Finding Missing Sides

Learn how to use trigonometric ratios to calculate unknown side lengths in right triangles.

Advanced25 minLesson
Printable worksheet

Definition

To find a missing side in a right triangle, we use the trigonometric ratios sine, cosine, and tangent along with one known angle and one known side.
The Process:
  1. 1.Identify the known angle (not the right angle)
  2. 2.Identify the known side and unknown side
  3. 3.Label them as opposite, adjacent, or hypotenuse relative to your angle
  4. 4.Choose the correct ratio that uses both sides
  5. 5.Set up the equation and solve
SOH CAH TOA:
sin⁡(θ)=oppositehypotenuse\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
cos⁡(θ)=adjacenthypotenuse\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}
tan⁡(θ)=oppositeadjacent\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}

Try it now

In a right triangle with angle θ\theta, if you know the hypotenuse and want to find the opposite side, which formula should you use?

Worked Examples

A ladder leans against a wall at an angle of 65°65° with the ground. If the ladder is 44 meters long, how high up the wall does it reach?
1

Draw and label the triangle

The ladder is the hypotenuse (44 m), and we want the height (opposite to the 65°65° angle) → Known: hypotenuse = 44 m, angle = 65°65°
2

Choose the right ratio

We have hypotenuse and want opposite, so use sin⁡(θ)=oppositehypotenuse\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} → Use sine
3

Set up the equation

sin⁡(65°)=h4\sin(65°) = \frac{h}{4} → Equation ready
4

Solve for the unknown

h=4×sin⁡(65°)=4×0.906=3.624h = 4 \times \sin(65°) = 4 \times 0.906 = 3.624 → h≈3.62h \approx 3.62 m

Common Mistakes

Using the wrong trigonometric ratio

Why it's wrong: Students often mix up which ratio to use when they have different combinations of sides.

Correct: Always label sides first (opposite, adjacent, hypotenuse), then choose the ratio that contains both your known and unknown sides.

Confusing which side is opposite vs adjacent

Why it's wrong: Opposite and adjacent depend on which angle you're using, not fixed positions in the triangle.

Correct: The opposite side is across from your angle. The adjacent side is next to your angle (but not the hypotenuse).

Calculator in wrong mode (radians instead of degrees)

Why it's wrong: Calculators can work in degrees or radians, and using the wrong mode gives completely wrong answers.

Correct: Make sure your calculator is set to DEG mode when working with degree angles.

Dividing instead of multiplying (or vice versa)

Why it's wrong: When solving sin⁡(θ)=xh\sin(\theta) = \frac{x}{h}, students sometimes set up the algebra incorrectly.

Correct: To find the numerator, multiply: x=h×sin⁡(θ)x = h \times \sin(\theta). To find the denominator, divide: h=xsin⁡(θ)h = \frac{x}{\sin(\theta)}.

Watch a video explanation

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

How To Solve Right Triangles Given Two Sides | Step-By-Step Example

by Understand The Math

Watch on YouTube

Interactive Visual

Trigonometry Problem Solver

Find Missing Side

Practice finding missing sides using trigonometric ratios.

Use sin, cos, or tan to find the missing side length.

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
In a right triangle with angle θ\theta, if you know the hypotenuse and want to find the opposite side, which formula should you use?

Why It Matters

Finding missing sides with trigonometry is essential in many real-world situations:
  • Construction: Calculating rafter lengths for roofs at specific angles
  • Navigation: Determining distances when only angles are known
  • Surveying: Measuring heights of buildings or mountains from a distance
  • Engineering: Designing ramps, bridges, and structures with precise measurements
  • Aviation: Calculating glide paths and approach angles
Anytime you know an angle and one side of a right triangle, trigonometry lets you find any other measurement!

Real World Applications

Architecture and Construction

Architects and builders use trigonometry to calculate roof pitches, ramp lengths, and support beam dimensions.

