Quotient Identities

Show your work for each problem.

1.
Which expression is equal to $\tan\theta$ ?
1. a) $\frac{\sin\theta}{\cos\theta}$
2. b) $\sin\theta\cos\theta$
3. c) $\frac{1}{\sin\theta}$
4. d) $\frac{\cos\theta}{\sin\theta}$
Answer: $\frac{\sin\theta}{\cos\theta}$
$\tan\theta = \frac{\sin\theta}{\cos\theta}$ : sine on top, cosine below. The fraction $\frac{\cos\theta}{\sin\theta}$ is the cotangent.
2.
Which expression is equal to $\cot\theta$ ?
1. a) $\frac{\sin\theta}{\cos\theta}$
2. b) $\sin\theta\cos\theta$
3. c) $\frac{\cos\theta}{\sin\theta}$
4. d) $\frac{1}{\cos\theta}$
Answer: $\frac{\cos\theta}{\sin\theta}$
$\cot\theta = \frac{1}{\tan\theta} = \frac{\cos\theta}{\sin\theta}$ . Note that $\frac{1}{\cos\theta}$ is $\sec\theta$ , not $\cot\theta$ .
3.
For an angle $\theta$ , $\sin\theta = 0.6$ and $\cos\theta = 0.8$ . What is $\tan\theta$ ?
Answer: 0.75
$\tan\theta = \frac{\sin\theta}{\cos\theta} = \frac{0.6}{0.8} = \frac{6}{8} = 0.75$ .
4.
For an angle $\theta$ , $\sin\theta = \frac{5}{13}$ and $\cos\theta = \frac{12}{13}$ . What is $\cot\theta$ ? Give a fraction.
Answer: 12/5
$\cot\theta = \frac{\cos\theta}{\sin\theta} = \frac{12/13}{5/13} = \frac{12}{5}$ .
5.
For an angle $\theta$ , $\sin\theta = \frac{8}{17}$ and $\cos\theta = \frac{15}{17}$ . Find $\tan\theta$ and $\cot\theta$ .
Answer: $\tan\theta = \frac{8}{15}$ , $\cot\theta = \frac{15}{8}$
1. $\tan\theta = \frac{\sin\theta}{\cos\theta}$ . What is $\tan\theta$ as a fraction? 8/15
2. $\cot\theta = \frac{\cos\theta}{\sin\theta}$ . What is $\cot\theta$ ? 15/8
3. Check: what is $\tan\theta \cdot \cot\theta$ ? 1
6.
For which angle is $\tan\theta$ undefined?
1. a) $0°$
2. b) $180°$
3. c) $90°$
4. d) $45°$
Answer: $90°$
$\tan 90° = \frac{\sin 90°}{\cos 90°} = \frac{1}{0}$ , which is undefined. At $0°$ and $180°$ the tangent is $0$ , and $\tan 45° = 1$ .
7.
Without a calculator: what is $\tan 37° \cdot \cot 37°$ ?
Answer: 1
$\tan 37° \cdot \cot 37° = \frac{\sin 37°}{\cos 37°} \cdot \frac{\cos 37°}{\sin 37°} = 1$ , because everything cancels.
8.
Simplify $\sin\theta \cot\theta$ , then evaluate it at $\theta = 60°$ .
Answer: $\sin\theta\cot\theta = \cos\theta$ , $\cos 60° = \frac{1}{2}$
1. Which fraction replaces $\cot\theta$ ? $\frac{\cos\theta}{\sin\theta}$
2. Cancel $\sin\theta$ in $\sin\theta \cdot \frac{\cos\theta}{\sin\theta}$ . What is left? $\cos\theta$
3. What is the value at $\theta = 60°$ ? 1/2
9.
Simplify $\frac{\cot\theta}{\cos\theta}$ .
1. a) $\sec\theta$
2. b) $\tan\theta$
3. c) $\sin\theta$
4. d) $\csc\theta$
Answer: $\csc\theta$
$\frac{\cot\theta}{\cos\theta} = \frac{\cos\theta}{\sin\theta} \cdot \frac{1}{\cos\theta} = \frac{1}{\sin\theta} = \csc\theta$ .
10.
For an angle $\theta$ , $\sin\theta = -0.8$ and $\cos\theta = 0.6$ . What is $\tan\theta$ ? Give a fraction.
Answer: -4/3
$\tan\theta = \frac{-0.8}{0.6} = -\frac{8}{6} = -\frac{4}{3}$ . The angle is in quadrant IV ( $x > 0$ , $y < 0$ ), where tangent is negative.
11.
Use $\sin 30° = \frac{1}{2}$ and $\cos 30° = \frac{\sqrt{3}}{2}$ to find the exact values of $\tan 30°$ and $\cot 30°$ .
