1. 2 Acute Angles and Right Triangles © 2008 Pearson Addison-Wesley.
All rights reserved

2. Acute Angles and Right Triangles2
2.1 Trigonometric Functions of Acute Angles
2.2 Trigonometric Functions of Non-Acute Angles
2.3 Finding Trigonometric Function Values Using a Calculator
2.4 Solving Right Triangles
2.5 Further Applications of Right Triangles
Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

3. Trigonometric Functions of Acute Angles
2.1
Trigonometric Functions of Acute Angles
Right-Triangle-Based Definitions of the Trigonometric Functions ▪ Cofunctions ▪ Trigonometric Function Values of Special Angles
Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

4. 2.1 Example 1 Finding Trigonometric Function Values of An Acute Angle (page 51)
Find the sine, cosine, and tangent values for angles D and E in the figure.
Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

5. 2.1 Example 1 Finding Trigonometric Function Values of An Acute Angle (cont.)
Find the sine, cosine, and tangent values for angles D and E in the figure.
Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

6. 2.1 Example 2 Writing Functions in Terms of Cofunctions (page 51)
Write each function in terms of its cofunction.
(a) sin 9°
= cos (90° – 9°) = cos 81°
(b) cot 76°
= tan (90° – 76°) = tan 14°
(c) csc 45°
= sec (90° – 45°) = sec 45°
Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

7. 2.1 Example 3(a) Solving Equations Using the Cofunction Identities (page 52)
Find one solution for the equation. Assume all angles involved are acute angles.
(a)
Since cotangent and tangent are cofunctions, the equation is true if the sum of the angles is 90º.
Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

8. 2.1 Example 3(b) Solving Equations Using the Cofunction Identities (page 52)
Find one solution for the equation. Assume all angles involved are acute angles.
(b)
Since secant and cosecant are cofunctions, the equation is true if the sum of the angles is 90º.
Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

9. 2.1 Example 5 Finding Trigonometric Values for 30° (page 54)
Find the six trigonometric function values for a 30° angle.
Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

10. 2.1 Example 5 Finding Trigonometric Values for 30° (cont.)
Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

11. Trigonometric Functions of Non-Acute Angles
2.2
Trigonometric Functions of Non-Acute Angles
Reference Angles ▪ Special Angles as Reference Angles ▪ Finding Angle Measures with Special Angles
Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

12. 2.2 Example 1(a) Finding Reference Angles (page 59)
Find the reference angle for 294°.
294 ° lies in quadrant IV.
The reference angle is 360° – 294° = 66°.
Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

13. 2.2 Example 1(b) Finding Reference Angles (page 59)
Find the reference angle for 883°.
Find a coterminal angle between 0° and 360° by dividing 883° by 360°. The quotient is about 2.5.
883° is coterminal with 163°.
The reference angle is 180° – 163° = 17°.
Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

14. Find the values of the six trigonometric functions for 135°.
2.2 Example 2 Finding Trigonometric Functions of a Quadrant II Angle (page 60)
Find the values of the six trigonometric functions for 135°.
The reference angle for 135° is 45°.
Choose point P on the terminal side of the angle. The coordinates of P are (1, –1).
Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

15. 2.2 Example 2 Finding Trigonometric Functions of a Quadrant II Angle (page 60)
Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

16. Find the exact value of sin(–150°).
2.2 Example 3(a) Finding Trigonometric Function Values Using Reference Angles (page 61)
Find the exact value of sin(–150°).
An angle of –150° is coterminal with an angle of –150° + 360° = 210°.
The reference angle is 210° – 180° = 30°.
Since an angle of –150° lies in quadrant III, its sine is negative.
Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

17. Find the exact value of cot(780°).
2.2 Example 3(b) Finding Trigonometric Function Values Using Reference Angles (page 61)
Find the exact value of cot(780°).
An angle of 780° is coterminal with an angle of 780° – 2 ∙ 360° = 60°.
The reference angle is 60°.
Since an angle of 780° lies in quadrant I, its cotangent is positive.
Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

18. 2.2 Example 4 Evaluating an Expression with Function Values of Special Angles (page 62)
Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

19. 2.2 Example 5 Using Coterminal Angles to Find Function Values (page 62)
Evaluate each function by first expressing the function in terms of an angle between 0° and 360°.
(a)
(b)
Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

20. Find all values of θ, if θ is in the interval [0°, 360°) and sin θ =
2.2 Example 6 Finding Angle Measures Given an Interval and a Function Value (page 63)
Find all values of θ, if θ is in the interval [0°, 360°) and sin θ =
Since sin θ is negative, θ must lie in quadrants III or IV.
The absolute value of sin θ is so the reference angle is 60°.
The angle in quadrant III is 60° + 180° = 240°.
The angle in quadrant IV is 360° – 60° = 300°.
Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

21. Finding Trigonometric Functions Using a Calculator
2.3
Finding Trigonometric Functions Using a Calculator
Finding Function Values Using a Calculator ▪ Finding Angle Measures Using a Calculator
Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

22. 2.3 Example 1 Finding Function Values with a Calculator (page 67)
Approximate the value of each expression.
Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

23. 2.3 Example 1 Finding Function Values with a Calculator (cont.)
Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

24. 2.3 Example 2 Using Inverse Trigonometric Functions to Find Angles (page 67)
Use a calculator to find an angle θ in the interval [0°, 90°] that satisfies each condition.
Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

25. 2.3 Example 3 Finding Grade Resistance (page 68)
The force F in pounds when an automobile travels uphill or downhill on a highway is called grade resistance and is modeled by the equation , where θ is the grade and W is the weight of the automobile.
If the automobile is moving uphill, then θ > 0°; if it is moving downhill, then θ < 0°.
(a) Calculate F to the nearest 10 pounds for a 5500-lb car traveling an uphill grade with θ = 3.9°.
Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

