1.
Without using pencil and paper or a calculator, give the supplement of each angle.
- [latex]30°[/latex]
- [latex]45°[/latex]
- [latex]120°[/latex]
- [latex]25°[/latex]
- [latex]165°[/latex]
- [latex]110°[/latex]
Chapter 3: Laws of Sines and Cosines
Exercises: 3.1 Obtuse Angles
Skills
Practice each skill in the Homework Problems listed.
- Use the coordinate definition of the trig ratios #3-20, 45-48
- Find the trig ratios of supplementary angles #7-10, 21-38
- Know the trig ratios of the special angles in the second quadrant #21, 41-44
- Find two solutions of the equation [latex]\sin \theta = k[/latex] #29-38
- Find the area of a triangle #49-58
Suggested homework problems
Problems: #4, 16, 20, 8, 26, 42, 32, 36, 54, 56, 58
Exercises Homework 3.1
2.
Without using pencil and paper or a calculator, give the complement of each angle.
- [latex]60°[/latex]
- [latex]80°[/latex]
- [latex]25°[/latex]
- [latex]18°[/latex]
- [latex]64°[/latex]
- [latex]47°[/latex]
Exercise Group
For Problems 3–6,
- Give the coordinates of point [latex]P[/latex] on the terminal side of the angle.
- Find the distance from the origin to point [latex]P{.}[/latex]
- Find [latex]\cos \theta,~\sin \theta,[/latex] and [latex]tan \theta.[/latex]
3.
4.
5.
6.
Exercise Group
For Problems 7–10,
- Find the sin and cosine of the angle.
- Sketch the supplement of the angle in standard position. (Use congruent triangles.)
- Find the sin and cosine of the supplement.
- Find the angle and its supplement, rounded to the nearest °.
7.
8.
9.
10.
Exercise Group
For Problems 11–20,
- Sketch an angle in standard position with the given properties.
- Find [latex]\cos \theta,~\sin \theta,[/latex] and [latex]tan \theta.[/latex]
- Find the angle [latex]\theta{,}[/latex] rounded to tenths of a °.
11.
The point [latex]-5, 12[/latex] is on the terminal side.
12.
The point [latex]12, 9[/latex] is on the terminal side.
13.
[latex]\cos \theta = -0.8[/latex]
14.
[latex]\cos \theta = \dfrac{5}{13}[/latex]
15.
[latex]\cos \theta = \dfrac{3}{11}[/latex]
16.
[latex]\cos \theta = \dfrac{-5}{6}[/latex]
17.
[latex]tan \theta = \dfrac{-1}{6}[/latex]
18.
[latex]tan \theta = \dfrac{9}{5}[/latex]
19.
[latex]tan \theta = 4[/latex]
20.
[latex]tan \theta = -1[/latex]
21.
Fill in exact values from memory without using a calculator.
| [latex]\theta[/latex] | [latex]~~~0°~~~[/latex] | [latex]~~~30°~~~[/latex] | [latex]~~~45°~~~[/latex] | [latex]~~~60°~~~[/latex] | [latex]~~~90°~~~[/latex] | [latex]~~~120°~~~[/latex] | [latex]~~~135°~~~[/latex] | [latex]~~~150°~~~[/latex] | [latex]~~~180°~~~[/latex] |
| [latex]\cos \theta[/latex] | [latex]~[/latex] | [latex]~[/latex] | [latex]~[/latex] | [latex]~[/latex] | [latex]~[/latex] | [latex]~[/latex] | [latex]~[/latex] | [latex]~[/latex] | [latex]~[/latex] |
| [latex]\sin \theta[/latex] | [latex]~[/latex] | [latex]~[/latex] | [latex]~[/latex] | [latex]~[/latex] | [latex]~[/latex] | [latex]~[/latex] | [latex]~[/latex] | [latex]~[/latex] | [latex]~[/latex] |
| [latex]tan \theta[/latex] | [latex]~[/latex] | [latex]~[/latex] | [latex]~[/latex] | [latex]~[/latex] | [latex]~[/latex] | [latex]~[/latex] | [latex]~[/latex] | [latex]~[/latex] | [latex]~[/latex] |
22.
