Trigonometry

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Questions 1 - 10

If $cos(x)=-\frac{\sqrt{3}}{2}$ and $0 \le x \le \pi$ , what is the value of $cos(x+\pi)$ ?

$\frac{1}{2}$

$\frac{2}{\sqrt{3}}$

$2\sqrt{5}$

$\frac{\sqrt{3}}{3}$

Explanation

Based on this data, we can make a little triangle that looks like:

Rt1

This is because $cos(x)=\frac{adjacent}{hypotenuse}$ .

Now, this means that must equal $\frac{\pi}{2}+\frac{\pi}{6}=\frac{4\pi}{6}=\frac{2\pi}{3}$ . (Recall that the cosine function is negative in the second quadrant.) Now, we are looking for:

$cos(\frac{2\pi}{3}+\pi)$ or $cos(\frac{5\pi}{3})$ . This is the cosine of a reference angle of:

$2\pi-\frac{5\pi}{3}=\frac{\pi}{3}$

Looking at our little triangle above, we can see that the cosine of $\frac{\pi}{3}$ is $\frac{1}{2}$ .

What is the period of the function $f(t) = -2\sin(4t)$ ?

$\frac{\pi}{2}$

$-\pi$

$2\pi$

Explanation

To find the period of Sine and Cosine functions you use the formula:
$\frac{2\pi}{\left | b\right |}$ where comes from $f(t) = a\sin(bt)$ . Looking at our formula you see b is 4 so
$\frac{2\pi}{\left | 4\right |}=\frac{\pi}{2}$

Given a function $y=2cos(\theta)+6$ , what is a valid domain?

$(-\infty, \infty)$

$(0,\infty)$

$[0,\infty)$

Explanation

The function $y=2cos(\theta)+6$ is related to the parent function $y=cos(\theta)$ .

The domain of the parent function is $(-\infty,\infty)$ . The values and will not affect the domain of the curve.

The answer is $(-\infty,\infty)$ .

Find the domain of $y=tan(x-\frac{3\pi}{4})$ . Assume is for all real numbers.

$\textup{Everywhere except: }x=\frac{5\pi}{4}\pm \pi k$

$x=\frac{5\pi}{4}\pm \pi k$

$\textup{Everywhere except: }x=\frac{3\pi}{4}\pm \pi k$

$x=\frac{3\pi}{4}\pm \pi k$

$\textup{Everywhere except: }x=\frac{5\pi}{4}\pm \frac{\pi}{2} k$

Explanation

The domain of does not exist at $\frac{\pi}{2}\pm \pi k$ , for is an integer.

The ends of every period approaches to either positive or negative infinity. Notice that for this problem, the entire graph shifts to the right $\frac{3\pi}{4}$ units. This means that the asymptotes would also shift right by the same distance.

The asymptotes will exist at:

$x=\frac{\pi}{2}\pm \pi k + \frac{3\pi}{4} = \frac{5\pi}{4}\pm \pi k$

Therefore, the domain of $y=tan(x-\frac{3\pi}{4})$ will exist anywhere EXCEPT:

$\frac{5\pi}{4}\pm \pi k$

Given a function $y=2cos(\theta)+6$ , what is a valid domain?

$(-\infty, \infty)$

$(0,\infty)$

$[0,\infty)$

Explanation

The function $y=2cos(\theta)+6$ is related to the parent function $y=cos(\theta)$ .

The domain of the parent function is $(-\infty,\infty)$ . The values and will not affect the domain of the curve.

The answer is $(-\infty,\infty)$ .

Using trig identities, simplify sinθ + cotθcosθ

tanθ

secθ

sin2θ

cos2θ

cscθ

Explanation

Cotθ can be written as cosθ/sinθ, which results in sinθ + cos2θ/sinθ.

Combining to get a single fraction results in (sin2θ + cos2θ)/sinθ.

Knowing that sin2θ + cos2θ = 1, we get 1/sinθ, which can be written as cscθ.

A man has a rope that is $40\textup{ feet}$ long, attached to the top of a small building. He pegs the rope into the ground at an angle of $14.5$^{\circ}$$ . How far away from the building did he walk horizontally to attach the rope to the ground? Round to the nearest inch.

$38\textup{ feet and }9\textup{ inches}$

$10\textup{ feet}$

$12\textup{ feet and }4\textup{ inches}$

$10\textup{ feet and 4 inches}$

$37\textup{ feet and 5 inches}$

Explanation

Begin by drawing out this scenario using a little right triangle:

Cos30

We know that the cosine of an angle is equal to the ratio of the side adjacent to that angle to the hypotenuse of the triangle. Thus, for our triangle, we know:

$cos(14.5) =\frac{x}{40}$

Using your calculator, solve for :

This is . Now, take the decimal portion in order to find the number of inches involved.

Thus, rounded, your answer is feet and inches.

$38\textup{ feet and }9\textup{ inches}$

$10\textup{ feet}$

$12\textup{ feet and }4\textup{ inches}$

$10\textup{ feet and 4 inches}$

$37\textup{ feet and 5 inches}$

Explanation

Begin by drawing out this scenario using a little right triangle:

Cos30

We know that the cosine of an angle is equal to the ratio of the side adjacent to that angle to the hypotenuse of the triangle. Thus, for our triangle, we know:

$cos(14.5) =\frac{x}{40}$

Using your calculator, solve for :

This is . Now, take the decimal portion in order to find the number of inches involved.

Thus, rounded, your answer is feet and inches.

Given a function $y=2cos(\theta)+6$ , what is a valid domain?

$(-\infty, \infty)$