Download Using Sine, Cosine, and Tangent

Geometry in the Trees
Using Sine, Cosine and Tangent
1. The students in Mrs. Cain’s class used a surveyor’s measuring device to find the angle from
their locations to the top of a building to be 72 degrees. They also measured their distance from
the bottom of the building to be 100 feet. The diagram shows the angle measure and the
distance. Find the height of the building to the nearest foot.
2. A flag pole is 100 feet tall. At a particular time of day, the flag pole casts a 249 foot shadow.
Find the angle of elevation from the tip of the shadow to the top of the flag pole. Round your
answer to the nearest degree.
3. A ramp, used to load items on a pick-up truck, is 4.1 meters long and makes an angle of 35
degrees with the ground. How far above the ground is the top of the ramp? (Round answer to the
nearest tenth of a meter)
4. Kim drives 170 meters up a hill that makes an angle of 6 degrees with the horizontal. What
horizontal distance has she covered? (Round answer to the nearest tenth of a meter)
5. In West Virginia, a certain interstate highway makes an angle of 6 degrees with the horizontal.
This angle is maintained for a horizontal distance of 8 miles.
a. Draw and label a diagram to represent this situation.
b. How high does the highway rise in this 8-mile section? (Round answer to the nearest
hundredth of a mile) Show all steps used in order to find the distance.
6. A triangular garden space has angle measures of 45 degrees, 45 degrees and 90 degrees.
One of the shorter sides of the triangle measures 15 feet.
a. Find the length of the longest side of the garden space. Then sketch and label the garden
space. Explain how you found this length.
b. Find the exact value of the sine and cosine of a 45 degree angle.
2
2
c. Show that (sin 45) + (cos 45) = 1. Justify your steps.
7. Sketch two triangles. Label the lengths of the sides of Triangle WVU as 3 inches, 4 inches,
and 5 inches. Label the lengths of the sides of Triangle PIT as 5 inches, 12 inches, and 13
inches.
a. What is the sum of the measures of the acute angles of any right triangle? Explain your
reasoning.
b. Write the tangent ratios for the acute angles of Triangle WVU.
c. Write the tangent ratios for the acute angles of Triangle PIT.
d. Write a rule describing the relationship between the tangents of the acute angles of any right
triangle.
Geometry in the Trees
Using Sine, Cosine and Tangent Answers
1. 308 ft
2. 22 degrees
3. 2.4 m
4. 169.1 m
5. a. diagram - check students’ drawings
b. tan 6 = x/8
use the tangent ratio
x = 8(tan 6)
solve for x
x = 0.84
the rise is about 0.84 miles
6. a. diagram- check students’ drawings
b. sin 45 =
15
2
15
2
=
and cos 45 =
=
15 2 2
15 2 2
22
22
2
2
c. (sin 45) + (cos 45) = ( 2 ) + ( 2 )
2 2
=4+4
=½+½
=1
7. a. The sum of the acute angles of any right triangle = 90. Since the sum of the three angles
of any triangle is 180, then the remaining angles of a right triangle must be 180- 90 = 90.
3 4
b. 4 , 3
5 12
c. 12, 5
d. The tangents of the acute angles of any right triangle are reciprocals