Name: Melany Jimenez Date: 10-7-2020

Student Exploration: Sine, Cosine, and Tangent Ratios

Directions: Follow the instructions to go through the simulation. Respond to the questions and

prompts in the orange boxes.

Vocabulary: angle of elevation, cosine, hypotenuse, leg, right triangle, sine, tangent, trigonometric ratio

Prior Knowledge Questions (Do these BEFORE using the Gizmo.)

Joseph’s math teacher challenges him to estimate the height of a pine tree next to

the school. Joseph walks 9.9 meters from the base of the trunk, lies on his belly, and

measures a 45° angle of elevation to the top of the tree.

1. Do you think Joseph has enough information to estimate of the height of the tree? Yes

Explain. he can use tan to find the height

2. What is your estimate of the height of the tree?

9.9 m

Gizmo Warm-up

There are several ways Joseph could estimate the height of the tree. He could

draw a right triangle (triangle with a 90° angle) with a side of 9.9 cm and an

angle of 45°. Another way to solve the problem is to use trigonometric ratios.

These ratios are the subject of the Sine, Cosine, and Tangent Ratios Gizmo.

You can use ΔABC to model how Joseph could measure the tree. To begin, check that m∠A is set to 45°.

(To quickly set a slider to a value, type the value in the box to the right of the slider and press Enter.)

1. The legs of a right triangle are the two sides that form the right angle, and . The hypotenuse is

the side opposite the right angle, .

A. Which side of the triangle represents the height of the tree? Line BC

B. Which side represents the distance from Joseph to the base of the tree? line AC

C. Which side represents the distance from Joseph to the top of the tree? Line AB

2. Turn on Show side lengths. Based on the lengths, what is the height of the tree?

9.9 meters

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Activity A:

Sine

Get the Gizmo ready:

● On the SINE tab, set m∠A to 30°.

● Check that Show side lengths is turned on.

● Drag point C as far as possible to the right.

1. In ΔABC, is the opposite leg because it is opposite ∠A.

A. What are the lengths of each side? AC = 12.12 BC = 7 AB = 14

B. When m∠A = 30°, what is the ratio of BC to AB? 7:14 or 1:2

C. Drag point C to the left. Notice that m∠A stays the same, so the new triangle is similar to the

original. For two different positions of point C, record BC, AB, and .

Position 1 Position 2

BC 6 AB 12.1

6

12.1 BC 4 AB 7.99

4

7.99

What do you notice? line BC is almost half the hypotenuse

2. Drag point C all the way to the right so that the length of the hypotenuse is 14. Turn on Show sine

computation. The sine of angle A (or “sin A”) is the ratio of the opposite leg to the hypotenuse:

A. What is sin 30°? 0.5

B. Turn off Show sine computation. Set m∠A to 20°. What is sin 20°? 0.34

Check your work by turning on Show sine computation.

3. With Show sine computation turned on, set m∠A to 0°.

A. What is sin 0°? 0

B. How will the length of change as m∠A increases?

the measurement will

increase by one as well

C. Slowly increase m∠A. What happens to sin A? it increases as well

D. Set m∠A to 90°. What is sin 90°? 1

E. Explain why the sine of an angle can never be greater than 1. because the adjacent side

can never be greater than

hypotenuse

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