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

A

Warm-Up

1.

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

A.

35°

B.

55°

C.

145°

D.

90°

2.

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

A.

$\alpha = \beta$

B.

$\alpha + \beta = 180\degree$

C.

$\alpha + \beta = 90\degree$

D.

$\alpha + \beta = 45\degree$

3.

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:

B

Fluency Practice

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

1.

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:

2.

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

A.

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

B.

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

C.

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

D.

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

3.

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:

4.

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:

5.

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

simplified expression:

C

Varied Practice

$Right triangle with acute angles alpha at bottom-left and beta at top-right, with sides labeled a, b, and c (hypotenuse)$

1.

The diagram shows a right triangle with both acute angles labeled. Which statement about $\sin(\alpha)$ and $\cos(\beta)$ is true?

A.

$\sin(\alpha) = \cos(\alpha)$

B.

$\sin(\alpha) = \cos(\beta)$ because the same side serves as opposite for $\alpha$ and adjacent for $\beta$

C.

$\sin(\alpha) = \sin(\beta)$ because the triangle is the same

D.

$\sin(\alpha) \neq \cos(\beta)$ unless $\alpha = 45\degree$

2.

Solve the equation $\sin(\theta) = \cos(40\degree)$ for $\theta$ where $0\degree < \theta < 90\degree$ . First, rewrite $\cos(40\degree)$ using the complementary relationship: $\cos(40\degree) = \sin(\underline{\hspace{5em}}\degree)$ . Then $\theta = \underline{\hspace{5em}}\degree$ .

equivalent sine angle:

value of theta:

3.

Nadia writes: ' $\sin(90\degree - \theta) = \sin(90\degree) - \sin(\theta) = 1 - \sin(\theta)$ .' What error did Nadia make?

A.

She confused sine with cosine

B.

She treated sine as if it distributes over subtraction: $\sin(A - B) \neq \sin(A) - \sin(B)$

C.

She used the wrong value for $\sin(90\degree)$

D.

She forgot that $\theta$ must be positive

4.

Using a right triangle diagram and the definitions of sine and cosine, explain in 2–3 sentences why $\sin(\theta) = \cos(90\degree - \theta)$ for any acute angle $\theta$ .

D

Word Problems

$Surveyor on the left at ground level looking up at a 53-degree angle of elevation to the top of a vertical cliff on the right$

1.

A surveyor stands at a point on flat ground and measures the angle of elevation to the top of a cliff. The angle of elevation from the surveyor to the cliff top is 53°.

1.

What is the angle between the cliff's vertical face and the line of sight from the surveyor to the cliff top? The angle at the cliff top is ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ °.

angle at cliff top:

2.

A classmate says: 'I can find $\sin(53\degree)$ without a calculator if I know $\cos(37\degree)$ .' Complete their reasoning: $\sin(53\degree) = \cos(\underline{\hspace{5em}}\degree)$ because ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ and ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ are complementary angles.

complementary angle:

first angle:

second angle:

$Side view of a wheelchair ramp showing a right triangle with a 5-degree angle between the ramp and the ground$

2.

Priya is designing a wheelchair ramp. Building codes require that the angle between the ramp surface and the ground is no more than 5°. Priya knows that $\cos(85\degree) \approx 0.087$ .

Using the complementary angle relationship, find $\sin(5\degree)$ without a calculator. Enter your answer as a decimal rounded to three decimal places.

3.

Marcus knows that $\sin(62\degree) \approx 0.883$ and $\cos(62\degree) \approx 0.469$ . He needs to evaluate the expression $\sin(62\degree) + \cos(28\degree)$ for a physics problem.

Find the value of $\sin(62\degree) + \cos(28\degree)$ without a calculator. (Hint: look for complementary angles before reaching for your calculator.)

E

Error Analysis

1.

Javier is working on this problem: "If $\sin(30\degree) = 0.5$ , find $\cos(30\degree)$ using the complementary relationship."
Javier writes:
$\cos(30\degree) = \sin(90\degree - 30\degree) = \sin(60\degree) \approx 0.866$
But then Javier says: "Wait — the complement of 30° is 30° itself, because complementary angles are equal. So $\cos(30\degree) = \sin(30\degree) = 0.5$ ."

Which part of Javier's work is correct, and what is his error?

A.

Javier's first calculation is correct: $\cos(30\degree) = \sin(60\degree) \approx 0.866$ . His error is confusing 'complementary' with 'equal' — the complement of 30° is 60°, not 30°.

B.

Javier's second statement is correct: $\cos(30\degree) = 0.5$ because sine and cosine always give the same value.

C.

Both calculations are wrong. Javier cannot use the complementary relationship at all.

D.

Javier's error is using 90° as the reference. He should subtract from 180°.

2.

Leila is solving: "Find $\sin(105\degree)$ using the complementary relationship."
Leila writes:
$\sin(105\degree) = \cos(90\degree - 105\degree) = \cos(-15\degree)$
Leila concludes: "So $\sin(105\degree) = \cos(-15\degree)$ . I'll use the complementary relationship for any angle."

What is the fundamental error in Leila's approach?

A.

Leila used the wrong formula. She should have written $\sin(105\degree) = \cos(105\degree - 90\degree) = \cos(15\degree)$ .

B.

The complementary angle relationship as derived from right triangles applies only to acute angles ( $0\degree < \theta < 90\degree$ ). Applying it to 105° goes beyond the right triangle context.

C.

Leila is correct. The relationship $\sin(\theta) = \cos(90\degree − \theta)$ holds for all angles.

D.

Leila made an arithmetic error: $90\degree − 105\degree$ should equal $-105\degree$ , not $-15\degree$ .

F

Challenge / Extension

1.

Simplify the expression $\sin(20\degree) + \cos(70\degree) + \sin(70\degree) + \cos(20\degree)$ without a calculator. Show your steps and explain which relationship you used.

2.

Write a formal proof of the statement: 'In any right triangle, the sine of each acute angle equals the cosine of the other acute angle.' Your proof should: (1) start by labeling the triangle and its sides, (2) use the definitions of sine and cosine, and (3) end with a clear conclusion. Use at least 4 numbered steps.

Exercises: Sine and Cosine of Complementary Angles

Warm-Up

Fluency Practice

Varied Practice

Word Problems

Error Analysis

Challenge / Extension