graph the following function from at least -2π to 2π radians. state the maximum and minimum value of the…

Explanation:

Step1: Recall properties of $y = \cos(\theta)$

The general form of a cosine - function is $y = A\cos(B\theta - C)+D$. For $y=\cos(\theta)$, $A = 1$, $B = 1$, $C = 0$, $D = 0$. The period of $y=\cos(\theta)$ is given by $T=\frac{2\pi}{|B|}$. Since $B = 1$, $T = 2\pi$. The maximum value of $y=\cos(\theta)$ is $1$ (when $\theta = 2k\pi,k\in\mathbb{Z}$) and the minimum value is $- 1$ (when $\theta=(2k + 1)\pi,k\in\mathbb{Z}$). The $x$ - intercepts: Set $y = 0$, so $\cos(\theta)=0$. Then $\theta=\frac{\pi}{2}+k\pi,k\in\mathbb{Z}$. In the interval $[-2\pi,2\pi]$, the $x$ - intercepts are $\theta=-\frac{3\pi}{2},-\frac{\pi}{2},\frac{\pi}{2},\frac{3\pi}{2}$. The $y$ - intercept: Set $\theta = 0$, then $y=\cos(0)=1$.

Step2: Recall properties of $y=\sin(\theta)$

The period of $y = \sin(\theta)$ is $T=\frac{2\pi}{|B|}$. Since $B = 1$, $T = 2\pi$. The maximum value of $y=\sin(\theta)$ is $1$ (when $\theta=\frac{\pi}{2}+2k\pi,k\in\mathbb{Z}$) and the minimum value is $-1$ (when $\theta=\frac{3\pi}{2}+2k\pi,k\in\mathbb{Z}$). The $x$ - intercepts: Set $y = 0$, so $\sin(\theta)=0$. Then $\theta=k\pi,k\in\mathbb{Z}$. In the interval $[-2\pi,2\pi]$, the $x$ - intercepts are $\theta=-2\pi,-\pi,0,\pi,2\pi$. The $y$ - intercept: Set $\theta = 0$, then $y=\sin(0)=0$.

Step3: Graph the functions

For $y = \cos(\theta)$: Plot the points for key values of $\theta$ such as $\theta=-2\pi,-\frac{3\pi}{2},-\pi,-\frac{\pi}{2},0,\frac{\pi}{2},\pi,\frac{3\pi}{2},2\pi$ and their corresponding $y$ - values ($1,0, - 1,0,1,0,-1,0,1$ respectively) and connect them with a smooth curve. For $y=\sin(\theta)$: Plot the points for key values of $\theta$ such as $\theta=-2\pi,-\frac{3\pi}{2},-\pi,-\frac{\pi}{2},0,\frac{\pi}{2},\pi,\frac{3\pi}{2},2\pi$ and their corresponding $y$ - values ($0,-1,0,1,0,1,0,-1,0$ respectively) and connect them with a smooth curve.

For $y = \cos(\theta)$:

• Maximum value: $1$
• Minimum value: $-1$
• Period: $2\pi$
• $x$ - intercepts in $[-2\pi,2\pi]$: $\theta=-\frac{3\pi}{2},-\frac{\pi}{2},\frac{\pi}{2},\frac{3\pi}{2}$
• $y$ - intercept: $y = 1$

For $y=\sin(\theta)$:

• Maximum value: $1$
• Minimum value: $-1$
• Period: $2\pi$
• $x$ - intercepts in $[-2\pi,2\pi]$: $\theta=-2\pi,-\pi,0,\pi,2\pi$
• $y$ - intercept: $y = 0$

Answer:

For $y = \cos(\theta)$: Maximum value is $1$, minimum value is $-1$, period is $2\pi$, $x$ - intercepts in $[-2\pi,2\pi]$ are $-\frac{3\pi}{2},-\frac{\pi}{2},\frac{\pi}{2},\frac{3\pi}{2}$, $y$ - intercept is $1$. For $y=\sin(\theta)$: Maximum value is $1$, minimum value is $-1$, period is $2\pi$, $x$ - intercepts in $[-2\pi,2\pi]$ are $-2\pi,-\pi,0,\pi,2\pi$, $y$ - intercept is $0$.