Sine and Cosine of Complementary Angles
The sine of an angle equals the cosine of its complement: . This identity explains why “co-sine” has “co” in its name — it's the sine of the complement.
Introduction
In a right triangle, the two acute angles always sum to 90° — they're complementary. Because one angle's opposite side is the other's adjacent side, their sine and cosine swap. This simple fact has deep consequences in trigonometry.
Past Knowledge
Sine & cosine (8.2.5). Complementary angles (3.1). Triangle angle sum.
Today's Goal
Prove and apply the cofunction identities for sine and cosine.
Future Success
Inverse trig (8.3.1), unit circle (PreCalc), Pythagorean identity.
Key Concepts
Cofunction Identities
Key Values
| Note | |||
|---|---|---|---|
| 30° | sin 30° = cos 60° | ||
| 45° | sin 45° = cos 45° | ||
| 60° | sin 60° = cos 30° |
Theorem & Proof
Two-Column Proof: Cofunction Identity
Given: Right with ,
Prove:
| # | Statement | Reason |
|---|---|---|
| 1 | , so | Acute angles in a right triangle are complementary |
| 2 | (opposite / hypotenuse) | Definition of sine |
| 3 | (adjacent to / hypotenuse) | Definition of cosine (the side opposite is adjacent to ) |
| 4 | Steps 2 and 3 give equal ratios; substitute |
∎ One angle's opposite is the other's adjacent - that's why the “co” prefix means “complement.”
Worked Examples
Direct Application
If , what is ?
25° and 65° are complementary (25 + 65 = 90). So .
Solving an Equation
Find if .
Using the cofunction identity:
So
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Pythagorean Identity Preview
Show that using a right triangle.
The numerator simplifies by the Pythagorean Theorem!
for all angles — the Pythagorean Identity.
Common Pitfalls
Thinking sin and cos of the SAME Angle Are Equal
. They're only equal when the angle is 45° (its own complement). The identity swaps to the complement.
Forgetting to Check Complementary
The identity only works for angles that sum to 90°. . It's .
Real-Life Applications
Navigation — Bearing Conversion
A ship heading at bearing 040° (40° from north) can convert to standard orientation using cofunctions: the east component is . Navigators use this to switch between reference systems.
Physics — Inclined Planes
When analyzing forces on a ramp, the component of gravity along the ramp is , while the normal force component is . The cofunction identity confirms these components are interconnected.
Practice Quiz
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