Use your calculator to fill in the table. Round values to four decimal places.
| [latex]\theta[/latex] | [latex]~~~15°~~~[/latex] | [latex]~~~25°~~~[/latex] | [latex]~~~65°~~~[/latex] | [latex]~~~75°~~~[/latex] | [latex]~~~105°~~~[/latex] | [latex]~~~115°~~~[/latex] | [latex]~~~155°~~~[/latex] | [latex]~~~165°~~~[/latex] |
| [latex]\cos \theta[/latex] | [latex]~[/latex] | [latex]~[/latex] | [latex]~[/latex] | [latex]~[/latex] | [latex]~[/latex] | [latex]~[/latex] | [latex]~[/latex] | [latex]~[/latex] |
| [latex]\sin \theta[/latex] | [latex]~[/latex] | [latex]~[/latex] | [latex]~[/latex] | [latex]~[/latex] | [latex]~[/latex] | [latex]~[/latex] | [latex]~[/latex] | [latex]~[/latex] |
| [latex]tan \theta[/latex] | [latex]~[/latex] | [latex]~[/latex] | [latex]~[/latex] | [latex]~[/latex] | [latex]~[/latex] | [latex]~[/latex] | [latex]~[/latex] | [latex]~[/latex] |
23.
For each angle [latex]\theta[/latex] in the table for Problem 22, the angle [latex]180° - \theta[/latex] is also in the table.
- What is true about [latex]\sin \theta[/latex] and [latex]\sin (180° - \theta){?}[/latex]
- What is true about [latex]\cos \theta[/latex] and [latex]\cos (180° - \theta){?}[/latex]
- What is true about [latex]tan \theta[/latex] and [latex]\tan (180° - \theta){?}[/latex]
24.
Describe and explain any patterns of equal values you see in the table for Problem 22.
Exercise Group
For Problems 25–28,
- Evaluate each pair of angles to the nearest [latex]0.1°[/latex] and show that they are supplements.
- Sketch both angles.
- Find the sine of each angle.
25.
[latex]\theta = \cos^{-1} (\dfrac{3}{4}){,}[/latex] [latex]\phi = \cos^{-1} (\dfrac{-3}{4})[/latex]
26.
[latex]\theta = \cos^{-1} (\dfrac{1}{5}){,}[/latex] [latex]\phi = \cos^{-1} (\dfrac{-1}{5})[/latex]
27.
[latex]\theta = \cos^{-1} (0.1525){,}[/latex] [latex]~ \phi = \cos^{-1} (-0.1525)[/latex]
28.
[latex]\theta = \cos^{-1} (0.6825){,}[/latex] [latex]\phi = \cos^{-1} (-0.6825)[/latex]
Exercise Group
For Problems 29–34, find two different angles that satisfy the equation. Round to the nearest [latex]0.1°{.}[/latex]
29.
[latex]\sin \theta = 0.7[/latex]
30.
[latex]\sin \theta = 0.1[/latex]
31.
[latex]\dfrac{sin \theta}{6} = 0.14[/latex]
32.
[latex]\dfrac{5}{sin \theta} = 6[/latex]
33.
[latex]4.8 = \dfrac{3.2}{sin \theta}[/latex]
34.
[latex]1.5 = \dfrac{sin \theta}{0.3}[/latex]
Exercise Group
For Problems 35–38, fill in the blanks with complements or supplements.
35.
If [latex]\sin 57° = q~{,}[/latex] then [latex]\sin \underline \qquad = q[/latex] also, [latex]\cos \underline \qquad = q{,}[/latex] and [latex]\cos \underline \qquad = -q{.}[/latex]
36.
If [latex]\sin 18° = w~{,}[/latex] then [latex]\sin \underline \qquad = w~[/latex] also, [latex]\cos \underline \qquad = w~{,}[/latex] and [latex]~ \cos \underline \qquad = -w{.}[/latex]
37.
If [latex]\cos 74° = m~{,}[/latex] then [latex]\cos \underline \qquad = -m~{,}[/latex] and [latex]\sin \underline \qquad~[/latex] and [latex]\sin \underline \qquad~[/latex] both equal [latex]m{.}[/latex]
38.
If [latex]\cos 36° = t~{,}[/latex] then [latex]\cos \underline \qquad = -t~{,}[/latex] and [latex]\sin \underline \qquad[/latex] and [latex]\sin \underline \qquad[/latex] both equal [latex]t{.}[/latex]
39.
- Sketch the line [latex]y = \dfrac{3}{4}x{.}[/latex]
- Find two points on the line with positive [latex]x[/latex]-coordinates.
- The line [latex]y = \dfrac{3}{4}x[/latex] makes an angle with the positive [latex]x[/latex]-axis. What is that angle?
- Repeat parts [latex]a[/latex] through [latex]c[/latex] for the line [latex]y = \dfrac{-3}{4}x{,}[/latex] except find two points with negative [latex]x[/latex]-coordinates.
40.
