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Chapter 5: Problem 3

For \(\theta=4\) radians, evaluate each of the following: (a) \(\sin ^{2} \theta\) (b) \(\sin \left(\theta^{2}\right)\)

Short Answer

Expert verified
(a) \(\sin^2(4) \approx 0.572753\) (b) \(\sin (16) \approx -0.2879\)

Step by step solution

01

Recall the sine function definition

The sine function is defined as the ratio of the opposite side to the hypotenuse in a right angle triangle. It can also be defined as the y-coordinate of a point on the unit circle. For any angle \(\theta\), it is represented as \(\sin{\theta}\).
02

Evaluate \(\sin^2 \theta\) with \(\theta = 4\) radians

We need to evaluate the expression: \(\sin^2 \theta\). Given that \(\theta = 4\), we can write: \[\sin^2 (4) \] Now, we can use a calculator or a mathematical software to find the sine of \(4\) radians: \[\sin(4) \approx -0.7568 \] Now we can square this value to get the result of \(\sin^2 \theta\): \[\sin^2 (4) \approx 0.572753 \]
03

Evaluate \(\sin (\theta^2)\) with \(\theta = 4\) radians

We need to evaluate the expression: \(\sin (\theta^2) \). Given that \(\theta = 4\), we can write: \[\sin (4^2) \] First, calculate the square of \(\theta\): \[4^2 = 16\] Now, we use the sine function to get the result: \[\sin(16) \approx -0.2879 \] Thus, we have evaluated the given expressions as follows: (a) \(\sin^2 (4) \approx 0.572753 \) (b) \(\sin (4^2) \approx -0.2879 \)

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