Graphs and Inverses of Trigonometric Functions - Pre-Calculus

Card 0 of 20

Question

Find angle A of the following triangle:

Answer

We are given the hypotenuse and the side opposite of the angle in question. The trig function that relates these two sides is SIN. Therefore, we can write:

$sin(A) = \frac{12}{22}$

In order to solve for A, we need to take the inverse sin of both sides:

$sin^{-1}(sin(A)) = sin^{-1}(\frac{12}{22})$

which becomes

$A = sin^{-1}\bigg(\frac{12}{22}\bigg) = 33^{\circ}$

Compare your answer with the correct one above

Question

Consider $sin(\theta)=-\frac{1}{2}$ , where theta is valid from $\left [ 0,2\pi\right ]$ . What is a possible value of theta?

Answer

Solve for theta by taking the inverse sine of both sides.

$\theta=sin^-^1\left(-\frac{1}{2}\right) = -\frac{\pi}{6}$

Since this angle is not valid for the given interval of theta, add $2\pi$ radians to this angle to get a valid answer in the interval.

$-\frac{\pi}{6}+2\pi = -\frac{\pi}{6}+\frac{12\pi}{6}= \frac{11\pi}{6}$

Compare your answer with the correct one above

Question

Evaluate:

$cos^-^1\left(-\frac{\sqrt3}{2}\right)+cos^-^1\left(\frac{\sqrt3}{2}\right)$

Answer

First evaluate $cos^-^1\left(-\frac{\sqrt3}{2}\right)$ .

To evaluate inverse cosine, it is necessary to know the domain and range of inverse cosine.

For: $\theta=cos^-^1(x)$

The domain is only valid from .

$\theta$ is only valid from $[0,\pi]$ .

The part is asking for the angle where the x-value of the coordinate is $-\frac{\sqrt3}{2}$ . The only possibility on the unit circle is the second quadrant.

$cos^-^1\left(-\frac{\sqrt3}{2}\right)= \frac{5}{6}\pi$

Next, evaluate $cos^-^1\left(\frac{\sqrt3}{2}\right)$ .

Using the same domain and range restrictions, the only valid angle for the given x-value is in the first quadrant on the unit circle.

$cos^-^1\left(\frac{\sqrt3}{2}\right)= \frac{\pi}{6}$

Therefore:

$cos^-^1\left(-\frac{\sqrt3}{2}\right)+cos^-^1\left(\frac{\sqrt3}{2}\right) = \frac{5}{6}\pi + \frac{\pi}{6}=\frac{6\pi}{6}=\pi$

Compare your answer with the correct one above

Question

Evaluate:

$cos^-^1\left(\frac{1}{2}\right)$

Answer

To find the correct value of $cos^-^1\left(\frac{1}{2}\right)$ , it is necessary to know the domain and range of inverse cosine.

Domain:

Range: $[0,\pi]$

The question is asking for the specific angle when the x-coordinate is half.

The only possibility is located in the first quadrant, and the point of the special angle is $\left(\frac{1}{2},\frac{\sqrt3}{2}\right)$

The special angle for this coordinate is $\frac{\pi}{3}$ .

Compare your answer with the correct one above

Question

Find the value of $\cos^-^1\left(\frac{1}{2}\right)$ .

Answer

In order to determine the value or values of $cos^-^1(\frac{1}{2})$ , it is necessary to know the domain and range of the inverse sine function.

Domain:

Range: $[0,\pi]$

The question is asking for the angle value of theta where the x-value is $\frac{1}{2}$ under the range restriction. Since $x=\frac{1}{2}$ is located in the first and fourth quadrants, the range restriction makes theta only allowable from $[0,\pi]$ . Therefore, the theta value must only be in the first quadrant.

The value of the angle when the x-value is $\frac{1}{2}$ is degrees.

Compare your answer with the correct one above

Question

Find the inverse of the function

.

Make sure the final notation is only in the forms including , , and .

