}\) find the missing sides and angles. show answer the answer is \(m\angle a\approx20^{\circ},\,\,\, a\approx 1.09,\,\,\, b\approx 3.19\) show work \(\text{solve for side } b\) \(\,\,\,\,\,\,\sin{\theta}=\frac{opposite}{hypotenuse}\) \(\,\,\,\,\,\,\sin{70^{\circ}}=\frac{3}{b}\) \(\,\,\,\,\,\,b=\displaystyle\frac{3}{\sin{70^{\circ}}}\) \(\,\,\,\,\,\,b\approx 3.19\) \(\text{solve for side a}\) \(\,\,\,\,\,\,a^2+c^2=b^2\) \(\,\,\,\,\,\,a^2+(3)^2=(3.19)^2\) \(\,\,\,\,\,\,a^2+9=10.18\) \(\,\,\,\,\,\,a^2=1.18\) \(\,\,\,\,\,\,a=\sqrt{1.18}\) \(\,\,\,\,\,\,a\approx 1.08\) \(\text{solve for }m\angle a\) \(\,\,\,\,\,\,m\angle a + m\angle b + m\angle c = 180^{\circ} \) \(\,\,\,\,\,\,m\angle a + 90^{\circ} + 70^{\circ} = 180^{\circ} \) \(\,\,\,\,\,\,m\angle a + 160^{\circ} = 180^{\circ} \) \(\,\,\,\,\,\,m\angle a = 20^{\circ} \), \(\textbf{4)}\) express \(\cos{32^{\circ}}\) in terms of sine. show answer the answer is \(\sin{58^{\circ}}\) show work \(\,\,\,\,\,\cos(x)=\sin(90-x)\) \(\,\,\,\,\,\cos(32)=\sin(90-32)\) \(\,\,\,\,\,\cos(32)=\sin(58)\), \(\textbf{5)}\) express \(\sin{48^{\circ}}\) in terms of cosine. show answer the answer is \(\cos{42^{\circ}}\) show work \(\,\,\,\,\,\sin(x)=\cos(90-x)\) \(\,\,\,\,\,\sin(48)=\cos(90-48)\) \(\,\,\,\,\,\sin(48)=\cos(42)\), see related pages\(\), \(\bullet\text{ geometry homepage}\) \(\,\,\,\,\,\,\,\,\text{all the best topics…}\), \(\bullet\text{ right triangle trigonometry}\) \(\,\,\,\,\,\,\,\,\sin{(x)}=\displaystyle\frac{\text{opp}}{\text{hyp}}…\), \(\bullet\text{ angle of depression and elevation}\) \(\,\,\,\,\,\,\,\,\text{angle of depression}=\text{angle of elevation}…\), \(\bullet\text{ convert to radians and to degrees}\) \(\,\,\,\,\,\,\,\,\text{radians} \rightarrow \text{degrees}, \times \displaystyle \frac{180^{\circ}}{\pi}…\), \(\bullet\text{ degrees, minutes and seconds}\) \(\,\,\,\,\,\,\,\,48^{\circ}34’21”…\), \(\bullet\text{ coterminal angles}\) \(\,\,\,\,\,\,\,\,\pm 360^{\circ} \text { or } \pm 2\pi n…\), \(\bullet\text{ reference angles}\) \(\,\,\,\,\,\,\,\,\) \(…\), \(\bullet\text{ find all 6 trig functions}\) \(\,\,\,\,\,\,\,\,\) \(…\), \(\bullet\text{ unit circle}\) \(\,\,\,\,\,\,\,\,\sin{(60^{\circ})}=\displaystyle\frac{\sqrt{3}}{2}…\), \(\bullet\text{ law of sines}\) \(\,\,\,\,\,\,\,\,\displaystyle\frac{\sin{a}}{a}=\frac{\sin{b}}{b}=\frac{\sin{c}}{c}\) \(…\), \(\bullet\text{ area of sas triangles}\) \(\,\,\,\,\,\,\,\,\text{area}=\frac{1}{2}ab \sin{c}\) \(…\), \(\bullet\text{ law of cosines}\) \(\,\,\,\,\,\,\,\,a^2=b^2+c^2-2bc \cos{a}\) \(…\), \(\bullet\text{ area of sss triangles (heron’s formula)}\) \(\,\,\,\,\,\,\,\,\text{area}=\sqrt{s(s-a)(s-b)(s-c)}\) \(…\), \(\bullet\text{ geometric mean}\) \(\,\,\,\,\,\,\,\,x=\sqrt{ab} \text{ or } \displaystyle\frac{a}{x}=\frac{x}{b}…\), \(\bullet\text{ geometric mean- similar right triangles}\) \(\,\,\,\,\,\,\,\,\) \(…\), \(\bullet\text{ inverse trigonmetric functions}\) \(\,\,\,\,\,\,\,\,\sin {\left(cos^{-1}\left(\frac{3}{5}\right)\right)}…\), \(\bullet\text{ sum and difference of angles formulas}\) \(\,\,\,\,\,\,\,\,\sin{(a+b)}=\sin{a}\cos{b}+\cos{a}\sin{b}…\), \(\bullet\text{ double-angle and half-angle formulas}\) \(\,\,\,\,\,\,\,\,\sin{(2a)}=2\sin{(a)}\cos{(a)}…\), \(\bullet\text{ trigonometry-pythagorean identities}\) \(\,\,\,\,\,\,\,\,\sin^2{(x)}+\cos^2{(x)}=1…\), \(\bullet\text{ product-sum identities}\) \(\,\,\,\,\,\,\,\,\cos{\alpha}\cos{\beta}=\left(\displaystyle\frac{\cos{(\alpha+\beta)}+\cos{(\alpha-\beta)}}{2}\right)…\), \(\bullet\text{ cofunction identities}\) \(\,\,\,\,\,\,\,\,\sin{(x)}=\cos{(\frac{\pi}{2}-x)}…\), \(\bullet\text{ proving trigonometric identities}\) \(\,\,\,\,\,\,\,\,\sec{x}-\cos{x}=\displaystyle\frac{\tan^2{x}}{\sec{x}}…\), \(\bullet\text{ graphing trig functions- sin and cos}\) \(\,\,\,\,\,\,\,\,f(x)=a \sin{b(x-c)}+d \) \(…\), \(\bullet\text{ solving trigonometric equations}\) \(\,\,\,\,\,\,\,\,2\cos{(x)}=\sqrt{3}…\), in summary…, right triangle trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of right triangles. a right triangle is a triangle with one right angle, and the side opposite the right angle is called the