1. Solving Trig Problems Precal – Sections 5.1-5.5

2. Reference Angles p. 280 Associated with every angle drawn in standard position there is another angle called the reference angle. The reference angle is the acute angle formed by the terminal side of the given angle and the x- axis. Reference angles may appear in all four quadrants. Angles in quadrant I are their own reference angles.

3. Calculating Reference Angles

4. The values of trigonometric functions of angles greater than 90° (or less than 0°) can be found using corresponding acute angles called reference angles. Let be an angle in standard position. Its reference angle is the acute angle (read theta prime) formed by the terminal side of and the x-axis. 0 0 ' 0 Reference Angles

5. 90° < < 180°; 0 π π 2 < 0 0 Radians: = π 0 – 0 ' 0 ' = 180° Degrees: 0 ' – 0

6. 0 0 ' – 180° 0 = Degrees: 0 ' Radians: = 0 – π 0 ' 180° < < 270°; π 3π 2 < 0 0

7. 0 0 ' – 0 360° = Degrees: 0 ' Radians: = 0 – 2π2π 0 ' 270° < < 360°; 2π 2π 3π 2 < 0 0

8. Using reference angles Use reference angles and the unit circle to find the six trig functions for a 135 angle:

9. 0 = 320° Because 270° < < 360°, the reference angle is = 360° – 320° = 40°. 0 0 ' 0 = – 5π 6 S OLUTION Find the reference angle for each angle. 0 0 ' Because is coterminal with and π < <, the reference angle is = – π =. 0 7π 6 6 π 6 6 3π 2 0 '

10. 0 Let (3, – 4) be a point on the terminal side of an angle in standard position. Evaluate the six trigonometric functions of. 0 0 S OLUTION Use the Pythagorean theorem to find the value of r. r = x 2 + y 2 = 3 2 + (– 4 ) 2 = 25 = 5 Evaluating Trigonometric Functions Given a Point r (3, – 4)

11. Using x = 3, y = – 4, and r = 5, you can write the following: 4 5 sin = = – y r 0 cos = = x r 3 5 0 tan = = – 0 y x 4 3 csc = = – r y 5 4 0 sec = = r x 5 3 0 cot = = – 0 x y 3 4 Evaluating Trigonometric Functions Given a Point 0 r (3, – 4)

12. EVALUATING TRIGONOMETRIC FUNCTIONS C ONCEPT S UMMARY Use these steps to evaluate a trigonometric function of any angle. 0 2 Use the quadrant in which lies to determine the sign of the trigonometric function value of. 0 0 3 1 Find the reference angle. 0 ' Evaluate the trigonometric function for angle. 0 '

13. EVALUATING TRIGONOMETRIC FUNCTIONS C ONCEPT S UMMARY Signs of Function Values Quadrant IQuadrant II Quadrant IIIQuadrant IV sin, csc : + 00 cos, sec : – 00 tan, cot : – 00 sin, csc : + 00 cos, sec : + 00 tan, cot : + 00 sin, csc : – 00 cos, sec : – 00 tan, cot : + 00 sin, csc : – 00 cos, sec : + 00 tan, cot : – 00

14. Evaluate tan (– 210°). S OLUTION The angle – 210° is coterminal with 150°. The tangent function is negative in Quadrant II, so you can write: tan (– 210°) = – tan 30° = – 3 3 0 ' = 30° 0 = – 210° The reference angle is = 180° – 150° = 30°. 0 '

15. HW: Page 296 (1-3, 5-12, 14-21)

16. Angle of Elevation and Depression The angle of elevation is measured from the horizontal up to the object. Imagine you are standing here.

17. Angle of Elevation and Depression The angle of depression is measured from the horizontal down to the object. Constructing a right triangle, we are able to use trig to solve the triangle. A second similar triangle may also be formed.

18. Angle of Elevation and Depression Example #1

19. Angle of Elevation and Depression Suppose the angle of depression from a lighthouse to a sailboat is 5.7 o. If the lighthouse is 150 ft tall, how far away is the sailboat? Construct a triangle and label the known parts. Use a variable for the unknown value. 5.7 o 150 ft. x

20. Angle of Elevation and Depression Suppose the angle of depression from a lighthouse to a sailboat is 5.7 o. If the lighthouse is 150 ft tall, how far away is the sailboat? 5.7 o 150 ft. x Set up an equation and solve.

21. Angle of Elevation and Depression 5.7 o 150 ft. x Remember to use degree mode! x is approximately 1,503 ft.

22. Angle of Elevation and Depression Example #2

23. Angle of Elevation and Depression A spire sits on top of the top floor of a building. From a point 500 ft. from the base of a building, the angle of elevation to the top floor of the building is 35 o. The angle of elevation to the top of the spire is 38 o. How tall is the spire? Construct the required triangles and label. 500 ft. 38 o 35 o

24. Angle of Elevation and Depression Write an equation and solve. Total height (t) = building height (b) + spire height (s) 500 ft. 38 o 35 o Solve for the spire height. t b s Total Height

25. Angle of Elevation and Depression Write an equation and solve. 500 ft. 38 o 35 o Building Height t b s

26. Angle of Elevation and Depression Write an equation and solve. 500 ft. 38 o 35 o The height of the spire is approximately 41 feet. t b s Total height (t) = building height (b) + spire height (s)

27. Angle of Elevation and Depression Example #3

28. Angle of Elevation and Depression A hiker measures the angle of elevation to a mountain peak in the distance at 28 o. Moving 1,500 ft closer on a level surface, the angle of elevation is measured to be 29 o. How much higher is the mountain peak than the hiker? Construct a diagram and label. 1 st measurement 28 o. 2 nd measurement 1,500 ft closer is 29 o.

29. Angle of Elevation and Depression Adding labels to the diagram, we need to find h. 28 o 29 o 1500 ftx ft h ft Write an equation for each triangle. Remember, we can only solve right triangles. The base of the triangle with an angle of 28 o is 1500 + x.

30. Angle of Elevation and Depression Now we have two equations with two variables. Solve by substitution. Solve each equation for h. Substitute.

31. Angle of Elevation and Depression Solve for x. Distribute. Get the x’s on one side and factor out the x. Divide. x = 35,291 ft.

32. Angle of Elevation and Depression However, we were to find the height of the mountain. Use one of the equations solved for “h” to solve for the height. x = 35,291 ft. The height of the mountain above the hiker is 19,562 ft.

33. Angle of Elevation and Depression Start homework on a new page. Assignment 4.8 / 1, 3, 8, 11, 14-16, 19, 26 Quiz 4.1-4.5 Friday Homework up to today is due Friday. Remember to change your calculator between radians and degrees when required. All graphing of trig functions is done in radians.