Question

Let \( z=\cos(4x+5y) \), where \( x=\frac{\pi}{2}+2\theta \), \( y=-\left(\frac{\pi}{4}+\theta\right) \).

Hint:

For composite trigonometric functions, first substitute all variables in terms of the parameter and then apply the chain rule carefully.

Answer 1

Correct Answer: 3

Step 1: Combine the angle.
Since $z=\cos(4x+5y)$, first build $4x+5y$ from $x=\frac{\pi}{2}+2\theta$ and $y=-\big(\frac{\pi}{4}+\theta\big)$.

Step 2: Simplify.
\[ 4x+5y=2\pi+8\theta-\frac{5\pi}{4}-5\theta=\frac{3\pi}{4}+3\theta \]
So $z=\cos\!\big(\frac{3\pi}{4}+3\theta\big)$, now a function of $\theta$ alone.

Step 3: Differentiate.
\[ \frac{dz}{d\theta}=-3\sin\!\Big(\frac{3\pi}{4}+3\theta\Big) \]

Step 4: Plug in $\theta=\frac{\pi}{4}$.
The angle becomes $\frac{3\pi}{4}+\frac{3\pi}{4}=\frac{3\pi}{2}$.

Step 5: Evaluate.
Since $\sin\frac{3\pi}{2}=-1$, we get $\frac{dz}{d\theta}=-3(-1)=3$.
\[ \boxed{3.0} \]