Answer

The easiest way to solve this problem is to simplify the original expression.

$y=csc(x)tan(x)=\frac{1}{sin(x)}\frac{sin(x)}{cos(x)}= \frac{1}{cos(x)}$

$y=\frac{1}{cos(x)}$

To find its inverse, let's exchange and ,

$x=\frac{1}{cos(y)}$

Solving for

$\frac{1}{x}=cos(y)$

$arccos(\frac{1}{x})=y$

Compare your answer with the correct one above

Question

Evaluate:

$cos^-^1\left(\frac{\sqrt3}{2}\right)+sin^-^1\left(\frac{\sqrt3}{2}\right)=$

Answer

To determine the value of $cos^-^1(\frac{\sqrt3}{2})+sin^-^1(\frac{\sqrt3}{2})$ , solve each of the terms first.

$cos^-^1(\frac{\sqrt3}{2})$

The inverse cosine has a domain and range restriction.

The domain exists from , and the range from $[0,\pi]$ . The inverse cosine asks for the angle when the x-value of the existing coordinate is $\frac{\sqrt3}{2}$ . The only possibility is $\frac{\pi}{6}$ since the coordinate can only exist in the first quadrant.

$sin^-^1(\frac{\sqrt3}{2})$

The inverse sine also has a domain and range restriction.

The domain exists from , and the range from $[-\frac{\pi}{2},\frac{\pi}{2}]$ . The inverse sine asks for the angle when the y-value of the existing coordinate is $\frac{\sqrt3}{2}$ . The only possibility is $\frac{\pi}{3}$ since the coordinate can only exist in the first quadrant.

Therefore:

$cos^-^1(\frac{\sqrt3}{2})+sin^-^1(\frac{\sqrt3}{2})= \frac{\pi}{6}+\frac{\pi}{3}=\frac{\pi}{2}$

Compare your answer with the correct one above

Question

Approximate:

Answer

:

There is a restriction for the range of the inverse tangent function from $(-\frac{\pi}{2},\frac{\pi}{2})$ .

The inverse tangent of a value asks for the angle where the coordinate lies on the unit circle under the condition that $\frac{y}{x}=99999$ . For this to be valid on the unit circle, the must be very close to 1, with an value also very close to zero, but cannot equal to zero since $\frac{y}{x}$ would be undefined.

The point is located on the unit circle when $\theta=\frac{\pi}{2}$ , but $tan(\frac{\pi}{2})$ is invalid due to the existent asymptote at this angle.

An example of a point very close to that will yield $\frac{y}{x}=99999$ can be written as:

Therefore, the approximated rounded value of is $\frac{\pi}{2}$ .

Compare your answer with the correct one above

Question

Determine the value of in degrees.

Answer

Rewrite and evaluate .

$csc^-^1(2)=sin^-^1(\frac{1}{2})=\frac{\pi}{6}$

The inverse sine of one-half is $\frac{\pi}{6}$ since $\frac{1}{2}$ is the y-value of the coordinate when the angle is $\frac{\pi}{6}$ .

$2csc^-^1(2)=2(\frac{\pi}{6})=\frac{\pi}{3}$

To convert from radians to degrees, replace $\pi$ with 180.

$\frac{\pi}{3}=\frac{180}{3}=60$

Compare your answer with the correct one above

Question

Evaluate the following:

$f(x)=cos(cos^{-1}(cos(\frac{\pi }{3})))$

Answer

For this particular problem we need to recall that the inverse cosine cancels out the cosine therefore,

$cos^{-1}(cos(\theta ))=\theta$ .

So the expression just becomes

$f(x)=cos\left(\frac{\pi }{3}\right)$

From here, recall the unit circle for specific angles such as $\frac{\pi}{3}$ .

Thus,

$f(x)=cos\left(\frac{\pi }{3}\right)=\frac{1}{2}$ .

