Solve for a value (Trig question)

Question

The question is phrased as, in the triangle $xyz$, $\cos(x)=\sin(z)$. If $x=3j-19$ and $z=5j-15$, what is the value of $j$?

Firstly, I'm not quite sure if the variables refer to the side length or the angle measurements.

I'm sure there's an identity that will solve this question in 1 step but I'm not sure which one it would be. I want to use sine law but that seems to only give me $$\frac{\sin(x)}{3j-19}=\frac{\cos(x)}{5j-15}$$

which doesn't seem helpful?

Answer 1

Usually, angles and sides are distinguishable by capital and lowercase letters. Here, $x$ and $y$ refer to angle measures.

Recalling $\cos \theta = \sin (90-\theta)$ in the first quadrant, you can conclude $\cos x = \sin z$ if $x+z = 90$, resulting in

$$3j-19+(5j-15) = 90 \iff 8j-34 = 90 \iff 8j = 124 \iff j = 15.5$$

Answer 2

Hint:

Use the identity

$$\cos(90^\circ-A)=\sin A$$