Maths Extension 1 Year 11 - Module 3 - Lesson 3
Graphs of Reciprocal Trig Functions
Name
Date
Class
1. Lesson focus
Graphs of reciprocal trigonometric functions. Reciprocal graphs have vertical asymptotes wherever their base sine or cosine is zero; their range excludes the interval (−1,1).
Success criteria
- Apply the lesson method accurately in a routine example.
- Show each algebraic or domain step and check the result.
- Explain why the method and restrictions fit the question.
2. Worked practice
2. State the range of y=2csc x−1.
3. For y=cot(2x), state its period, vertical asymptotes and zeros on 0≤x<π.
3. Challenge
For y=3sec x+2, state the range and the value at x=0.
4. Multiple choice
What is the period of y=csc(3x)?
A. π/3
B. 3π
C. 2π/3
D. 2π
Worked answer key
Compare each step with your working. Keep stated intervals, domains and restrictions in your final answer.
1. Give the vertical asymptotes of y=csc x for 0≤x<2π.
csc x=1/sin x. In the stated half-open interval, sin x=0 at x=0 and x=π, so these are the asymptotes; x=2π is the next periodic one but is excluded.
2. State the range of y=2csc x−1.
cscx≤−1 or cscx≥1, so y≤−3 or y≥1.
3. For y=cot(2x), state its period, vertical asymptotes and zeros on 0≤x<π.
The period is π/2. Asymptotes satisfy sin(2x)=0: x=0,π/2 in the interval. Zeros satisfy cos(2x)=0: x=π/4,3π/4.
Challenge. For y=3sec x+2, state the range and the value at x=0.
secx≤−1 or ≥1; therefore y≤−1 or y≥5. At x=0, y=3(1)+2=5.
Multiple choice. What is the period of y=csc(3x)?
Correct answer: C. The cosecant graph has the sine period 2π; replacing x by 3x compresses the period to 2π/3.
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