Use relationships between sine and cosine of complementary angles (SAT). | Right triangles and trigonometry | SAT Math

When sine and cosine are set equal, the two angles should add to 90 degrees.

Core Idea

$s i n (𝑥 °) = c o s (90 ° - 𝑥 °)$ and $c o s (𝑥 °) = s i n (90 ° - 𝑥 °)$ . When the SAT sets a sine expression equal to a cosine expression, the arguments must add up to 90°.

Understanding

Rule: Sine and cosine of complementary angles match.

$s i n 𝜃 = c o s (90 \circ - 𝜃)$ .
$c o s 𝜃 = s i n (90 \circ - 𝜃)$ .
For equations, make the two angle expressions add to $90 \circ$ .

That turns a trig identity into a simple linear equation.

Step by Step

Recognize the pattern: $s i n (𝐴) = c o s (𝐵)$ .
Set $𝐴 + 𝐵 = 90 °$ .
Solve for the variable.
Check that both angle expressions give valid (positive) degree measures.

Misconceptions

Setting $𝐴 = 𝐵$ instead of $𝐴 + 𝐵 = 90 °$ .
Forgetting the complementary rule and trying to use identities that aren't needed.
Not checking that the resulting angles are between 0° and 90° — if they aren't, the equation has no solution in context.

Question

Worked Example

If $s i n (4 𝑥 + 10) ° = c o s (2 𝑥 - 4) °$ , what is the value of $𝑥$ ?

Select an answer to see the explanation