Graphing Secant and Cosecant

Show your work for each problem.

1.
What is equal to?
1. a)
2. b)
3. c)
4. d)
Answer:
The secant function is defined as the reciprocal of cosine: .
2.
What is equal to?
1. a)
2. b)
3. c)
4. d)
Answer:
The cosecant function is the reciprocal of sine: .
3.
What is ?
Answer: 1
4.
What is ?
Answer: 1
5.
What is the period of ?
1. a)
2. b)
3. c)
4. d)
Answer:
The period of is , the same as since .
6.
Where does have vertical asymptotes?
1. a)
2. b)
3. c)
4. d)
Answer:
Since , asymptotes occur where , which is at for any integer .
7.
Where does have vertical asymptotes?
1. a)
2. b)
3. c)
4. d)
Answer:
Since , asymptotes occur where , which is at for any integer .
8.
What is ?
Answer: -1
9.
What is ?
Answer: 2
10.
What is the range of ?
1. a)
2. b)
3. c)
4. d)
Answer:
Since , the reciprocal can only take values or . It never takes values between and .
11.
Evaluate
Answer: 2
1. What is ? 1/2
2. What is the reciprocal of ? 2
12.
Find the range of
Answer: (-inf, -1] U [3, inf)
1. What is the range of ? (-inf, -1] U [1, inf)
2. After multiplying by 2, what is the range of ? (-inf, -2] U [2, inf)
3. After adding 1, what is the range of ? (-inf, -1] U [3, inf)
13.
Sketch on by identifying key features
Answer: asymptotes at 0, pi, 2pi; min at pi/2 = 1; max at 3pi/2 = -1
1. Where are the asymptotes (where )? 0, pi, 2pi
2. What is the minimum value and where does it occur? 1 at pi/2
3. What is the maximum value and where does it occur? -1 at 3pi/2
4. Does the curve open upward or downward on ? upward
14.
If , what is ?
Answer: -0.5
Since , if , then .
15.
Find all asymptotes of on
Answer: pi/4, 3pi/4, 5pi/4, 7pi/4
1. Asymptotes occur where . For , pi/2 + n*pi
2. So . Solving for ? pi/4 + n*pi/2
3. List all values in pi/4, 3pi/4, 5pi/4, 7pi/4
16.
What is the period of ?
1. a)
2. b)
3. c)
4. d)
Answer:
The period of is . For , the period is .