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: Write the given function.
\[ z=\cos(4x+5y) \] where
\[ x=\frac{\pi}{2}+2\theta \] and
\[ y=-\left(\frac{\pi}{4}+\theta\right) \]

Step 2: Substitute \(x\) and \(y\) into \(z\).
\[ 4x+5y = 4\left(\frac{\pi}{2}+2\theta\right) + 5\left(-\frac{\pi}{4}-\theta\right) \]

Step 3: Simplify the expression.
\[ 4x+5y = 2\pi+8\theta-\frac{5\pi}{4}-5\theta \] \[ = \frac{3\pi}{4}+3\theta \]
Hence,
\[ z=\cos\left(\frac{3\pi}{4}+3\theta\right) \]

Step 4: Differentiate with respect to \(\theta\).
Using chain rule,
\[ \frac{dz}{d\theta} = -\sin\left(\frac{3\pi}{4}+3\theta\right)\cdot 3 \] \[ \frac{dz}{d\theta} = -3\sin\left(\frac{3\pi}{4}+3\theta\right) \]

Step 5: Put \(\theta=\frac{\pi}{4}\).
\[ \left.\frac{dz}{d\theta}\right|_{\theta=\frac{\pi}{4}} = -3\sin\left(\frac{3\pi}{4}+\frac{3\pi}{4}\right) \]

Step 6: Simplify the angle.
\[ \frac{3\pi}{4}+\frac{3\pi}{4} = \frac{6\pi}{4} = \frac{3\pi}{2} \] Therefore,
\[ \left.\frac{dz}{d\theta}\right|_{\theta=\frac{\pi}{4}} = -3\sin\frac{3\pi}{2} \]

Step 7: Evaluate the value.
\[ \sin\frac{3\pi}{2}=-1 \] Hence,
\[ \left.\frac{dz}{d\theta}\right|_{\theta=\frac{\pi}{4}} = -3(-1)=3 \] \[ \boxed{3.0} \]