Question

θ is an acute angle and sin θ and cos θ are given. use identities to find tan θ, csc θ, sec θ, and cot θ. where necessary, rationalize denominators. sin θ = $\frac{4}{5}$, cos θ = $\frac{3}{5}$

Explanation:

Step1: Find $\tan\theta$

Recall the identity $\tan\theta=\frac{\sin\theta}{\cos\theta}$. Substitute $\sin\theta = \frac{4}{5}$ and $\cos\theta=\frac{3}{5}$ into the formula.
$\tan\theta=\frac{\frac{4}{5}}{\frac{3}{5}}=\frac{4}{3}$

Step2: Find $\csc\theta$

Recall the identity $\csc\theta=\frac{1}{\sin\theta}$. Substitute $\sin\theta=\frac{4}{5}$ into the formula.
$\csc\theta=\frac{1}{\frac{4}{5}}=\frac{5}{4}$

Step3: Find $\sec\theta$

Recall the identity $\sec\theta=\frac{1}{\cos\theta}$. Substitute $\cos\theta = \frac{3}{5}$ into the formula.
$\sec\theta=\frac{1}{\frac{3}{5}}=\frac{5}{3}$

Step4: Find $\cot\theta$

Recall the identity $\cot\theta=\frac{\cos\theta}{\sin\theta}$. Substitute $\sin\theta=\frac{4}{5}$ and $\cos\theta=\frac{3}{5}$ into the formula.
$\cot\theta=\frac{\frac{3}{5}}{\frac{4}{5}}=\frac{3}{4}$

Answer:

$\tan\theta=\frac{4}{3}$, $\csc\theta=\frac{5}{4}$, $\sec\theta=\frac{5}{3}$, $\cot\theta=\frac{3}{4}$