Question

using lengths a, b, and c in the right triangle shown, how are the trigonometric functions of θ defined?
sin θ =
cos θ =
tan θ =
csc θ =
sec θ =
cot θ =

Explanation:

Step1: Recall sine definition

$\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{a}{c}$

Step2: Recall cosine definition

$\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{b}{c}$

Step3: Recall tangent definition

$\tan\theta=\frac{\text{opposite}}{\text{adjacent}}=\frac{a}{b}$

Step4: Recall cosecant definition

$\csc\theta=\frac{1}{\sin\theta}=\frac{c}{a}$

Step5: Recall secant definition

$\sec\theta=\frac{1}{\cos\theta}=\frac{c}{b}$

Step6: Recall cotangent definition

$\cot\theta=\frac{1}{\tan\theta}=\frac{b}{a}$

Answer:

$\sin\theta=\frac{a}{c}$, $\csc\theta=\frac{c}{a}$, $\cos\theta=\frac{b}{c}$, $\sec\theta=\frac{c}{b}$, $\tan\theta=\frac{a}{b}$, $\cot\theta=\frac{b}{a}$