hypotenuse. the other two sides are called the legs of the triangle. the trigonometric functions are used to describe the relationships between the sides and angles of a right triangle. the three main trigonometric functions are sine, cosine, and tangent. these functions are often abbreviated as sin, cos, and tan, respectively. the sine function is defined as the ratio of the length of the side opposite an angle to the length of the hypotenuse \(\left(\frac{\text{opposite}}{\text{hypotenuse}}\right)\). the cosine function is defined as the ratio of the length of the adjacent side to the length of the hypotenuse \(\left(\frac{\text{adjacent}}{\text{hypotenuse}}\right)\). and the tangent function is defined as the ratio of the length of the opposite side to the length of the adjacent side \(\left(\frac{\text{opposite}}{\text{adjacent}}\right)\). soh cah toa is a popular way to remember these relationships. another one of the important tools in right triangle trigonometry is the pythagorean theorem, which states that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides. this can be written as \(a^2 + b^2 = c^2\), where \(c\) is the hypotenuse and \(a\) and \(b\) are the legs. to solve for the acute angles in right triangle trigonometry, it is often necessary to use the inverse trigonometric functions, which are the inverse of the sine, cosine, and tangent functions. these functions are abbreviated as \(\sin^{-1}\), \(\cos^{-1}\), and \(\tan^{-1}\), respectively. right triangle trigonometry is used in a variety of fields, including geometry, engineering, and physics. it is also a useful tool for solving real-world problems., about andymath.com, andymath.com is a free math website with the mission of helping students, teachers and tutors find helpful notes, useful sample problems with answers including step by step solutions, and other related materials to supplement classroom learning. if you have any requests for additional content, please contact andy at [email protected] . he will promptly add the content. topics cover elementary math , middle school , algebra , geometry , algebra 2/pre-calculus/trig , calculus and probability/statistics . in the future, i hope to add physics and linear algebra content. visit me on youtube , tiktok , instagram and facebook . andymath content has a unique approach to presenting mathematics. the clear explanations, strong visuals mixed with dry humor regularly get millions of views. we are open to collaborations of all types, please contact andy at [email protected] for all enquiries. to offer financial support, visit my patreon page. let’s help students understand the math way of thinking thank you for visiting. how exciting, right triangle trigonometry is a branch of mathematics that deals with the study of triangles, specifically right triangles. a right triangle is a triangle in which one of the angles is a right angle (90 degrees). right triangle trigonometry is based on the principles of geometry and algebra, and it is used to find the lengths of the sides and the measure of the angles of a right triangle. it is a crucial tool for solving problems in various fields, such as engineering, physics, and architecture. right triangle trigonometry is often taught in high school math classes, usually in the geometry or algebra ii curriculum. it is an essential foundation for further study in mathematics and related fields. one common mistake students make when learning right triangle trigonometry is confusing the three primary trigonometric functions: sine, cosine, and tangent. these functions are used to relate the sides of a right triangle to the angles, and it is important to understand how to use them correctly. an interesting fact about right triangle trigonometry is that it was first developed by the ancient greeks, who used it to study the properties of triangles and to solve problems in geometry. the greek mathematician pythagoras is credited with discovering the famous pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. in addition to the pythagorean theorem, there are several other important theorems and