Question

Just the answer please.

6: http://imgur.com/iOdRCwy

7: http://imgur.com/HXcr1Bj

8: http://imgur.com/hXA28tQ

9: http://imgur.com/ZgnoiNe

10: http://imgur.com/5h6sQJn

Answer 1

Answer on Question #58376 – Math – Trigonometry

Question

1. Let the function $f(x)$ have the form $f(x) = A\cos (x + C)$ . To produce a graph that matches the one shown below? What must the value of $C$ be?

2

3

4

1

Solution

Parameter C in such kind of function means a horizontal shift. According to graph, shift value is equal to 2.

Answer: 2.

Question

2. For the function $y = -1 + 6\cos \left(\frac{2\pi}{7} (x - 5)\right)$ , what is the maximum value?

Solution

Function $\cos(x)$ has a maximum value of 1, then maximum value of the given function is

$$- 1 + 6 \cdot 1 = - 1 + 6 = 5$$

Answer: 5.

Question

3. For the function $y = -1 + 6\cos \left(\frac{2\pi}{7} (x - 5)\right)$ , what is the minimum value?

Solution

Function $\cos(x)$ has a minimum value of $-1$, then minimum value of given function is

$$-1 + 6 \cdot (-1) = -1 - 6 = -7$$

Answer: -7.

Question

4. Which of the following are vertical asymptotes of the function $y = 3 \cot(2x) - 4$?

Check all that apply.

$$x = \pi$$ $$x = 2\pi$$ $$x = \frac{\pi}{3}$$ $$x = \pm \frac{\pi}{2}$$

Answer: $x = \pi$, $x = 2\pi$, $x = \pm \frac{\pi}{2}$, i.e. all except for $x = \frac{\pi}{3}$.

Question

5. Which of the following are equivalent to the function $y = -3 \sin x + 2$?

Check all that apply

$$y = 3 \cos \left(x + \frac{\pi}{2}\right) + 2$$ $$y = -3 \cos \left(x - \frac{\pi}{2}\right) + 2$$ $$y = -3 \sin x - 2$$ $$y = 3 \sin(-x) + 2$$

Solution

There are reduction formula in trigonometry. It says Any trigonometric function whose argument is $\frac{\pi}{2} \pm x$, $\pi \pm x$, $2\pi \pm x$ can be written simply in terms of $x$, in particular $\cos \left(x + \frac{\pi}{2}\right) = -\sin x$, and $\cos \left(x - \frac{\pi}{2}\right) = \sin x$. This means the first and the second options are equivalent to the function $y = -3 \sin x + 2$.

Function $\sin x$ is symmetric with respect to the origin $(0; 0)$, it means $\sin(-x) = -\sin x$, so the last option is correct too.

Answer: $y = 3\cos \left(x + \frac{\pi}{2}\right) + 2, y = -3\cos \left(x - \frac{\pi}{2}\right) + 2, y = 3\sin (-x) + 2.$

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