Task Sets: Explain sine and cosine relationship - Common Core Math - HS Geometry

Warm-Up

Two angles are complementary. One angle measures 35°. What is the measure of the other angle?

35°

55°

145°

90°

In a right triangle with acute angles $\alpha$ and $\beta$ , which statement must be true?

$\alpha = \beta$

$\alpha + \beta = 180\degree$

$\alpha + \beta = 90\degree$

$\alpha + \beta = 45\degree$

In a right triangle with legs of length 3 and 4 and hypotenuse of length 5, if $\theta$ is the angle opposite the side of length 3, then $\sin(\theta) = \dfrac{\text{opposite}}{\text{hypotenuse}} = \dfrac{\hspace{0.2em}\fbox{\phantom{000000}}\hspace{0.2em}}{\hspace{0.2em}\fbox{\phantom{000000}}\hspace{0.2em}}$ and $\cos(\theta) = \dfrac{\text{adjacent}}{\text{hypotenuse}} = \dfrac{\hspace{0.2em}\fbox{\phantom{000000}}\hspace{0.2em}}{\hspace{0.2em}\fbox{\phantom{000000}}\hspace{0.2em}}$ .

opposite side:

hypotenuse for sin:

adjacent side:

hypotenuse for cos:

Fluency Practice

$Right triangle ABC with right angle at C, angle A labeled 47 degrees, and angle B unknown$

In right triangle $ABC$ with right angle at $C$ , if $\angle A = 47\degree$ , then $\angle B = \underline{\hspace{5em}}\degree$ . The two acute angles are ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ angles.

measure of angle B:

relationship type:

Which equation correctly states the complementary angle relationship for sine and cosine?

$\sin(\theta) = \cos(\theta)$

$\sin(\theta) = \cos(90\degree + \theta)$

$\sin(\theta) = \cos(90\degree - \theta)$

$\sin(\theta) = 1 - \cos(\theta)$

Given that $\sin(32\degree) \approx 0.53$ , use the complementary angle relationship to find $\cos(58\degree)$ without a calculator. $\cos(58\degree) = \sin(\underline{\hspace{5em}}\degree) \approx \underline{\hspace{5em}}$ .

complementary angle:

value:

Given that $\cos(71\degree) \approx 0.326$ , find $\sin(19\degree)$ without a calculator. $\sin(19\degree) = \cos(\underline{\hspace{5em}}\degree) \approx \underline{\hspace{5em}}$ .

complementary angle:

value:

Simplify $\cos(90\degree - x)$ . Your answer should be a single trig function of $x$ : $\cos(90\degree - x) = \underline{\hspace{5em}}$ .

simplified expression:

Exercises: Sine and Cosine of Complementary Angles

Warm-Up

Fluency Practice