formulas in right triangle trigonometry, such as the law of sines and the law of cosines. these theorems allow us to solve for unknown quantities in right triangles, even when we do not have a right angle. other related topics in mathematics that are closely connected to right triangle trigonometry include trigonometry in general, complex numbers, and vector calculus. these topics build on the concepts introduced in right triangle trigonometry and are often studied at the college level. in conclusion, right triangle trigonometry is a fundamental branch of mathematics that deals with the properties of right triangles. it is used in various fields and has a long history dating back to the ancient greeks. understanding right triangle trigonometry is essential for further study in mathematics and related fields. andy math 5 real world examples of right triangle trigonometry here are five real-world examples of right triangle trigonometry: surveying: surveyors use right triangle trigonometry to measure distances and angles on the land. for example, they may use a theodolite, a tool that measures angles in both the horizontal and vertical planes, to determine the elevation of a certain point. navigation: navigators and pilots use right triangle trigonometry to find their position and to determine the distance and direction to their destination. they use instruments like sextants, which measure the angle between two objects, and calculate their position using trigonometric formulas. construction: in construction, right triangle trigonometry is used to calculate the dimensions of buildings and other structures. for example, architects and engineers may use trigonometry to determine the slope of a roof or the height of a tower. sports: right triangle trigonometry is also used in sports, such as golf and baseball. golfers use trigonometry to calculate the distance to the hole and to determine the best club to use for a shot. in baseball, trigonometry is used to calculate the trajectory of a pitched ball and the distance it will travel. space exploration: right triangle trigonometry is essential for space exploration, as it is used to calculate the orbits of satellites and to navigate spacecraft. it is also used to determine the distance and size of celestial objects, such as planets and stars. andy math 5 other math topics that use right triangle trigonometry here are five other math topics that use right triangle trigonometry: trigonometry: right triangle trigonometry is a subfield of trigonometry, which is the study of triangles in general. trigonometry involves the study of the relationships between the sides and angles of triangles, and it includes concepts such as the sine, cosine, and tangent functions, which are used to relate the sides and angles of a right triangle. complex numbers: complex numbers are numbers that consist of a real part and an imaginary part. they can be represented in the form a + bi, where a and b are real numbers and i is the imaginary unit. complex numbers can be used to represent points in the complex plane, and trigonometry is used to perform geometric operations on these points. vector calculus: vector calculus is a branch of mathematics that deals with vector fields, which are functions that assign a vector to every point in space. vector calculus uses trigonometry to perform operations such as dot products, cross products, and gradient, divergence, and curl. differential equations: differential equations are equations that involve derivatives, which are used to describe how a function changes over time. trigonometry is often used to solve differential equations, especially in engineering and physics. fourier analysis: fourier analysis is a technique used to decompose a function into a sum of simpler periodic functions, called sine and cosine waves. trigonometry is used to perform fourier analysis, which is used in fields such as signal processing, image processing, and data compression. i've put out a lot of practice problems, notes and videos related to right triangle trigonometry on andymath.com. i hope it helps thank you.

