Question

if tan(θ) = 1/3 and 0 < θ < π/2, what is csc(θ)? write your answer in simplified, rationalized form. csc(θ) =

Explanation:

Step1: Recall the relationship between tangent and sine/cosine

$\tan(\theta)=\frac{\sin(\theta)}{\cos(\theta)}=\frac{1}{3}$, so $\cos(\theta) = 3\sin(\theta)$

Step2: Use the Pythagorean identity

$\sin^{2}(\theta)+\cos^{2}(\theta)=1$. Substitute $\cos(\theta)=3\sin(\theta)$ into it: $\sin^{2}(\theta)+(3\sin(\theta))^{2}=1$.

Step3: Simplify the equation

$\sin^{2}(\theta)+9\sin^{2}(\theta)=1$, $10\sin^{2}(\theta)=1$, $\sin^{2}(\theta)=\frac{1}{10}$.

Step4: Determine the value of sine

Since $0 < \theta<\frac{\pi}{2}$, $\sin(\theta)>0$. So $\sin(\theta)=\frac{1}{\sqrt{10}}$.

Step5: Recall the definition of cosecant

$\csc(\theta)=\frac{1}{\sin(\theta)}$. Substitute $\sin(\theta)=\frac{1}{\sqrt{10}}$ into it, we get $\csc(\theta)=\sqrt{10}$.

Answer:

$\sqrt{10}$