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Teacher Guide: Cosine Ratio (CAH)

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

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All practice problems on paper, with a separate answer key.

Worksheet Answer key

Class quiz

10 questions on Trigonometric Ratios. Students join with a name, you see everyone's score.

For Teachers

Learning Objectives
  • Define the cosine ratio as adjacent over hypotenuse
  • Identify the adjacent side and hypotenuse relative to a given angle
  • Use the cosine ratio to find missing side lengths
  • Use inverse cosine to find missing angles
  • Apply the cosine ratio to solve real-world problems
Prerequisites
  • • Understanding of right triangles and their properties
  • • Knowledge of the Pythagorean theorem
  • • Familiarity with angles measured in degrees
  • • Basic understanding of ratios and proportions
  • • Introduction to trigonometric ratios (SOH CAH TOA)
Discussion Starters
  • 1. Why does cosine start at 1 (when the angle is 0°) and decrease to 0 (at 90°)?
  • 2. How is cosine related to shadows? Think about how shadow length changes as the sun moves.
  • 3. If you know cos⁡(30°)=32\cos(30°) = \frac{\sqrt{3}}{2}, what is sin⁡(60°)\sin(60°)? Why?
  • 4. When would you choose to use cosine instead of sine to solve a real-world problem?
Common Misconceptions

Thinking cosine and sine work the same way

Remediation: Emphasize that sine uses the opposite side while cosine uses the adjacent side. Use a labeled diagram to show the difference clearly.

Believing cosine values increase as the angle increases

Remediation: Show that cosine decreases from 1 to 0 as the angle goes from 0° to 90°. This is the opposite of sine! Use a graph to visualize.

Confusing the adjacent side with the base of the triangle

Remediation: The adjacent side depends on which angle you're using. Always label relative to your reference angle, not the triangle's position.

Differentiation Ideas

For Struggling Students:

  • • Provide color-coded diagrams: hypotenuse in red, adjacent in blue, opposite in green
  • • Start with only special angles (30°, 45°, 60°) where values can be verified
  • • Create a step-by-step checklist: 1) Draw triangle, 2) Label angle, 3) Identify A and H, 4) Write formula, 5) Solve
  • • Use physical models with labeled sides

For On-Level Students:

  • • Mix problems finding sides and finding angles
  • • Include word problems with real-world contexts
  • • Practice distinguishing when to use sine vs cosine
  • • Work with non-special angles using calculators

For Advanced Students:

  • • Explore the relationship sin⁡2(θ)+cos⁡2(θ)=1\sin^2(\theta) + \cos^2(\theta) = 1
  • • Derive why cos⁡(60°)=0.5\cos(60°) = 0.5 using an equilateral triangle
  • • Solve problems requiring both sine and cosine together
  • • Investigate cosine values for angles greater than 90° using the unit circle
Standards Alignment
  • HSG-SRT.C.6 (CCSS.MATH.CONTENT.HSG.SRT.C.6)

    Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles

  • HSG-SRT.C.7 (CCSS.MATH.CONTENT.HSG.SRT.C.7)

    Explain and use the relationship between the sine and cosine of complementary angles

  • HSG-SRT.C.8 (CCSS.MATH.CONTENT.HSG.SRT.C.8)

    Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems

Lesson Resources
  • visualInteractive Right Triangle

    Drag the angle to see how cosine changes

  • activityCAH Practice Problems

    Progressive difficulty finding sides and angles

  • worksheetReal-World Cosine Applications

    Word problems involving horizontal distances and angles

Lesson Content

Everything students see: definition, examples, common mistakes, applications. Tap to open.

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.

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.

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

Why is it called 'cosine'?

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.

What is the relationship between sine and cosine?

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).

When should I use cosine instead of sine or tangent?

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).

What does cos⁡−1\cos^{-1} mean?

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

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