1. Applications of Trigonometric Functions
Chapter 7
TexPoint fonts used in EMF.
Read the TexPoint manual before you delete this box.: AAAAAAAA

2. Right Triangle Trigonometry; Applications
Section 7.1

3. Trigonometric Functions of Acute Angles
Right triangle: Triangle in which one angle is a right angle
Hypotenuse: Side opposite the right angle in a right triangle
Legs: Remaining two sides in a right triangle

4. Trigonometric Functions of Acute Angles
Non-right angles in a right triangle must be acute (0± < µ < 90±)
Pythagorean Theorem: a2 + b2 = c2

5. Trigonometric Functions of Acute Angles
These functions will all be positive

6. Trigonometric Functions of Acute Angles
Example.
Problem: Find the exact value of the six trigonometric functions of the angle µ
Answer:

7. Complementary Angle Theorem
Complementary angles: Two acute angles whose sum is a right angle
In a right triangle, the two acute angles are complementary

8. Complementary Angle Theorem

9. Complementary Angle Theorem
Cofunctions:
sine and cosine
tangent and cotangent
secant and cosecant
Theorem. [Complementary Angle Theorem] Cofunctions of complementary angles are equal

10. Complementary Angle Theorem
Example
Problem: Find the exact value of tan 12± { cot 78± without using a calculator
Answer:

11. Solving Right Triangles
Convention:
® is always the angle opposite side a
¯ is always the angle opposite side b
Side c is the hypotenuse
Solving a right triangle: Finding the missing lengths of the sides and missing measures of the angles
Express lengths rounded to two decimal places
Express angles in degrees rounded to one decimal place

12. Solving Right Triangles
We know:
a2 + b2 = c2
® + ¯ = 90±

13. Solving Right Triangles
Example.
Problem: If b = 6 and ¯ = 65±, find a, c and ®
Answer:

14. Solving Right Triangles
Example.
Problem: If a = 8 and b = 5, find c, ® and ¯
Answer:

15. Applications of Right Triangles
Angle of Elevation
Angle of Depression

16. Applications of Right Triangles
Example.
Problem: The angle of elevation of the Sun is 35.1± at the instant it casts a shadow 789 feet long of the Washington Monument. Use this information to calculate the height of the monument.
Answer:

17. Applications of Right Triangles
Direction or Bearing from a point O to a point P : Acute angle µ between the ray OP and the vertical line through O

18. Key Points Trigonometric Functions of Acute Angles
Complementary Angle Theorem
Solving Right Triangles
Applications of Right Triangles

19. The Law of Sines
Section 7.2

20. Solving Oblique Triangles
Oblique Triangle: A triangle which is not a right triangle
Can have three acute angles, or
Two acute angles and one obtuse angle (an angle between 90± and 180±)

21. Solving Oblique Triangles
Convention:
® is always the angle opposite side a
¯ is always the angle opposite side b
° is always the angle opposite side c

22. Solving Oblique Triangles
Solving an oblique triangle: Finding the missing lengths of the sides and missing measures of the angles
Must know one side, together with
Two angles
One angle and one other side
The other two sides

23. Solving Oblique Triangles
Known information:
One side and two angles: (ASA, SAA)
Two sides and angle opposite one of them: (SSA)
Two sides and the included angle (SAS)
All three sides (SSS)

24. Law of Sines Theorem. [Law of Sines]
For a triangle with sides a, b, c and opposite angles ®, ¯, °, respectively
Law of Sines can be used to solve ASA, SAA and SSA triangles
Use the fact that ® + ¯ + ° = 180±

25. Solving SAA Triangles Example.
Problem: If b = 13, ® = 65±, and ¯ = 35±, find a, c and °
Answer:

26. Solving ASA Triangles Example.
Problem: If c = 2, ® = 68±, and ¯ = 40±, find a, b and °
Answer:

27. Solving SSA Triangles Ambiguous Case Information may result in
One solution
Two solutions
No solutions

28. Solving SSA Triangles Example.
Problem: If a = 7, b = 9 and ¯ = 49±, find c, ® and °
Answer:

29. Solving SSA Triangles Example.
Problem: If a = 5, b = 4 and ¯ = 80±, find c, ® and °
Answer:

30. Solving SSA Triangles Example.
Problem: If a = 17, b = 14 and ¯ = 25±, find c, ® and °
Answer:

31. Solving Applied Problems
Example.
Problem: An airplane is sighted at the same time by two ground observers who are 5 miles apart and both directly west of the airplane. They report the angles of elevation as 12± and 22±. How high is the airplane?
Solution:

32. Key Points Solving Oblique Triangles Law of Sines
Solving SAA Triangles
Solving ASA Triangles
Solving SSA Triangles
Solving Applied Problems