- Sketch the line [latex]y = \dfrac{5}{3}x{.}[/latex]
- Find two points on the line with positive [latex]x[/latex]-coordinates.
- The line [latex]y = \dfrac{5}{3}x[/latex] makes an angle with the positive [latex]x[/latex]-axis. What is that angle?
- Repeat parts [latex]a[/latex] through [latex]c[/latex] for the line [latex]y = \dfrac{-5}{3}x{,}[/latex] except find two points with negative [latex]x[/latex]-coordinates.
Exercise Group
For Problems 41–44,
- Find exact values for the base and height of the triangle.
- Compute an exact value for the area of the triangle.
41.
42.
43.
44.
45.
Sketch an angle of [latex]120°[/latex] in standard position. Find the missing coordinates of the points on the terminal side.
- [latex](-1, ?)[/latex]
- [latex](?, 3)[/latex]
46.
Sketch an angle of [latex]150°[/latex] in standard position. Find the missing coordinates of the points on the terminal side.
- [latex](?, 2)[/latex]
- [latex](-4, ?)[/latex]
47.
Sketch an angle of [latex]135°[/latex] in standard position. Find the missing coordinates of the points on the terminal side.
- [latex](?, 3)[/latex]
- [latex](-\sqrt{5}, ?)[/latex]
48.
- Use a sketch to explain why [latex]\cos 90° = 0{.}[/latex]
- Use a sketch to explain why [latex]\cos 180° = 1{.}[/latex]
Exercise Group
For Problems 49–54, find the area of the triangle with the given properties. Round your answer to two decimal places.
49.
50.
51.
52.
53.
[latex]b = 2.5[/latex] in, [latex]c = 7.6[/latex] in, [latex]A = 138°[/latex]
54.
[latex]a = 0.8[/latex] m, [latex]c = 0.15[/latex] m, [latex]B = 15°[/latex]
55.
Find the area of the regular pentagon shown at right. (Hint: The pentagon can be divided into five congruent triangles.)
56.
Find the area of the regular hexagon shown at right. (Hint: The hexagon can be divided into six congruent triangles.)
Exercise Group
For Problems 57 and 58, lots from a housing development have been subdivided into triangles. Find the total area of each lot by computing and adding the areas of each triangle.
57.
58.
Exercise Group
For Problems 59 and 60,
- Find the coordinates of point [latex]P{.}[/latex] Round to two decimal places.
- Find the sides [latex]BC[/latex] and [latex]PC[/latex] of [latex]\triangle PCB{.}[/latex]
- Find side [latex]PB{.}[/latex]
59.
60.
61.
Later we will be able to show that [latex]\sin 18° = \dfrac{\sqrt{5} - 1}{4}{.}[/latex] What is the exact value of [latex]\sin 162°?[/latex] (Hint: Sketch both angles in standard position.)
62.
Later we will be able to show that [latex]\cos 36° = \dfrac{\sqrt{5} + 1}{4}{.}[/latex] What is the exact value of [latex]\cos 144°?[/latex] (Hint: Sketch both angles in standard position.)
63.
Alice wants an obtuse angle [latex]\theta[/latex] that satisfies [latex]\sin \theta = 0.3{.}[/latex] Bob presses some buttons on his calculator and reports that [latex]\theta = 17.46°{.}[/latex] Explain Bob’s error and give a correct approximation of [latex]\theta[/latex] accurate to two decimal places.
64.
Yaneli finds that the angle [latex]\theta[/latex] opposite the longest side of a triangle satisfies [latex]\sin \theta = 0.8{.}[/latex] Zelda reports that [latex]\theta = 53.13°{.}[/latex] Explain Zelda’s error and give a correct approximation of [latex]\theta[/latex] accurate to two decimal places.
Exercise Group
For Problems 65–70,
- Sketch an angle [latex]\theta[/latex] in standard position, [latex]0° \le \theta \le 180°{,}[/latex] with the given properties.
- Find expressions for [latex]\cos \theta, \sin \theta{,}[/latex] and [latex]tan \theta[/latex] in terms of the given variable.
65.
[latex]\cos \theta = \dfrac{x}{3}, ~ x \lt 0[/latex]
66.
[latex]tan \theta = \dfrac{4}{\alpha}, \alpha \lt 0[/latex]
67.
[latex]\theta[/latex] is obtuse and [latex]\sin \theta = \dfrac{y}{2}[/latex]
68.
[latex]\theta[/latex] is obtuse and [latex]tan \theta = \dfrac{q}{-7}[/latex]
69.
[latex]\theta[/latex] is obtuse and [latex]tan \theta = m[/latex]
70.
[latex]\cos \theta = h[/latex]