Reciprocal trigonometric functions and excluded angles
| English | 中文 | Pinyin |
|---|---|---|
| secant/ˈsiːkənt/ | 正割 | zhèng gē |
A formula divides by a rotating point’s horizontal coordinate. What happens as that coordinate approaches zero?
- A formula divides by a rotating point’s horizontal coordinate. What happens as that coordinate approaches zero?
- This lesson studies secant 正割: The reciprocal of cosine wherever cosine is nonzero.
Choose the mathematical structure
- secθ=1/cosθ and cosecθ=1/sinθ. Define cotθ=cosθ/sinθ, so it exists when sinθ≠0. Secant excludes θ=π/2+kπ; cosecant and cotangent exclude θ=kπ, for integer k. Secant/cosecant have period 2π and range (−∞,−1]∪[1,∞). Cotangent has period π and every real output. Secant is even; cosecant/cotangent are odd.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines secant?
The reciprocal of cosine wherever cosine is nonzero.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
At θ=2π/3, cosθ=−1/2 and sinθ=√3/2, so secθ=−2, cosecθ=2√3/3 and cotθ=−√3/3. At θ=π/2, cotθ=0/1=0 and cosecθ=1, but secθ is undefined. Draw secant branches outside the strip −1<y<1: y=1 at 0, y=−1 at π, with vertical asymptotes at π/2 and 3π/2. Cosecant reaches 1 at π/2 and −1 at 3π/2, with asymptotes at 0, π and 2π. Cotangent decreases from positive to negative infinity on (0,π), crossing zero at π/2; repeat after π.
Reciprocal trigonometric functions and excluded angles
secθ=1/cosθ and cosecθ=1/sinθ
Explain which denominator or branch restriction makes each step valid.
Find sec(2π/3).
cos(2π/3)=−1/2, so its reciprocal is −2.
Test a tempting shortcut
- cotθ=1/tanθ is usable only where both expressions exist and tanθ≠0. It cannot define cot(π/2), because tan(π/2) is undefined while cot(π/2)=0. The exponent −1 in sin⁻¹ often denotes inverse sine, not the reciprocal; write cosecθ when you mean 1/sinθ.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
Cotangent is undefined at every angle where tangent is undefined. This claim is false. Explain which definition or assumption it violates.
Find cot(π/2).
cot(π/2)=cos(π/2)/sin(π/2)=0/1=0.
Cotangent is undefined at every angle where tangent is undefined.
cotθ=1/tanθ is usable only where both expressions exist and tanθ≠0. It cannot define cot(π/2), because tan(π/2) is undefined while cot(π/2)=0. The exponent −1 in sin⁻¹ often denotes inverse sine, not the reciprocal; write cosecθ when you mean 1/sinθ.
Interpret a new situation
- Sketch the base sine/cosine zeros first: these locate the reciprocal asymptotes. Values 1 and −1 are unchanged by taking a reciprocal. Split the domain at each excluded angle and never join branches through an asymptote. State the function’s domain before rearranging an equation involving its denominator.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find cosec(π/2).
cosec(π/2)=1/sin(π/2)=1.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
Use this in your course
- 7357 · A-level · E. Match the target tier and specification before assigning extensions.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
The reciprocal of cosine wherever cosine is nonzero. Choose the relationship, show the method, check its assumptions and interpret the result.