Answer
$a)$ $\sin\alpha=\cos\beta=\dfrac{3\sqrt{34}}{34}$
$b)$ $\tan\alpha=\cot\beta=\dfrac{3}{5}$
$c)$ $\sec\alpha=\csc\beta=\dfrac{\sqrt{34}}{5}$
Work Step by Step
The triangle is shown below.
Let $h$ be the hypotenuse of the triangle. Find it using the Pythagorean Theorem:
$h=\sqrt{5^{2}+3^{2}}=\sqrt{25+9}=\sqrt{34}$
$a)$ $\sin\alpha$ and $\cos\beta$
$\sin\alpha=\dfrac{opposite}{hypotenuse}=\dfrac{3}{\sqrt{34}}=\dfrac{3\sqrt{34}}{34}$
$\cos\beta=\dfrac{adjacent}{hypotenuse}=\dfrac{3}{\sqrt{34}}=\dfrac{3\sqrt{34}}{34}$
$b)$ $\tan\alpha$ and $\cot\beta$
$\tan\alpha=\dfrac{opposite}{adjacent}=\dfrac{3}{5}$
$\cot\beta=\dfrac{adjacent}{opposite}=\dfrac{3}{5}$
$c)$ $\sec\alpha$ and $\csc\beta$
$\sec\alpha=\dfrac{hypotenuse}{adjacent}=\dfrac{\sqrt{34}}{5}$
$\csc\beta=\dfrac{hypotenuse}{opposite}=\dfrac{\sqrt{34}}{5}$