33. The Law of Cosines
Section 7.3

34. Law of Cosines Theorem. [Law of Cosines]
For a triangle with sides a, b, c and opposite angles ®, ¯, °, respectively
Law of Cosines can be used to solve SAS and SSS triangles

35. Law of Cosines Theorem. [Law of Cosines - Restated]
The square of one side of a triangle equals the sum of the squares of the two other sides minus twice their product times the cosine of the included angle.
The Law of Cosines generalizes the Pythagorean Theorem
Take ° = 90±

36. Solving SAS Triangles Example.
Problem: If a = 5, c = 9, and ¯ = 25±, find b, ® and °
Answer:

37. Solving SSS Triangles Example.
Problem: If a = 7, b = 4, and c = 8, find ®, ¯ and °
Answer:

38. Solving Applied Problems
Example. In flying the 98 miles from Stevens Point to Madison, a student pilot sets a heading that is 11± off course and maintains an average speed of 116 miles per hour. After 15 minutes, the instructor notices the course error and tells the student to correct the heading.
(a) Problem: Through what angle will the plane move to correct the heading?
Answer:
(b) Problem: How many miles away is Madison when the plane turns?

39. Key Points Law of Cosines Solving SAS Triangles Solving SSS Triangles
Solving Applied Problems

40. Area of a Triangle
Section 7.4

41. Area of a Triangle Theorem. The area A of a triangle is
where b is the base and h is an altitude drawn to that base

42. Area of SAS Triangles
If we know two sides a and b and the included angle °, then
Also,
Theorem. The area A of a triangle equals one-half the product of two of its sides times the sine of their included angle.

43. Area of SAS Triangles Example.
Problem: Find the area A of the triangle for which a = 12, b = 15 and ° = 52±
Solution:

44. Area of SSS Triangles
Theorem. [Heron’s Formula] The area A of a triangle with sides a, b and c is where

45. Area of SSS Triangles Example.
Problem: Find the area A of the triangle for which a = 8, b = 6 and c = 5
Solution:

46. Key Points Area of a Triangle Area of SAS Triangles
Area of SSS Triangles

47. Simple Harmonic Motion; Damped Motion; Combining Waves
Section 7.5

48. Simple Harmonic Motion
Equilibrium (rest) position
Amplitude: Distance from rest position to greatest displacement
Period: Length of time to complete one vibration

49. Simple Harmonic Motion
Simple harmonic motion: Vibrational motion in which acceleration a of the object is directly proportional to the negative of its displacement d from its rest position
a = {kd, k > 0
Assumes no friction or other resistance

50. Simple Harmonic Motion
Simple harmonic motion is related to circular motion

51. Simple Harmonic Motion
Theorem. [Simple Harmonic Motion] An object that moves on a coordinate axis so that the distance d from its rest position at time t is given by either
d = a cos(!t) or d = a sin(!t)
where a and ! > 0 are constants, moves with simple harmonic motion. The motion has amplitude jaj and period

52. Simple Harmonic Motion
Frequency of an object in simple harmonic motion: Number of oscillations per unit time
Frequency f is reciprocal of period

53. Simple Harmonic Motion
Example. Suppose that an object attached to a coiled spring is pulled down a distance of 6 inches from its rest position and then released.
Problem: If the time for one oscillation is 4 seconds, write an equation that relates the displacement d of the object from its rest position after time t (in seconds). Assume no friction.
Answer:

54. Simple Harmonic Motion
Example. Suppose that the displacement d (in feet) of an object at time t (in seconds) satisfies the equation
d = 6 sin(3t)
(a) Problem: Describe the motion of the object.
Answer:
(b) Problem: What is the maximum displacement from its resting position?

55. Simple Harmonic Motion
Example. (cont.)
(c) Problem: What is the time required for one oscillation?
Answer:
(d) Problem: What is the frequency?

56. Damped Motion
Most physical systems experience friction or other resistance

57. Damped Motion
Theorem. [Damped Motion] The displacement d of an oscillating object from its at-rest position at time t is given by
where b is a damping factor (damping coefficient) and m is the mass of the oscillating object.

58. Damped Motion
Here jaj is the displacement at t = 0 and is the period under simple harmonic motion (no damping).

59. Damped Motion
Example. A simple pendulum with a bob of mass 15 grams and a damping factor of 0.7 grams per second is pulled 11 centimeters from its at-rest position and then released. The period of the pendulum without the damping effect is 3 seconds.
Problem: Find an equation that describes the position of the pendulum bob.
Answer:

60. Graphing the Sum of Two Functions
Example. f(x) = x + cos(2x)
Problem: Use the method of adding y-coordinates to graph y = f(x)
Answer:

61. Key Points Simple Harmonic Motion Damped Motion
Graphing the Sum of Two Functions