Cosine Ratio (CAH)

Master the cosine ratio and learn how to find missing sides and angles using adjacent and hypotenuse.

Advanced25 minLesson
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Definition

The cosine of an angle in a right triangle is the ratio of the adjacent side to the hypotenuse.
cos⁡(θ)=adjacenthypotenuse\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}
The mnemonic CAH helps you remember:
  • C = Cosine
  • A = Adjacent
  • H = Hypotenuse
For example, in a right triangle where the angle is 60°60°:
cos⁡(60°)=12=0.5\cos(60°) = \frac{1}{2} = 0.5
This means the adjacent side is exactly half the length of the hypotenuse.

Try it now

In a right triangle, the cosine of an angle equals:

Worked Examples

A support cable is 2020 meters long and makes a 40°40° angle with the ground. How far from the base of the pole does it anchor into the ground?
1

Identify the known values

Angle = 40°40°, Hypotenuse (cable) = 2020 m, Unknown = Adjacent (ground distance) → Set up the problem
2

Write the cosine formula

cos⁡(40°)=adjacent20\cos(40°) = \frac{\text{adjacent}}{20} → Formula ready
3

Find cos⁡(40°)\cos(40°)

cos⁡(40°)≈0.766\cos(40°) \approx 0.766
4

Solve for the adjacent side

adjacent=20×0.766=15.32\text{adjacent} = 20 \times 0.766 = 15.32 m → 15.3215.32 meters

Common Mistakes

Confusing adjacent and opposite sides

Why it's wrong: The adjacent and opposite sides depend on which angle you're working with. The adjacent side is always next to your angle (not the hypotenuse).

Correct: Always identify your reference angle first. Adjacent is the side that forms the angle with the hypotenuse (touches the angle but isn't the hypotenuse).

Using cosine when you should use a different ratio

Why it's wrong: Cosine only involves the adjacent and hypotenuse. If you know the opposite side, you need sine or tangent.

Correct: Check which sides you know: Adjacent + Hypotenuse → Cosine, Opposite + Hypotenuse → Sine, Opposite + Adjacent → Tangent.

Forgetting that cos⁡(0°)=1\cos(0°) = 1 and cos⁡(90°)=0\cos(90°) = 0

Why it's wrong: At 0°, the adjacent side equals the hypotenuse (ratio = 1). At 90°, the adjacent side has length 0.

Correct: Remember: as the angle increases from 0° to 90°, cosine decreases from 1 to 0. This is opposite to sine!

Mixing up sine and cosine values for complementary angles

Why it's wrong: sin⁡(30°)=cos⁡(60°)\sin(30°) = \cos(60°) and cos⁡(30°)=sin⁡(60°)\cos(30°) = \sin(60°) because they are complementary angles.

Correct: For complementary angles: cos⁡(θ)=sin⁡(90°−θ)\cos(\theta) = \sin(90° - \theta). This is called the cofunction identity.

Interactive Visual

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

Explore how the cosine ratio (adjacent/hypotenuse) changes with the angle.

Interactive Sandbox

Expression Calculator

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Practice Problems

15 problems
Problem 1 of 15
Easy
In a right triangle, the cosine of an angle equals:

Why It Matters

The cosine ratio is essential for solving problems involving horizontal distances and angles:
  • Architecture: Calculating horizontal spans of roofs and bridges
  • Navigation: Finding horizontal distances when given an angle and total distance
  • Physics: Analyzing horizontal components of forces and motion
  • Engineering: Designing support structures and calculating load distributions
  • Surveying: Measuring horizontal distances across terrain
Whenever you need to find the horizontal component or work with the adjacent side in a right triangle, cosine is your key tool.

Real World Applications

Shadow Length Calculations

Architects and solar engineers use the cosine ratio to calculate shadow lengths cast by buildings and structures.

Example:

When the sun is at a 70°70° angle of elevation, a 5050 meter tall building casts a shadow. The horizontal distance from the building to the shadow tip involves the cosine ratio.

1Try It Yourself

A tree is 1515 meters tall. The sun is at a 55°55° angle of elevation. A line from the treetop to the tip of its shadow is 18.318.3 meters long.

What is the length of the shadow?

Step 1: Write the mathematical expression

Use cosine: shadow = 18.3×cos⁡(55°)18.3 \times \cos(55°)

Navigation and Distance

Pilots and ship captains use cosine to calculate horizontal distances when traveling at an angle.

