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$g (x) = \cos x$

A) What is the $y$ intercept?

I got $(0,1)$.

B) For what numbers $x$ is $-\pi \leq x \leq \pi$ find where the graph is decreasing?

I got $0

C) What is the absolute minimum?

I got $-1$.

D) For what numbers $x$, $0 \leq x \leq 2\pi$, does the function $=0$?

I got $x= \pi/2$, $x=3\pi/2$, and $x=5\pi/2$.

E) For what numbers $x$, $-2\pi \leq x \leq 2\pi$ does $g(x)=1$? Where does it equal $-1$?

I got it equals $1$ when $x=-2\pi$, $0$, and $2\pi$. It equals $-1$ when $x= \pi, -\pi$.

F) For what numbers $x$, $-2\pi \leq x \leq 2\pi$ does $g(x)= \sqrt{3}/2$?

I got $k\pi/2$ and $k$ is an integer, and that's all.

G) What are the $x$ intercepts of $g$?

I got $(-\pi/2,0)$, $(\pi/2,0)$, $(3\pi/2, 0)$, $(-3\pi/2, 0)$, $(5\pi/2,0)$ $(-5\pi/2,0)$.

  • 0
    Please read this [tutorial](http://meta.math.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference) on how to typeset mathematics on this site. – 2017-02-09
  • 0
    In part F, $\cos(\dfrac{\pi}{6})=\dfrac{\sqrt3}{2}$. In general $\cos(2k\pi\pm\dfrac{\pi}{6})=\dfrac{\sqrt3}{2}$ – 2017-02-09

1 Answers 1

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Your answers to A, C, and E are correct.

$g(x) = \cos x$

(b) In the interval $-\pi \leq x \leq \pi$, where is the graph decreasing?

A function $f$ is (strictly) decreasing on an interval $I$ if for each $x_1, x_2 \in I$, with $x_1 < x_2$, $f(x_1) > f(x_2)$.

For the function $g(x) = \cos x$ restricted to the domain $[-\pi, \pi]$, the function is decreasing in the interval $[0, \pi] = \{x \in \mathbb{R} \mid 0 \leq x \leq \pi\}$.

(d) For which numbers $x$, $0 \leq x \leq 2\pi$, does $g(x) = 0$?

You answers $\pi/2$ and $3\pi/2$ are both correct. While $\cos\left(\frac{5\pi}{2}\right) = 0$, $\frac{5\pi}{2} > \frac{4\pi}{2} = 2\pi$, so $5\pi/2$ is not a valid solution in this interval.

(f) For which numbers $x$, $-2\pi \leq x \leq 2\pi$, does $g(x) = \dfrac{\sqrt{3}}{2}$?

A particular solution of the equation $$\cos x = \frac{\sqrt{3}}{2}$$ is $$x = \arccos\left(\frac{\sqrt{3}}{2}\right) = \frac{\pi}{6}$$ For the others, we can use symmetry.

symmetry_diagram_for_sine_and_cosine

Two angles have the same cosine if the $x$-coordinates of the points where there terminal sides intersect the unit circle are equal. Hence, $\cos\theta = \cos\varphi$ if $\varphi = \theta$ or $\varphi = -\theta$.

Thus, $$x = -\frac{\pi}{6}$$ is a solution.

Coterminal angles have the same cosine. Hence, $\cos\theta = \cos\varphi$ if $$\varphi = \theta + 2k\pi, k \in \mathbb{Z}$$ or $$\varphi = -\theta + 2m\pi, m \in \mathbb{Z}$$

We can express the two equations above in the form $$\varphi = \pm\theta + 2n\pi, n \in \mathbb{Z}$$

In this problem, we know that $x = \frac{\pi}{6}$ is a solution. Hence, any solution to the equation $\cos x = \frac{\sqrt{3}}{2}$ has the form $$x = \pm \frac{\pi}{6} + 2n\pi, n \in \mathbb{Z}$$ In the interval $[-2\pi, 2\pi]$, the only solutions are \begin{align*} x & = \frac{\pi}{6}\\ x & = -\frac{\pi}{6}\\ x & = \frac{\pi}{6} - 2\pi = -\frac{11\pi}{6}\\ x & = -\frac{\pi}{6} + 2\pi = \frac{11\pi}{6} \end{align*}

(g) What are the $x$-intercepts of $g(x) = \cos x$?

All of the numbers you listed are valid $x$-intercepts. However, no interval was specified. Therefore, the solution set consists of all angles whose terminal side lies on the positive or negative $y$-axis. The angles whose terminal sides lie on the positive $y$-axis are $$x = \frac{\pi}{2} + 2k\pi, k \in \mathbb{Z}$$ The angles whose terminal sides lie on the negative $y$-axis are $$x = -\frac{\pi}{2} + 2m\pi, m \in \mathbb{Z}$$ Notice that we could also use symmetry. A particular solution of the equation $$\cos x = 0$$ is $$x = \arccos(0) = \frac{\pi}{2}$$ By symmetry, $$x = -\frac{\pi}{2}$$ is also a solution. Since any angle coterminal with these angles is a solution, the general solution is $$x = \pm \frac{\pi}{2} + 2n\pi, n \in \mathbb{Z}$$