Finding Missing Angles

Learn how to use inverse trigonometric functions to find unknown angles in right triangles.

Advanced25 minLesson
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Definition

When we know two sides of a right triangle but need to find an angle, we use inverse trigonometric functions (also called arc functions).
The three inverse trig functions are:
  • sin⁡−1\sin^{-1} or arcsin⁡\arcsin (inverse sine)
  • cos⁡−1\cos^{-1} or arccos⁡\arccos (inverse cosine)
  • tan⁡−1\tan^{-1} or arctan⁡\arctan (inverse tangent)
Key Concept: If sin⁡(θ)=x\sin(\theta) = x, then θ=sin⁡−1(x)\theta = \sin^{-1}(x)
θ=sin⁡−1(oppositehypotenuse)\theta = \sin^{-1}\left(\frac{\text{opposite}}{\text{hypotenuse}}\right)
θ=cos⁡−1(adjacenthypotenuse)\theta = \cos^{-1}\left(\frac{\text{adjacent}}{\text{hypotenuse}}\right)
θ=tan⁡−1(oppositeadjacent)\theta = \tan^{-1}\left(\frac{\text{opposite}}{\text{adjacent}}\right)

Try it now

If sin⁡(θ)=0.5\sin(\theta) = 0.5, what is θ\theta?

Worked Examples

In a right triangle, the side opposite to angle θ\theta is 5 cm and the hypotenuse is 10 cm. Find angle θ\theta.
1

Identify the known sides relative to the angle

Opposite = 5 cm, Hypotenuse = 10 cm → We have opposite and hypotenuse
2

Choose the appropriate ratio

sin⁡(θ)=oppositehypotenuse\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} → Use sine (SOH)
3

Set up the equation

sin⁡(θ)=510=0.5\sin(\theta) = \frac{5}{10} = 0.5 → sin⁡(θ)=0.5\sin(\theta) = 0.5
4

Apply the inverse function

θ=sin⁡−1(0.5)\theta = \sin^{-1}(0.5) → θ=30°\theta = 30°

Common Mistakes

Confusing inverse trig with reciprocal trig functions

Why it's wrong: sin⁡−1(x)\sin^{-1}(x) is NOT the same as 1sin⁡(x)\frac{1}{\sin(x)}. The notation sin⁡−1\sin^{-1} means the inverse function (arcsin), not the reciprocal.

Correct: sin⁡−1(0.5)=30°\sin^{-1}(0.5) = 30° because sin⁡(30°)=0.5\sin(30°) = 0.5. The reciprocal 1sin⁡(30°)=10.5=2\frac{1}{\sin(30°)} = \frac{1}{0.5} = 2 is completely different.

Using the wrong ratio for the given sides

Why it's wrong: Students often forget which sides correspond to which ratio. SOH-CAH-TOA only works when you correctly identify opposite and adjacent relative to the angle.

Correct: Always draw the triangle and label: the side across from the angle is opposite, the side touching the angle (not the hypotenuse) is adjacent.

Calculator in wrong mode (radians vs degrees)

Why it's wrong: If your calculator is in radian mode, sin⁡−1(0.5)=0.524\sin^{-1}(0.5) = 0.524 radians, not 30°30°.

Correct: Always check that your calculator is in degree mode (DEG) before calculating. Look for the mode indicator on your display.

Forgetting that inverse trig outputs are limited

Why it's wrong: The outputs of sin⁡−1\sin^{-1} and tan⁡−1\tan^{-1} are between −90°-90° and 90°90°. The output of cos⁡−1\cos^{-1} is between 0°0° and 180°180°.

Correct: For right triangle problems, this is usually fine since all angles are between 0°0° and 90°90°.

Watch a video explanation

The same topic explained by another teacher, if a video helps you more.

How To Solve Right Triangles Given A Side And Angle | Step-By-Step Example

by Understand The Math

Watch on YouTube

Interactive Visual

Trigonometry Problem Solver

Find Missing Angle

Practice finding missing angles using inverse trigonometric functions.

Use inverse trig functions to find the missing angle.

Right Triangle Trigonometry

θ =30°
5°45°85°
sin(θ)
0.5
Opp / Hyp
cos(θ)
√3/2
Adj / Hyp
tan(θ)
√3/3
Opp / Adj

Move the slider to change the angle and see how trigonometric ratios change.

Interactive Sandbox

Expression Calculator

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Practice Problems

15 problems
Problem 1 of 15
Easy
If sin⁡(θ)=0.5\sin(\theta) = 0.5, what is θ\theta?

