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
Practice each skill in the Homework Problems listed.
Suggested homework problems
Without using pencil and paper or a calculator, give the supplement of each angle.
Without using pencil and paper or a calculator, give the complement of each angle.
For Problems 3–6,
For Problems 7–10,
For Problems 11–20,
The point [latex]-5, 12[/latex] is on the terminal side.
The point [latex]12, 9[/latex] is on the terminal side.
[latex]\cos \theta = -0.8[/latex]
[latex]\cos \theta = \dfrac{5}{13}[/latex]
[latex]\cos \theta = \dfrac{3}{11}[/latex]
[latex]\cos \theta = \dfrac{-5}{6}[/latex]
[latex]tan \theta = \dfrac{-1}{6}[/latex]
[latex]tan \theta = \dfrac{9}{5}[/latex]
[latex]tan \theta = 4[/latex]
[latex]tan \theta = -1[/latex]
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] |
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] |
For each angle [latex]\theta[/latex] in the table for Problem 22, the angle [latex]180° - \theta[/latex] is also in the table.
Describe and explain any patterns of equal values you see in the table for Problem 22.
For Problems 25–28,
[latex]\theta = \cos^{-1} (\dfrac{3}{4}){,}[/latex] [latex]\phi = \cos^{-1} (\dfrac{-3}{4})[/latex]
[latex]\theta = \cos^{-1} (\dfrac{1}{5}){,}[/latex] [latex]\phi = \cos^{-1} (\dfrac{-1}{5})[/latex]
[latex]\theta = \cos^{-1} (0.1525){,}[/latex] [latex]~ \phi = \cos^{-1} (-0.1525)[/latex]
[latex]\theta = \cos^{-1} (0.6825){,}[/latex] [latex]\phi = \cos^{-1} (-0.6825)[/latex]
For Problems 29–34, find two different angles that satisfy the equation. Round to the nearest [latex]0.1°{.}[/latex]
[latex]\sin \theta = 0.7[/latex]
[latex]\sin \theta = 0.1[/latex]
[latex]\dfrac{sin \theta}{6} = 0.14[/latex]
[latex]\dfrac{5}{sin \theta} = 6[/latex]
[latex]4.8 = \dfrac{3.2}{sin \theta}[/latex]
[latex]1.5 = \dfrac{sin \theta}{0.3}[/latex]
For Problems 35–38, fill in the blanks with complements or supplements.
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]
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]
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]
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]
For Problems 41–44,
Sketch an angle of [latex]120°[/latex] in standard position. Find the missing coordinates of the points on the terminal side.
Sketch an angle of [latex]150°[/latex] in standard position. Find the missing coordinates of the points on the terminal side.
Sketch an angle of [latex]135°[/latex] in standard position. Find the missing coordinates of the points on the terminal side.
For Problems 49–54, find the area of the triangle with the given properties. Round your answer to two decimal places.
[latex]b = 2.5[/latex] in, [latex]c = 7.6[/latex] in, [latex]A = 138°[/latex]
[latex]a = 0.8[/latex] m, [latex]c = 0.15[/latex] m, [latex]B = 15°[/latex]
Find the area of the regular pentagon shown at right. (Hint: The pentagon can be divided into five congruent triangles.)
Find the area of the regular hexagon shown at right. (Hint: The hexagon can be divided into six congruent triangles.)
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.
For Problems 59 and 60,
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.)
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.)
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.
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.
For Problems 65–70,
[latex]\cos \theta = \dfrac{x}{3}, ~ x \lt 0[/latex]
[latex]tan \theta = \dfrac{4}{\alpha}, \alpha \lt 0[/latex]
[latex]\theta[/latex] is obtuse and [latex]\sin \theta = \dfrac{y}{2}[/latex]
[latex]\theta[/latex] is obtuse and [latex]tan \theta = \dfrac{q}{-7}[/latex]
[latex]\theta[/latex] is obtuse and [latex]tan \theta = m[/latex]
[latex]\cos \theta = h[/latex]