Example:

When a helicopter travels 55 km at a 30°30° climb angle, the horizontal distance covered is calculated using cosine.

2Try It Yourself

A hiker walks 22 km up a hill that slopes at 25°25° to the horizontal. How much horizontal distance has the hiker covered?

What is the horizontal distance?

Step 1: Write the mathematical expression

Horizontal distance = 2×cos⁡(25°)2 \times \cos(25°)

Key Takeaways

  • 1Cosine equals adjacent divided by hypotenuse: cos⁡(θ)=adjhyp\cos(\theta) = \frac{\text{adj}}{\text{hyp}}
  • 2Remember CAH: Cosine = Adjacent / Hypotenuse
  • 3To find the adjacent: multiply hypotenuse by cos⁡(θ)\cos(\theta)
  • 4To find the hypotenuse: divide adjacent by cos⁡(θ)\cos(\theta)
  • 5To find the angle: use inverse cosine θ=cos⁡−1(adjhyp)\theta = \cos^{-1}\left(\frac{\text{adj}}{\text{hyp}}\right)
  • 6Special values: cos⁡(0°)=1\cos(0°) = 1, cos⁡(30°)=32\cos(30°) = \frac{\sqrt{3}}{2}, cos⁡(45°)=22\cos(45°) = \frac{\sqrt{2}}{2}, cos⁡(60°)=0.5\cos(60°) = 0.5, cos⁡(90°)=0\cos(90°) = 0

Frequently Asked Questions

The word 'cosine' is short for 'complementary sine.' It was named because cos⁡(θ)=sin⁡(90°−θ)\cos(\theta) = \sin(90° - \theta). The cosine of an angle equals the sine of its complement.
The word 'cosine' is short for 'complementary sine.' It was named because cos⁡(θ)=sin⁡(90°−θ)\cos(\theta) = \sin(90° - \theta). The cosine of an angle equals the sine of its complement.
Sine and cosine are related through complementary angles: cos⁡(θ)=sin⁡(90°−θ)\cos(\theta) = \sin(90° - \theta) and sin⁡(θ)=cos⁡(90°−θ)\sin(\theta) = \cos(90° - \theta). Also, sin⁡2(θ)+cos⁡2(θ)=1\sin^2(\theta) + \cos^2(\theta) = 1 (Pythagorean identity).
Use cosine when you're working with the adjacent side and the hypotenuse. If you have (or need) the opposite side, use sine (with hypotenuse) or tangent (with adjacent).
cos⁡−1\cos^{-1} is the inverse cosine function, also called arccosine (arccos). It answers: 'What angle has this cosine value?' For example, cos⁡−1(0.5)=60°\cos^{-1}(0.5) = 60° because cos⁡(60°)=0.5\cos(60°) = 0.5.

Glossary

Cosine
The ratio of the adjacent side to the hypotenuse in a right triangle: cos⁡(θ)=adjhyp\cos(\theta) = \frac{\text{adj}}{\text{hyp}}
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, always opposite the right angle
Inverse cosine
The function cos⁡−1\cos^{-1} (arccos) that finds an angle when you know its cosine value
Cofunction identity
The relationship cos⁡(θ)=sin⁡(90°−θ)\cos(\theta) = \sin(90° - \theta), showing that cosine and sine are complementary functions

Formula Card

Cosine definition

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

The ratio of the adjacent side to the hypotenuse

Find adjacent

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

Multiply hypotenuse by cosine to find the adjacent side

Find hypotenuse

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

Divide adjacent by cosine to find the hypotenuse

Find angle

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

Use inverse cosine (arccos) to find the angle

Special value: cos(0)

cos⁡(0°)=1\cos(0°) = 1

Cosine of 0 degrees equals 1

Special value: cos(30)

cos⁡(30°)=32≈0.866\cos(30°) = \frac{\sqrt{3}}{2} \approx 0.866

Cosine of 30 degrees equals square root of 3 over 2

Special value: cos(45)

cos⁡(45°)=22≈0.707\cos(45°) = \frac{\sqrt{2}}{2} \approx 0.707

Cosine of 45 degrees equals square root of 2 over 2

Special value: cos(60)

cos⁡(60°)=12=0.5\cos(60°) = \frac{1}{2} = 0.5

Cosine of 60 degrees equals one-half

Today's challengeOne problem, three tries

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