Answer
$\sin\theta=\dfrac{3}{4}$
$\cos\theta=\dfrac{\sqrt 7}{4}$
$\tan\theta=\dfrac{3\sqrt 7}{7}$
$\cot\theta=\dfrac{\sqrt 7}{3}$
$\sec\theta=\dfrac{4\sqrt 7}{7}$
$\csc\theta=\dfrac{4}{3}$
Work Step by Step
Let's note:
$h$=the hypotenuse
$o$=the opposite side of angle $\theta$
$a$=the adjacent side of angle $\theta$
We are given:
$h=4$
$o=3$
Determine the adjacent side, using the Pythagorean Theorem:
$h^2=o^2+a^2$
$4^2=3^2+a^2$
$16=9+a^2\\
a^2=7$
$a=\pm\sqrt{7}$
Since $a$ is never negative. then $a=\sqrt7$.
Determine the $6$ trigonometric functions of angle $\theta$:
$\sin\theta=\dfrac{o}{h}=\dfrac{3}{4}$
$\cos\theta=\dfrac{a}{h}=\dfrac{\sqrt 7}{4}$
$\tan\theta=\dfrac{o}{a}=\dfrac{3}{\sqrt 7}=\dfrac{3\sqrt 7}{7}$
$\cot\theta=\dfrac{a}{o}=\dfrac{\sqrt 7}{3}$
$\sec\theta=\dfrac{h}{a}=\dfrac{4}{\sqrt 7}=\dfrac{4\sqrt 7}{7}$
$\csc\theta=\dfrac{h}{o}=\dfrac{4}{3}$
