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All courses Geometry · G-SRT.7 37 of 57
Explain and use the complementary-angle relationship between sine and cosine.

Find a complement and one matching cofunction identity

Problem
Acute angles \(α\) and \(β\) are complementary, and \(α=35°\). Calculate \(β\) and state the corresponding sine-cosine cofunction identity.
Prompt diagram of a right triangle with acute angle alpha marked 35 degrees. The other acute angle beta, its measure, and the cofunction identity are withheld. Open full size
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Hint

Write the two-acute-angle sum equation.

The acute angles sum to90°, and sine of one equals cosine of its complement.

Solution walkthrough

01

Use the complementary-angle equation

\[\text{alpha}+\text{beta}=90^{\circ};~\text{alpha}=35^{\circ}\]

The two acute angles of a right triangle sum to 90 degrees.

02

Solve for beta

\[\text{beta}=90^{\circ}-35^{\circ}=55^{\circ}\]

Subtracting the known acute angle from 90 degrees gives its complement.

03

Apply the sine-cosine cofunction rule

\[\sin(\text{alpha})=\cos(\text{beta})\]

For complementary angles, the side opposite alpha is adjacent to beta, and the hypotenuse is the same.

04

Substitute and check

\[\sin(35^{\circ})=\cos(55^{\circ});~35+55=90\]

The angle sum verifies complementarity, so the numerical cofunction identity applies.

+

Another way

  1. Use beta=90 degrees-alpha directly, then substitute alpha=35 degrees into sin(alpha)=cos(90 degrees-alpha).

!

Common mistake

Do not use 180-35. Acute angles in a right triangle complement to 90 degrees, not supplement to 180 degrees.

Answer diagram: beta = 55 degrees because 35 degrees + 55 degrees = 90 degrees. Opposite alpha is adjacent beta, so sin 35 degrees = cos 55 degrees; also cos 35 degrees = sin 55 degrees.
The answer traces the complementary sum and shared opposite/adjacent leg over one hypotenuse.

Solution walkthrough video