Lesson 5.6

Cofunction Identities

Sine and cosine are cofunctions — the sine of an angle equals the cosine of its complement. This elegant relationship extends to all three trig pairs and unlocks a new simplification strategy.

Introduction

The prefix “co-” in cosine literally means “complement.” In a right triangle, the two acute angles are complementary (they sum to ), so the sine of one equals the cosine of the other. This pattern holds for all three pairs: sin/cos, tan/cot, sec/csc.

Past Knowledge

Complementary angles sum to (or radians). Right triangle trig ratios.

Today's Goal

State and apply all six cofunction identities.

Future Success

Cofunction identities help derive sum & difference formulas in Lessons 5.10–5.12.

Key Concepts

The Right Triangle Proof

In a right triangle with acute angles and :

  • The side opposite is adjacent to
  • So

The Cofunction Identities

Degrees FormRadians Form

Pattern to Remember

Each function equals its co-partner evaluated at the complement. The “co-” prefix literally means complement: cosine = complement of sine, cotangent = complement of tangent, cosecant = complement of secant.

Worked Examples

Basic

Direct Cofunction Evaluation

Question: Evaluate using a cofunction identity.

Step 1: Apply the cofunction identity:

Step 2: Evaluate:

Final Answer:

Intermediate

Solving a Cofunction Equation

Question: Find if and .

Step 1: By the cofunction identity, .

Step 2: So , giving .

Final Answer:

Advanced

Radian Cofunction with Simplification

Question: Simplify .

Step 1: Apply the cofunction identity:

Step 2: Substitute:

Final Answer:

Common Pitfalls

Subtracting from the Wrong Value

The complement is (or ), not . The order matters — always subtract the angle from 90°.

Real-Life Applications

Surveying & Construction

When a surveyor measures an angle of elevation, the complementary angle of depression is automatically known. Cofunction identities let them switch between horizontal and vertical distance calculations seamlessly, which is essential for grading roads, positioning foundations, and setting roof pitches.

Practice Quiz

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