Answer
$sin\theta=\frac{6}{11}$
$cos\theta=\frac{\sqrt{85}}{11}$
$tan\theta=\frac{6\sqrt{85}}{85}$
$cot\theta=\frac{\sqrt{85}}{6}$
$sec\theta=\frac{11\sqrt{85}}{85}$
Work Step by Step
To identify all the remaining trigonometric ratios we will use the trigonometric identities.
Step by step, we will first identify one of the trigonometric identities and then calculate:
$csc\theta = \frac{11}{6}$;
$\frac{11}{6}=\frac{1}{sin\theta}$
$sin\theta=\frac{6}{11}$; Which is $\frac{Opposite}{Hypotenuse}$. Using Pythagoras Theorem we will calculate $Adjacent$:
$Adjacent=\sqrt{11^2-6^2}=\sqrt{121-36}=\sqrt{85}$
$cos\theta=\frac{Adjacent}{Hypotenuse}=\frac{\sqrt{85}}{11}$
$tan\theta=\frac{sin\theta}{cos\theta}=\frac{6}{11}\times\frac{11}{\sqrt{85}}= \frac{6}{\sqrt{85}}=\frac{6\sqrt{85}}{85}$
$cot\theta=\frac{cos\theta}{sin\theta}=\frac{\sqrt{85}}{11}\times\frac{11}{6}=\frac{\sqrt{85}}{6}$
$sec\theta=\frac{1}{cos\theta}=\frac{11}{\sqrt{85}}=\frac{11\sqrt{85}}{85}$