Why It Matters

Finding missing angles is essential in many real-world applications:
  • Construction: Determining roof pitch angles from rise and run measurements
  • Navigation: Calculating the heading angle to reach a destination
  • Physics: Finding launch angles for projectiles
  • Engineering: Designing ramps, stairs, and support structures
  • Surveying: Measuring angles of elevation and depression
Inverse trigonometry reverses the process: instead of finding a ratio from an angle, we find an angle from a ratio!

Real World Applications

Roof Pitch Calculation

Builders use inverse tangent to determine roof angles from measurements of rise (vertical) and run (horizontal).

Example:

A roof rises 4 meters over a horizontal distance of 6 meters. The pitch angle is tan⁡−1(4/6)≈33.7°\tan^{-1}(4/6) \approx 33.7°.

1Try It Yourself

A carpenter measures that a roof rises 5 feet for every 12 feet of horizontal run.

What is the angle of the roof pitch?

Step 1: Write the mathematical expression

Use inverse tangent: tan⁡−1(rise/run)\tan^{-1}(\text{rise}/\text{run})

Aircraft Navigation

Pilots use inverse trigonometry to determine heading angles and descent paths.

Example:

A plane needs to descend 3000 feet while traveling 5 miles (26,400 feet) horizontally. The descent angle is tan⁡−1(3000/26400)≈6.5°\tan^{-1}(3000/26400) \approx 6.5°.

2Try It Yourself

A plane must descend from 10,000 feet to land, starting 40,000 feet away horizontally.

What descent angle should the pilot use?

Step 1: Write the mathematical expression

Calculate: tan⁡−1(altitude/distance)\tan^{-1}(\text{altitude}/\text{distance})

Key Takeaways

  • 1Inverse trig functions find angles when we know side ratios: sin⁡−1\sin^{-1}, cos⁡−1\cos^{-1}, tan⁡−1\tan^{-1}
  • 2Use sin⁡−1\sin^{-1} when you know opposite and hypotenuse
  • 3Use cos⁡−1\cos^{-1} when you know adjacent and hypotenuse
  • 4Use tan⁡−1\tan^{-1} when you know opposite and adjacent
  • 5Always ensure your calculator is in degree mode for angle measurements
  • 6Remember: sin⁡−1(x)\sin^{-1}(x) is the angle whose sine is xx, not the reciprocal of sine

Frequently Asked Questions

They mean exactly the same thing! Both notations represent the inverse sine function. Scientific calculators often use sin⁡−1\sin^{-1}, while mathematicians prefer arcsin⁡\arcsin.
They mean exactly the same thing! Both notations represent the inverse sine function. Scientific calculators often use sin⁡−1\sin^{-1}, while mathematicians prefer arcsin⁡\arcsin.
While sin⁡(150°)=0.5\sin(150°) = 0.5 is true, inverse functions must give a single output. By convention, sin⁡−1\sin^{-1} returns angles between −90°-90° and 90°90°. So sin⁡−1(0.5)=30°\sin^{-1}(0.5) = 30°, not 150°150°.
Look at which two sides you know relative to the angle you're finding. Use SOH-CAH-TOA backward: if you have opposite and hypotenuse, use sin⁡−1\sin^{-1}; adjacent and hypotenuse, use cos⁡−1\cos^{-1}; opposite and adjacent, use tan⁡−1\tan^{-1}.

Glossary

Inverse sine
The function sin⁡−1(x)\sin^{-1}(x) or arcsin⁡(x)\arcsin(x) that returns the angle whose sine is xx
Inverse cosine
The function cos⁡−1(x)\cos^{-1}(x) or arccos⁡(x)\arccos(x) that returns the angle whose cosine is xx
Inverse tangent
The function tan⁡−1(x)\tan^{-1}(x) or arctan⁡(x)\arctan(x) that returns the angle whose tangent is xx
Angle of elevation
The angle formed between the horizontal and a line of sight looking upward
Angle of depression
The angle formed between the horizontal and a line of sight looking downward

Formula Card

Inverse Sine

θ=sin⁡−1(oppositehypotenuse)\theta = \sin^{-1}\left(\frac{\text{opposite}}{\text{hypotenuse}}\right)

Use when you know the opposite side and hypotenuse

Inverse Cosine

θ=cos⁡−1(adjacenthypotenuse)\theta = \cos^{-1}\left(\frac{\text{adjacent}}{\text{hypotenuse}}\right)

Use when you know the adjacent side and hypotenuse

Inverse Tangent

θ=tan⁡−1(oppositeadjacent)\theta = \tan^{-1}\left(\frac{\text{opposite}}{\text{adjacent}}\right)

Use when you know the opposite and adjacent sides

Today's challengeOne problem, three tries

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