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. Vocabula ry: angle of elevation, cosine, hypotenuse, leg, right triangle, sine, tangent, trigonometric ratio Student Expl...
physics 101 gizmo student exploration sine cosine and tangent ratios answer key angle of elevation cosine hypotenuse leg right triangle sine tangent trigonometric ratio
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Student Exploration: Sine, Cosine, and Tangent Ratios
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?
Explain.
2. What is your estimate of the height of the tree?
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 mA 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, AC and BC . The
hypotenuse is the side opposite the right angle, AB .
A. Which side of the triangle represents the height of the tree?
B. Which side represents the distance from Joseph to the base of the tree?
C. Which side represents the distance from Joseph to the top of the tree?
2019
, 2. Turn on Show side lengths. Based on the lengths, what is the height of the tree?
Get the Gizmo ready:
Activity A:
On the SINE tab, set mA to 30°.
Sine Check that Show side lengths is turned on.
Drag point C as far as possible to the right.
1. In ΔABC, BC is the opposite leg because it is opposite A.
A. What are the lengths of each side? AC = BC = AB =
B. When mA = 30°, what is the ratio of BC to AB?
C. Drag point C to the left. Notice that mA stays the same, so the new triangle is
similar to the original. For two different positions of point C, record BC, AB, and
BC
AB .
Position 1 Position 2
BC BC
BC AB AB BC AB AB
What do you notice?
2. Drag point C all the way to the right so that the length of the hypotenuse AB is 14. Turn on
Show sine computation. The sine of angle A (or “sin A”) is the ratio of the opposite leg to
opposite
the hypotenuse: sin A = hypotenuse .
A. What is sin 30°?
B. Turn off Show sine computation. Set mA to 20°. What is sin 20°?
Check your work by turning on Show sine computation.
3. With Show sine computation turned on, set mA to 0°.
A. What is sin 0°?
B. How will the length of BC change as mA increases?
2019
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