This document contains multiple types of math problems involving trigonometric functions: 1) Conversions between radians and degrees for various angle measures. 2) Right triangle problems estimating heights or distances given angle measures and one side of a triangle. 3) Expressing trigonometric expressions as integers using Pythagorean identities. 4) Verifying trigonometric identities by algebraic manipulation.

1. A. Conversion problems: Convert radians to degrees or degrees to radians as applicable.
2π 4π π
1. 1500 5. −600 9. 2250 13. 3 17. 3 21. 9
11π 11π 5π
2. 1200 6. −1350 10. −2100 14. 6 18. 4 22. 2
3π
3. 4500 7. 720 11. 1000 15. 4 19. − 7π
2 23. 9π
5π π
4. 6300 8. 540 12. 950 16. 6 20. 7π 24. 16
B. Right angle problems.
1. A forester, 200 feet from the base of a redwood tree, observes that the angle between the ground and the top
of the tree is 600 . Estimate the height of the tree.
2. The peak of Mt. Fuji in Japan is approximately 12,400 feet high. A trigonometry student, several miles away,
notes that the angle between level ground and the peak is 300 . Estimate the distance from the student to the
point on level ground directly beneath the peak.
3. Stonehenge in Salisbury Plains, England, was constructed using solid stone blocks weighing over 99,000 pounds
each. Lifting a single stone required 550 people, who pulled the stone up a ramp inclined at an angle of 90 .
Approximate the distance that a stone was moved in order to raise it to a height of 30 feet.
4. The highest advertising sign in the world is a large letter I situated at the top of the 73-storey First Interstate
World Center building in Los Angeles. At a distance of 200 feet from a point directly below the sign, the angle
between the ground and the top of the sign is 78.870 . Approximate the height of the top of the sign.
C. Use Pythagorean identities to write the expression as an integer.
1. tan2 4β − sec2 4β 4. 3 csc2 α − 3 cot2 α 7. 7 sec2 γ − 7 tan2 γ
2. 4 tan2 β − 4 sec2 β 5. 5 sin2 θ + 5 cos2 θ 8. 7 sec2 (γ/3) − 7 tan2 (γ/3)
3. csc2 3α − cot2 3α 6. 5 sin2 (θ/4) + 5 cos2 (θ/4)
D. Verify the identity by transforming the left-hand side into the right-hand side.
1. cos θ sec θ = 1 12. 1 − 2 sin2 (θ/2) = 2 cos2 (θ/2) − 1
2. tan θ cot θ = 1 1
13. (1 + sin θ)(1 − sin θ) =
sec2 θ
3. sin θ sec θ = tan θ
14. (1 − sin2 θ)(1 + tan2 θ) = 1
4. sin θ cot θ = cos θ
15. sec θ − cos θ = tan θ sin θ
csc θ
5. = cot θ sin θ + cos θ
sec θ 16. = 1 + tan θ
6. cot θ sec θ = csc θ cos θ
17. (cot θ + csc θ)(tan θ − sin θ) = sec θ − cos θ
7. (1 + cos 2θ)(1 − cos 2θ) = sin2 2θ
18. cot θ + tan θ = csc θ sec θ
8. cos2 2θ − sin2 2θ = 2 cos2 2θ − 1
19. sec2 3θ csc2 3θ = sec2 3θ + csc2 3θ
9. (cos2 θ)(sec2 θ − 1) = sin2 θ
1 + cos2 3θ
2
10. (tan θ + cot θ) tan θ = sec θ 20. = 2 csc2 3θ − 1
sin2 3θ
sin(θ/2) cos(θ/2)
11. + =1
csc(θ/2) sec(θ/2)