Right Triangle Trigonometry

1. Right Triangle Trigonometry By: Jeffrey Bivin Lake Zurich High School jeff.bivin@lz95.org Last Updated: December 1, 2010

2. SOH CAH TOA hypotenuse opposite θ adjacent

3. Reciprocal Identities hypotenuse opposite θ adjacent

4. Find the sides. B 2 c 1 a A C b

5. Find the sides. B c 10 a A C b

6. Find the sides. B c 17 a A C b

7. Find the sides. B c a A C b 9

8. Find the sides. Use your calculator! B 15 c 68o a 22o A C b

9. Find the sides. B c 56o 25 a 34o A C b

10. Find the angles and the 3rd side. 25 β θ 21

11. Find the angles and the 3rd side. β 6 θ 11

13. Quotient Identities hypotenuse opposite We Know θ adjacent

14. Pythagorean Identities hypotenuse opposite θ adjacent Divide by hyp2 Divide by adj2 Divide by opp2

15. Using Pythagorean Identities • Find cosθ if sinθ =

16. Using Pythagorean Identities • Find secθ if tanθ =

17. Using Pythagorean Identities • Find sinθ if cotθ =

18. Co-function Identities Use your calculators to evaluate each of the following. cos(θ) = sin(90o – θ) and sin(θ) = cos(90o – θ) Complimentary Angles

19. Co-function Identities cos(θ) = sin(90o – θ) and sin(θ) = cos(90o – θ) sec(θ) = csc(90o – θ) and csc(θ) = sec(90o – θ) Complimentary Angles

20. Co-function Identities cos(θ) = sin(90o – θ) and sin(θ) = cos(90o – θ) sec(θ) = csc(90o – θ) and csc(θ) = sec(90o – θ) tan(θ) = cot(90o – θ) and cot(θ) = tan(90o – θ) Complimentary Angles

21. Co-function Identities cos(θ) = sin(90o – θ) and sin(θ) = cos(90o – θ) sec(θ) = csc(90o – θ) and csc(θ) = sec(90o – θ) tan(θ) = cot(90o – θ) and cot(θ) = tan(90o – θ) Complimentary Angles 90o- θ c a θ b

22. t (b, a) (a, b) 90o-t t -t (a, -b)

23. Applications

24. A surveyor is standing 45 feet from the base of a large tree. The surveyor measures the angle of elevation from the ground to the top of the tree to be 67.5o. Find the height of the tree. h 67.5o 45 feet

25. An airplane flying at 4500 feet is on a flight path directly toward an observer. If 30o is the angle of elevation from the observer to the plane, find the distance from the observer to the plane. d 4500 feet 30o

26. In traveling across flat land a driver noticed a mountain directly in front of the car. The angle of elevation to the peek is 4o. After the driver traveled 10 miles, the angle of elevation was 11o. Approximate the height of the mountain. h 11o 4o x 10 mi 10 + x

27. A flagpole at the top of a tall building (and at the edge of the building) is know to be 45 feet tall. If a man standing down the street from the building calculates the angle of elevation to the top of the building to be 55o and the angle of elevation to the top of the flagpole to be 57o. Find the height of the building. 45 h 57o 55o d

28. An observer standing on the cliff adjacent to the ocean looks out and sees an airplane flying directly over a ship. The observer calculates the angle of elevation to the plane to be 14o and the angle of depression to the ship to be 27o. How high above the ship is the airplane if we know that the ship is 1.5 miles from shore? p 14o 1.5 mi 27o b Distance of plane above ship = p +b=

29. In Washington, D.C., the Washington Monument is situated between the Capitol and the Lincoln Memorial. A tourist standing at the Lincoln Memorial tilts her head at an angle of 7.491° in order to look up to the top of the Washington Monument. At the same time, another tourist standing at the Capitol steps tilts his head at a 5.463° to also look at the top of the Washington Monument. Find the distance from the Lincoln Memorial to the Washington Monument.