# Algebra 2 and Trigonometry

## Educators

Problem 1

In any right triangle, the acute angles are complementary. What is the relationship between the sine of the measure of an angle and the cosine of the measure of the complement of that angle? Justify your answer.

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Problem 2

Bebe said that if $A$ is the measure of an acute angle of a right triangle, $0<\sin A<1 .$ Do you agree with Bebe? Justify your answer.

Jeremy S.

Problem 3

The lengths of the sides of $\triangle A B C$ are given. For each triangle, $\angle C$ is the right angle and $\mathrm{m} \angle A<\mathrm{m} \angle B .$ Find: a. $\sin A$ b. $\cos A$ c. $\tan A$.
$6,8,10$

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Problem 4

The lengths of the sides of $\triangle A B C$ are given. For each triangle, $\angle C$ is the right angle and $\mathrm{m} \angle A<\mathrm{m} \angle B .$ Find: a. $\sin A$ b. $\cos A$ c. $\tan A$.
$5,12,13$

Jeremy S.

Problem 5

The lengths of the sides of $\triangle A B C$ are given. For each triangle, $\angle C$ is the right angle and $\mathrm{m} \angle A<\mathrm{m} \angle B .$ Find: a. $\sin A$ b. $\cos A$ c. $\tan A$.
$11,60,61$

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Problem 6

The lengths of the sides of $\triangle A B C$ are given. For each triangle, $\angle C$ is the right angle and $\mathrm{m} \angle A<\mathrm{m} \angle B .$ Find: a. $\sin A$ b. $\cos A$ c. $\tan A$.
$8,17,15$

Jeremy S.

Problem 7

The lengths of the sides of $\triangle A B C$ are given. For each triangle, $\angle C$ is the right angle and $\mathrm{m} \angle A<\mathrm{m} \angle B .$ Find: a. $\sin A$ b. $\cos A$ c. $\tan A$.
$16,30,34$

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Problem 8

The lengths of the sides of $\triangle A B C$ are given. For each triangle, $\angle C$ is the right angle and $\mathrm{m} \angle A<\mathrm{m} \angle B .$ Find: a. $\sin A$ b. $\cos A$ c. $\tan A$.
$2 \sqrt{5}, 2,4$

Jeremy S.

Problem 9

The lengths of the sides of $\triangle A B C$ are given. For each triangle, $\angle C$ is the right angle and $\mathrm{m} \angle A<\mathrm{m} \angle B .$ Find: a. $\sin A$ b. $\cos A$ c. $\tan A$.
$\sqrt{2}, 3, \sqrt{7}$

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Problem 10

The lengths of the sides of $\triangle A B C$ are given. For each triangle, $\angle C$ is the right angle and $\mathrm{m} \angle A<\mathrm{m} \angle B .$ Find: a. $\sin A$ b. $\cos A$ c. $\tan A$.
$6,3 \sqrt{5}, 9$

Jeremy S.

Problem 11

Two of the answers to Exercises 3–10 are the same. What is the relationship between the triangles described in these two exercises? Justify your answer.

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Problem 12

Use an isosceles right triangle with legs of length 3 to find the exact values of $\sin 45^{\circ},$ $\cos 45^{\circ},$ and $\tan 45^{\circ} .$

Jeremy S.

Problem 13

Use an equilateral triangle with sides of length 4 to find the exact values of $\sin 30^{\circ}, \cos 30^{\circ},$ and $\tan 30^{\circ} .$

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Problem 14

A 20-foot ladder leaning against a vertical wall reaches to a height of 16 feet. Find the sine, cosine, and tangent values of the angle that the ladder makes with the ground.

Jeremy S.

Problem 15

An access ramp reaches a doorway that is 2.5 feet above the ground. If the ramp is 10 feet long, what is the sine of the angle that the ramp makes with the ground?

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Problem 16

The bed of a truck is 5 feet above the ground. The driver of the truck uses a ramp 13 feet long to load and unload the truck. Find the sine, cosine, and tangent values of the angle that the ramp makes with the ground.

Jeremy S.

Problem 17

A 20-meter line is used to keep a weather balloon in place. The sine of the angle that the line makes with the ground is $\frac{3}{4}$ . How high is the balloon in the air?

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Problem 18

From a point on the ground that is 100 feet from the base of a building, the tangent of the angle of elevation of the top of the building is $\frac{5}{4} .$ To the nearest foot, how tall is the building?

Jeremy S.
From the top of a lighthouse 75 feet high, the cosine of the angle of depression of a boat out at sea is $\frac{4}{5} .$ To the nearest foot, how far is the boat from the base of the lighthouse?