Answer: $\tan 30° = \frac{\sqrt{3}}{3}$ , $\cot 30° = \sqrt{3}$
1. Which value is $\tan 30° = \frac{1/2}{\sqrt{3}/2}$ (with a rational denominator)? $\frac{\sqrt{3}}{3}$
2. Which value is $\cot 30° = \frac{\sqrt{3}/2}{1/2}$ ? $\sqrt{3}$
3. Check: what is $\tan 30° \cdot \cot 30°$ ? 1
12.
Simplify $\tan\theta \csc\theta$ , then evaluate it at $\theta = 60°$ .
Answer: $\tan\theta\csc\theta = \sec\theta$ , $\sec 60° = 2$
1. Which fraction replaces $\tan\theta$ ? $\frac{\sin\theta}{\cos\theta}$
2. Which fraction replaces $\csc\theta$ ? $\frac{1}{\sin\theta}$
3. After cancelling $\sin\theta$ you get $\frac{1}{\cos\theta}$ . Which function is that? $\sec\theta$
4. What is the value at $\theta = 60°$ ? 2
13.
For $0° \le \theta < 360°$ , at how many angles is $\cot\theta$ undefined?
Answer: 2
1. Which value must be $0$ for $\cot\theta$ to be undefined? $\sin\theta$
2. What is the smallest such angle in this interval (in degrees)? 0
3. At which other angle (in degrees) is $\sin\theta = 0$ ? 180
4. How many angles is that? ( $360°$ is not in the interval.) 2
14.
The angle $\theta$ lies in quadrant II and $\sin\theta = \frac{3}{5}$ . What is $\tan\theta$ ?
1. a) $-\frac{4}{3}$
2. b) $\frac{4}{3}$
3. c) $\frac{3}{4}$
4. d) $-\frac{3}{4}$
Answer: $-\frac{3}{4}$
$\cos^2\theta = 1 - \frac{9}{25} = \frac{16}{25}$ , and in quadrant II $\cos\theta = -\frac{4}{5}$ . So $\tan\theta = \frac{3/5}{-4/5} = -\frac{3}{4}$ .
15.
The angle $\theta$ lies in quadrant IV and $\cos\theta = \frac{5}{13}$ . Find $\tan\theta$ and $\cot\theta$ .
Answer: $\tan\theta = -\frac{12}{5}$ , $\cot\theta = -\frac{5}{12}$
1. Use $\sin^2\theta = 1 - \cos^2\theta$ . What is $\sin^2\theta$ ? 144/169
2. What is $\sin\theta$ in quadrant IV? -12/13
3. What is $\tan\theta = \frac{\sin\theta}{\cos\theta}$ ? -12/5
4. What is $\cot\theta$ ? -5/12
16.
The angle $\theta$ lies in quadrant III and $\cos\theta = -0.28$ . What is $\cot\theta$ ? Give a fraction.
Answer: 7/24
$\sin^2\theta = 1 - 0.0784 = 0.9216$ , so $\sin\theta = -0.96$ . Then $\cot\theta = \frac{-0.28}{-0.96} = \frac{28}{96} = \frac{7}{24}$ (positive, as in quadrant III).
17.
Simplify $\frac{\sin\theta}{\tan\theta} + \frac{\cos\theta}{\cot\theta}$ , then evaluate it for $\sin\theta = 0.6$ and $\cos\theta = 0.8$ .
Answer: $\cos\theta + \sin\theta = \frac{7}{5}$
1. What does $\frac{\sin\theta}{\tan\theta}$ simplify to? $\cos\theta$
2. What does $\frac{\cos\theta}{\cot\theta}$ simplify to? $\sin\theta$
3. What is the value for $\sin\theta = 0.6$ and $\cos\theta = 0.8$ ? 1.4
18.
An acute angle $\theta$ satisfies $3\sin\theta = 4\cos\theta$ . Find $\tan\theta$ , $\cot\theta$ , $\cos\theta$ and $\sin\theta$ .
Answer: $\tan\theta = \frac{4}{3}$ , $\cot\theta = \frac{3}{4}$ , $\cos\theta = \frac{3}{5}$ , $\sin\theta = \frac{4}{5}$
1. Divide both sides by $3\cos\theta$ . What is $\tan\theta$ ? 4/3
2. What is $\cot\theta$ ? 3/4
3. Put $\sin\theta = \frac{4}{3}\cos\theta$ into $\sin^2\theta + \cos^2\theta = 1$ . What is $\cos\theta$ ( $\theta$ is acute)? 3/5
4. What is $\sin\theta = \frac{4}{3}\cos\theta$ ? 4/5