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What are the differences between the domain of tangent and cotangent?
Tangent: Defined everywhere except where $cos(x) = 0$ | Cotangent: Defined everywhere except where $sin(x) = 0$
What are the differences between the range of secant and tangent?
Secant: $(-infty, -1] cup [1, infty)$ | Tangent: $(-infty, infty)$
Compare the periods of sin(x) and csc(x).
sin(x): Period is $2pi$ | csc(x): Period is $2pi$
Compare the periods of cos(x) and sec(x).
cos(x): Period is $2pi$ | sec(x): Period is $2pi$
Compare the periods of tan(x) and cot(x).
tan(x): Period is $pi$ | cot(x): Period is $pi$
Define cosecant (csc x).
$csc(x) = frac{1}{sin(x)}$
Define secant (sec x).
$sec(x) = frac{1}{cos(x)}$
Define cotangent (cot x).
$cot(x) = frac{1}{ an(x)} = frac{cos(x)}{sin(x)}$
Explain the relationship between sine and cosecant.
Cosecant is the reciprocal of sine. When sine is zero, cosecant is undefined (vertical asymptote). When sine is at its max/min, cosecant is at its min/max.
Explain the relationship between cosine and secant.
Secant is the reciprocal of cosine. When cosine is zero, secant is undefined (vertical asymptote). When cosine is at its max/min, secant is at its min/max.
Explain the relationship between tangent and cotangent.
Cotangent is the reciprocal of tangent. When tangent is zero or undefined, cotangent is undefined (vertical asymptote). Cotangent is decreasing while tangent is increasing.
Describe the domain of the cosecant function.
All real numbers except $x = npi$, where n is an integer. (Where $sin(x) = 0$)
Describe the domain of the secant function.
All real numbers except $x = frac{(2n+1)pi}{2}$, where n is an integer. (Where $cos(x) = 0$)
Describe the domain of the cotangent function.
All real numbers except $x = npi$, where n is an integer. (Where $sin(x) = 0$)
Describe the range of the cosecant function.
$(-infty, -1] cup [1, infty)$
Describe the range of the secant function.
$(-infty, -1] cup [1, infty)$
Describe the range of the cotangent function.
$(-infty, infty)$
What are the vertical asymptotes of csc(x)?
At $x = npi$, where n is an integer, because $sin(x) = 0$ there.
What are the vertical asymptotes of sec(x)?
At $x = frac{(2n+1)pi}{2}$, where n is an integer, because $cos(x) = 0$ there.
What are the vertical asymptotes of cot(x)?
At $x = npi$, where n is an integer, because $sin(x) = 0$ there.