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 Texas Essential Knowledge and Skills (Geometry, §111.41). It is aligned to Standard G.9(A), which requires students to use the relationship between the sine and cosine of complementary angles in right triangles.
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.