Example:

A roof must have a 30°30° pitch and span 88 meters horizontally. Using cosine, the rafter length is 8cos⁡(30°)≈9.24\frac{8}{\cos(30°)} \approx 9.24 meters.

1Try It Yourself

You need to build a roof with a 25°25° pitch. The horizontal span is 66 meters.

How long should each rafter be?

Step 1: Write the mathematical expression

Use cosine: rafter = horizontal / cos(angle)

Navigation and Aviation

Pilots and navigators calculate distances and altitudes using trigonometry when they know angles and partial distances.

Example:

A plane descends at a 3°3° glide angle from 10,00010{,}000 feet altitude. The horizontal distance to the runway is 10000tan⁡(3°)≈190,810\frac{10000}{\tan(3°)} \approx 190{,}810 feet (about 36 miles).

2Try It Yourself

A helicopter needs to reach a rooftop 200200 meters away horizontally. The angle of elevation from the helipad is 15°15°.

How high is the rooftop above the helipad?

Step 1: Write the mathematical expression

Use tangent: height = horizontal distance times tan(angle)

Key Takeaways

  • 1To find a missing side, identify which sides you have (opposite, adjacent, hypotenuse) relative to the known angle
  • 2Use SOH CAH TOA to select the correct ratio for your known and unknown sides
  • 3To find the numerator of a ratio, multiply: side=other side×trig ratio\text{side} = \text{other side} \times \text{trig ratio}
  • 4To find the denominator, divide: side=other sidetrig ratio\text{side} = \frac{\text{other side}}{\text{trig ratio}}
  • 5Always check that your calculator is in degree mode when using degree measurements

Frequently Asked Questions

First label your sides relative to the angle: opposite (across from angle), adjacent (next to angle, not hypotenuse), and hypotenuse (longest side). Then pick the ratio that contains your known side and the side you want to find.
First label your sides relative to the angle: opposite (across from angle), adjacent (next to angle, not hypotenuse), and hypotenuse (longest side). Then pick the ratio that contains your known side and the side you want to find.
Check your calculator mode! If it's in radians instead of degrees, your answers will be wrong. Also verify you're using the correct ratio and doing the algebra correctly.
You should use one of the acute angles (not the 90-degree angle). Either acute angle will work, but your 'opposite' and 'adjacent' sides will switch depending on which angle you choose.

Glossary

Opposite side
The side of a right triangle that is across from (not touching) the reference angle
Adjacent side
The side of a right triangle that is next to the reference angle (not the hypotenuse)
Hypotenuse
The longest side of a right triangle, opposite the right angle
Angle of elevation
The angle measured upward from the horizontal to a line of sight
Angle of depression
The angle measured downward from the horizontal to a line of sight

Formula Card

Opposite from Hypotenuse

opposite=hypotenuse×sin⁡(θ)\text{opposite} = \text{hypotenuse} \times \sin(\theta)

When you know the hypotenuse and need the opposite side

Adjacent from Hypotenuse

adjacent=hypotenuse×cos⁡(θ)\text{adjacent} = \text{hypotenuse} \times \cos(\theta)

When you know the hypotenuse and need the adjacent side

Opposite from Adjacent

opposite=adjacent×tan⁡(θ)\text{opposite} = \text{adjacent} \times \tan(\theta)

When you know the adjacent side and need the opposite side

Hypotenuse from Opposite

hypotenuse=oppositesin⁡(θ)\text{hypotenuse} = \frac{\text{opposite}}{\sin(\theta)}

When you know the opposite side and need the hypotenuse

Hypotenuse from Adjacent

hypotenuse=adjacentcos⁡(θ)\text{hypotenuse} = \frac{\text{adjacent}}{\cos(\theta)}

When you know the adjacent side and need the hypotenuse

Adjacent from Opposite

adjacent=oppositetan⁡(θ)\text{adjacent} = \frac{\text{opposite}}{\tan(\theta)}

When you know the opposite side and need the adjacent side

Today's challengeOne problem, three tries

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