26. 2.3 Example 3 Finding Grade Resistance (cont.)
(b) Calculate F to the nearest 10 pounds for a 2800-lb car traveling a downhill grade with θ = –4.8°.
(c) A 2400-lb car traveling uphill has a grade resistance of 288 pounds. What is the angle of the grade?
Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

27. Solving Right Triangles
2.4
Solving Right Triangles
Significant Digits ▪ Solving Triangles ▪ Angles of Elevation or Depression
Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

28. Solve right triangle ABC, if B = 28°40′ and a = 25.3 cm.
2.4 Example 1 Solving a Right Triangle Given an Angle and a Side (page 75)
Solve right triangle ABC, if B = 28°40′ and a = 25.3 cm.
Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

29. Three significant digits
2.4 Example 1 Solving a Right Triangle Given an Angle and a Side (cont.)
Three significant digits
Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

30. Three significant digits
2.4 Example 1 Solving a Right Triangle Given an Angle and a Side (cont.)
Three significant digits
Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

31. 2.4 Example 1 Solving a Right Triangle Given an Angle and a Side (cont.)
Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

32. 2.4 Example 2 Solving a Right Triangle Given Two Sides (page 76)
Solve right triangle ABC, if a = cm and b = cm.
Use the Pythagorean theorem to find c:
Four significant digits
Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

33. 2.4 Example 2 Solving a Right Triangle Given Two Sides (cont.)
Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

34. 2.4 Example 2 Solving a Right Triangle Given Two Sides (cont.)
Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

35. 2.4 Example 3 Finding a Length When the Angle of Elevation is Known (page 77)
The angle of depression from the top of a tree to a point on the ground 15.5 m from the base of the tree is 60.4°. Find the height of the tree.
The measure of equals the measure of the angle of depression because the two angles are alternate interior angles, so
Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

36. 2.4 Example 3 Finding a Length When the Angle of Elevation is Known (cont.)
Three significant digits
The tree is about 27.3 m tall.
Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

37. The angle of elevation of the sun is 63.39°.
2.4 Example 4 Finding a Length When the Angle of Elevation is Known (page 78)
The length of a shadow of a flagpole ft tall is ft. Find the angle of elevation of the sun.
Four significant digits
The angle of elevation of the sun is 63.39°.
Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

38. Further Applications of Right Triangles
2.5
Further Applications of Right Triangles
Bearing ▪ Further Applications
Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

39. Angle C is a right angle because angles CAB and CBA are complementary.
2.5 Example 1 Solving Problem Involving Bearing (First Method) (page 83)
Radar stations A and B are on an east-west line, 8.6 km apart. Station A detects a plane at C, on a bearing of 53°. Stations B simultaneously detects the same plane, on a bearing of 323°. Find the distance from B to C.
A line drawn due north is perpendicular to an east-west line, so right angles are formed at A and B, and angles CAB and CBA can be found as shown in the figure.
Angle C is a right angle because angles CAB and CBA are complementary.
Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

40. 2.5 Example 1 Solving Problem Involving Bearing (First Method) (cont.)
Find distance a by using the cosine function for angle A.
The distance from B to C is about 5.2 km.
Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

41. The information in the example gives and
2.5 Example 2 Solving Problem Involving Bearing (Second Method) (page 84)
The bearing from A to C is N 64° W. The bearing from A to B is S 82° W. The bearing from B to C is N 26° E. A plane flying at 350 mph take 1.8 hours to go from A to B. Find the distance from B to C.
The information in the example gives and
Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

42. 2.5 Example 2 Solving Problem Involving Bearing (Second Method) (page 84)
The sum of the measures of angles EAB and FBA is 180º because they are interior angles on the same side of a transversal.
90°
34°
56°
98°
Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

43. The distance from B to C is about 350 miles.
2.5 Example 2 Solving Problem Involving Bearing (Second Method) (page 84)
It takes 1.8 hours at 350 mph to fly from A to B, so
90°
34°
56°
630 mi
98°
The distance from B to C is about 350 miles.
Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

44. 2.5 Example 4 Solving a Problem Involving Angles of Elevation (page 85)
Marla needs to find the height of a building. From a given point on the ground, she finds that the angle of elevation to the top of the building is 74.2°. She then walks back 35 feet. From the second point, the angle of elevation to the top of the building is 51.8°. Find the height of the building.
Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

45. 2.5 Example 4 Solving a Problem Involving Angles of Elevation (cont.)
There are two unknowns, the distance from the base of the building, x, and the height of the building, h.
In triangle ABC
In triangle BCD
Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

46. 2.5 Example 4 Solving a Problem Involving Angles of Elevation (cont.)
Set the two expressions for h equal and solve for x.
Since h = x tan 74.2°, substitute the expression for x to find h.
The building is about 69 feet tall.
Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

47. 2.5 Example 4 Solving a Problem Involving Angles of Elevation (cont.)
Graphing Calculator Solution
Superimpose coordinate axes on the figure with D at the origin.
The coordinates of A are (35, 0).
The tangent of the angle between the x-axis and the graph of a line with equation y = mx + b is the slope of the line. For line DB, m = tan 51.8º.
Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

48. 2.5 Example 4 Solving a Problem Involving Angles of Elevation (cont.)
Since b = 0, the equation of line DB is
The equation of line AB is
Use the coordinates of A and the point-slope form to find the equation of AB:
Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

49. 2.5 Example 4 Solving a Problem Involving Angles of Elevation (cont.)
Graph y1 and y2, then find the point of intersection. The y-coordinate gives the height, h.
The building is about 69 feet tall.
Copyright © 2008 Pearson Addison-Wesley. All rights reserved.