Sine and cosine of complementary angles
Sine and Cosine of Complementary Angles
Sine and Cosine of Complementary Angles is a topic in Right Triangles & Trigonometry in the California Common Core State Standards. It is aligned to Standard G-SRT.7, which requires students to explain and use the relationship between the sine and cosine of complementary angles.
Sine and cosine of complementary angles are equal: \(\sin\theta=\cos(90^\circ-\theta)\).
Theory
The two acute angles of a right triangle are complementary — they add to \(90^\circ\). This creates the cofunction relationship:
The reason is that the side opposite one acute angle is the side adjacent to the other, so the same ratio is a sine for one angle and a cosine for its complement.
The cofunction identities:
How to use the relationship
- Rewrite a sine as a cosine of the complement (or vice versa).
- To solve \(\sin A=\cos B\), set \(A+B=90^\circ\).
- Check with a known pair like \(\sin 30^\circ=\cos 60^\circ\).
Since \(30^\circ\) and \(60^\circ\) are complementary, sine of one equals cosine of the other.
| \(\sin 30^\circ\) | \(=\) | \(\cos(90^\circ-30^\circ)=\cos 60^\circ\) |
Use \(\cos\theta=\sin(90^\circ-\theta)\).
| \(\cos 25^\circ\) | \(=\) | \(\sin 65^\circ\) |
Sine equals cosine when the angles are complementary.
| \(2x+50^\circ\) | \(=\) | \(90^\circ\) |
| \(2x\) | \(=\) | \(40^\circ\) |
| \(x\) | \(=\) | \(20^\circ\) |
The side opposite \(\theta\) is the side adjacent to the other acute angle \(90^\circ-\theta\); the same ratio is sine for one and cosine for the other.
Common pitfalls
Frequently asked questions
What is the cofunction relationship?
\(\sin\theta=\cos(90^\circ-\theta)\): the sine of an angle equals the cosine of its complement.
Why does sin 30 equal cos 60?
Because \(30^\circ\) and \(60^\circ\) are complementary, and sine and cosine are cofunctions.
How do you solve sin A = cos B?
Set the angles complementary: \(A+B=90^\circ\), then solve.
Why are sine and cosine called cofunctions?
Because they relate complementary angles — “co” stands for complement.