Right Triangle Trigonometry Calculator

Table of contents

The right triangle trigonometry calculator can help you with problems where angles and triangles meet: keep reading to find out:

• The basics of trigonometry;
• How to calculate a right triangle with trigonometry;
• A worked example of how to use trigonometry to calculate a right triangle with steps;

And much more!

Basics of trigonometry

Trigonometry is a branch of mathematics that relates angles to the length of specific segments . We identify multiple trigonometric functions: sine, cosine, and tangent, for example. They all take an angle as their argument, returning the measure of a length associated with the angle itself. Using a trigonometric circle , we can identify some of the trigonometric functions and their relationship with angles.

As you can see from the picture, sine and cosine equal the projection of the radius on the axis, while the tangent lies outside the circle. If you look closely, you can identify a right triangle using the elements we introduced above: let's discover the relationship between trigonometric functions and this shape.

Right triangles trigonometry calculations

Consider an acute angle in the trigonometric circle above: notice how you can build a right triangle where:

• The radius is the hypotenuse; and
• The sine and cosine are the catheti of the triangle.

α \alpha α is one of the acute angles, while the right angle lies at the intersection of the catheti (sine and cosine)

Let this sink in for a moment: the length of the cathetus opposite from the angle α \alpha α is its sine , sin ⁡ ( α ) \sin(\alpha) sin ( α ) ! You just found an easy and quick way to calculate the angles and sides of a right triangle using trigonometry.

The complete relationships between angles and sides of a right triangle need to contain a scaling factor, usually the radius (the hypotenuse). Identify the opposite and adjacent . We can then write:

By switching the roles of the legs, you can find the values of the trigonometric functions for the other angle.

Taking the inverse of the trigonometric functions , you can find the values of the acute angles in any right triangle.

Using the three equations above and a combination of sides, angles, or other quantities, you can solve any right triangle . The cases we implemented in our calculator are:

• Solving the triangle knowing two sides ;
• Solving the triangle knowing one angle and one side ; and
• Solving the triangle knowing the area and one side .

Example of right triangle trigonometry calculations with steps

Take a right triangle with hypotenuse c = 5 c = 5 c = 5 and an angle α = 38 ° \alpha=38\degree α = 38° . Surprisingly enough, this is enough data to fully solve the right triangle! Follow these steps:

• Calculate the third angle: β = 90 ° − α \beta = 90\degree - \alpha β = 90° − α .
• sin ⁡ ( α ) = 0.61567 \sin(\alpha) = 0.61567 sin ( α ) = 0.61567 .
• o p p o s i t e = sin ⁡ ( α ) ⋅ h y p o t e n u s e = 0.61567 ⋅ 5 = 3.078 \mathrm{opposite} = \sin(\alpha)\cdot\mathrm{hypotenuse} = 0.61567 \cdot 5 = 3.078 opposite = sin ( α ) ⋅ hypotenuse = 0.61567 ⋅ 5 = 3.078 .
• a d j a c e n t = 0.788 ⋅ 5 = 3.94 \mathrm{adjacent} = 0.788\cdot 5 = 3.94 adjacent = 0.788 ⋅ 5 = 3.94 .

More trigonometry and right triangles calculators (and not only)

If you liked our right triangle trigonometry calculator, why not try our other related tools? Here they are:

• The trigonometry calculator ;
• The cosine triangle calculator ;
• The sine triangle calculator ;
• The trig triangle calculator ;
• The trig calculator ;
• The sine cosine tangent calculator ;
• The tangent ratio calculator ; and
• The tangent angle calculator .

How do I apply trigonometry to a right triangle?

To apply trigonometry to a right triangle, remember that sine and cosine correspond to the legs of a right triangle . To solve a right triangle using trigonometry:

• sin(α) = opposite/hypotenuse ; and
• cos(α) = adjacent/hypotenuse .
• By taking the inverse trigonometric functions , we can find the value of the angle α .
• You can repeat the procedure for the other angle.

What is the hypotenuse of a triangle with α = 30° and opposite leg a = 3?

The length of the hypotenuse is 6 . To find this result:

• Calculate the sine of α : sin(α) = sin(30°) = 1/2 .
• Apply the following formula: sin(α) = opposite/hypotenuse hypotenuse = opposite/sin(α) = 3 · 2 = 6 .

Can I apply right-triangle trigonometric rules in a non-right triangle?

Not directly: to apply the relationships between trigonometric functions and sides of a triangle, divide the shape alongside one of the heights lying inside it. This way, you can split the triangle into two right triangles and, with the right combination of data, solve it!

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Trig Word Problems #1

Now that we have a basic understanding of what the trig functions sine, cosine, and tangent represent, and we can use our calculators to find values of trig functions, we can use all of this to solve some word problems. In this reading we'll simply look at examples of word problems, and then let you give them a try. Sample #1 The sun's angle of inclination is 20 degrees, and a pole casts a 40 foot shadow. How tall is the pole?  

Solution Using the image above, X = 20 degrees, and y = 40 ft. tan X = x / y 0.3640 = x / 40 x = 14.56 ft Sample #2 A ramp is 50 feet long, and it is set at a 30 degree angle of inclination. If you walk up the ramp, how high off the ground will you be? Solution Using the image above, X = 30 degrees and z = 50 ft. sin X = x / z 0.5 = x / 50 x = 25 Sample #3 A man walks 5 miles at 60 degrees north of east. How far east of his starting point is he? Solution Using the image above, with y representing the eastern travel, x representing the northern travel, and z representing the actual path of the man, sin X = x / z 0.8660 = x / 5 x=4.33 miles.

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