Reciprocal trigonometric functions and excluded angles
Đã giới thiệu| English | Ý nghĩa |
|---|---|
| secant/ˈsiːkənt/ | hàm secant |
What happens as that coordinate approaches zero?
- A formula divides by a rotating point’s horizontal coordinate.
- This lesson studies secant 正割: The reciprocal of cosine wherever cosine is nonzero.
The method
- 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.
Why is cot(π/2)=0 while tan(π/2) is undefined?
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Defining cotangent as cos/sin rather than 1/tan keeps it valid exactly where tangent fails. At π/2, cos=0 kills tangent but gives cot=0/1=0.
A worked case
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).
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
cos(2π/3)=−1/2, so its reciprocal is −2.
Where it goes wrong
- 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.
Not true: Cotangent is undefined at every angle where tangent is undefined. cot θ=cos θ/sin θ: where tan is undefined, cos θ=0, so cot θ=0.
Find cot(π/2).
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
cot(π/2)=cos(π/2)/sin(π/2)=0/1=0.
Cotangent is undefined at every angle where tangent is undefined.
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
cot θ=cos θ/sin θ: where tan is undefined, cos θ=0, so cot θ=0.
Applying it
- 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.
Find cosec(π/2).
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
cosec(π/2)=1/sin(π/2)=1.
Match each reciprocal function to its excluded angles.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Each reciprocal inherits its domain from the zeros of its denominator; |sec| and |cosec| never fall below 1.
In your course
- AQA A-level Mathematics (7357), specification section E.
- Every paper allows a calculator, but method marks still need the expression or equation written before its value.
- Secant: The reciprocal of cosine wherever cosine is nonzero.
- secθ=1/cosθ and cosecθ=1/sinθ.
- Watch for: cotθ=1/tanθ is usable only where both expressions exist and tanθ≠0.