Chapter 6.3, Problem 130AYU

Prove the quotient identities given in formula (3).

To determine

To prove: The quotient identities $\text{tan}\theta$ and $\text{cot}\theta$.

Answer to Problem 130AYU

Solution:

$\text{tan}\theta =\frac{y}{x}=\frac{\text{sin}\theta }{\text{cos}\theta }$ if $x\ne 0$

$\text{cot}\theta =\frac{x}{y}=\frac{\text{cos}\theta }{\text{sin}\theta }$ if $y\ne 0$

Explanation of Solution

If $P\text{ = }\left(x,y\right)$ is the point on the unit circle corresponding to $\text{θ}$, then

$\text{tan}\theta =\frac{y}{x}=\frac{\text{sin}\theta }{\text{cos}\theta }$ if $x\ne 0$

$\text{cot}\theta =\frac{x}{y}=\frac{\text{cos}\theta }{\text{sin}\theta }$ if $y\ne 0$