Math / Trigonometry

QuestionFind an expression that describes the location of each of the following values for $y=\cos \theta$ , where $n \in \mathbf{I}$ and $\theta$ is in radians. a) $\theta$ -intercepts b) maximum values c) minimum values

Studdy Solution

STEP 1

1. The function is $y = \cos \theta$ .
2. The variable $\theta$ is in radians.
3. $n$ is an integer, $n \in \mathbf{I}$ .

STEP 2

1. Determine the $\theta$ -intercepts of $y = \cos \theta$ .
2. Determine the maximum values of $y = \cos \theta$ .
3. Determine the minimum values of $y = \cos \theta$ .

STEP 3

The $\theta$ -intercepts occur where $y = \cos \theta = 0$ .
The cosine function is zero at odd multiples of $\frac{\pi}{2}$ .
Therefore, the $\theta$ -intercepts are at $\theta = \frac{\pi}{2} + n\pi$ , where $n$ is an integer.

STEP 4

The maximum value of the cosine function is $1$ .
This occurs at even multiples of $\pi$ .
Therefore, the maximum values occur at $\theta = 2n\pi$ , where $n$ is an integer.

STEP 5

The minimum value of the cosine function is $-1$ .
This occurs at odd multiples of $\pi$ .
Therefore, the minimum values occur at $\theta = (2n+1)\pi$ , where $n$ is an integer.
The expressions for each are:
a) $\theta$ -intercepts: $\theta = \frac{\pi}{2} + n\pi$
b) Maximum values: $\theta = 2n\pi$
c) Minimum values: $\theta = (2n+1)\pi$

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