Compare your answer with the correct one above

Question

Evaluate the following expression: $f(x)=tan(arctan(tan(\frac{\pi}{4})))$

Answer

This one seems complicated, but becomes considerably easier once you implement the fact that the composite cancels out to 1 and you are left with $tan(\frac{\pi}{4})$ which is equal to 1

Compare your answer with the correct one above

Question

Evaluate: $5sin^{-1}(\frac{\sqrt{3}}{2})$

Answer

$arcsin(\frac{\sqrt{3}}{2})=60^\circ\rightarrow \frac{\pi}{3}\rightarrow 5\cdot \frac{\pi}{3}$

Compare your answer with the correct one above

Question

Approximate the following:

Answer

This one is rather simple with knowledge of the unit circle: the value is extremely close to zero, of which always

Compare your answer with the correct one above

Question

Given that $sin\theta=\frac{3}{4}$ and that $\theta$ is acute, find the value of $cot\theta$ without using a calculator.

Answer

Given the value of the opposite and hypotenuse sides from the sine expression (3 and 4 respectively) we can use the Pythagorean Theorem to find the 3rd side (we’ll call it “t”): $t=\sqrt{4^{2}-3^{2}}\rightarrow t=\sqrt{7}$ . From here we can easily deduce the value of $cot\theta=\frac{\sqrt{7}}{3}$ (the adjacent side over the opposite side)

Compare your answer with the correct one above

Question

Evaluate the following expression: $f\left ( x \right )=\tan (\arctan (\tan (\frac{\pi }{4})))$

Answer

This one seems complicated but becomes considerably easier once you implement the fact that the composite $\tan \left ( \arctan \right )$ cancels out to and you are left with $\tan \left ( \frac{\pi }{4} \right )$ which is equal to , and so the answer is .

Compare your answer with the correct one above

Question

Evaluate: $5\sin ^{-1}\left ( \frac{\sqrt{3}}{2} \right )$

Answer

$\arcsin \left ( \frac{\sqrt{3}}2{} \right )=60^{\circ}\rightarrow \frac{\pi }{3}\rightarrow 5\cdot \frac{\pi }{3}$ and so the credited answer is $\frac{5\pi }{3}$ .

Compare your answer with the correct one above

Question

Approximate the following: $\arcsin \left ( 0.00002 \right )$ is closest in value to which of the following?

Answer

This problem is quite manageable with knowledge of the unit circle: the value is extremely close to zero, of which $\arcsin \left ( 0 \right )= 0$ always, so the only reasonable estimation of this value is 0.

Compare your answer with the correct one above

Question

Given that $\sin \theta =\frac{3}{4}$ and that $\theta$ is acute, find the value of $\cot \theta$ without using a calculator.

Answer

Given the value of the opposite and hypotenuse sides from the sine expression (3 and 4 respectively) we can use the Pythagorean Theorem to find the 3rd side (we’ll call it “t”): $t=\sqrt{4^{2}-3^{2}}\rightarrow t=\sqrt{7}$ .

From here we can deduce the value of $\cot \theta =\frac{\sqrt{7}}{3}$ (the adjacent side over the opposite side) and so the answer is $\frac{\sqrt{7}}{3}$ .

Compare your answer with the correct one above

Question

Which of the given functions has the greatest amplitude?

Answer

The amplitude of a function is the amount by which the graph of the function travels above and below its midline. When graphing a sine function, the value of the amplitude is equivalent to the value of the coefficient of the sine. Similarly, the coefficient associated with the x-value is related to the function's period. The largest coefficient associated with the sine in the provided functions is 2; therefore the correct answer is $2\sin(x)$ .

The amplitude is dictated by the coefficient of the trigonometric function. In this case, all of the other functions have a coefficient of one or one-half.

Compare your answer with the correct one above

Question

What is the amplitude of $y=3\sin\left(\frac{x}{5}\right)}$ ?

Answer

For any equation in the form $y=a\sin(bx)$ , the amplitude of the function is equal to $\left | a\right |$ .

In this case, and $b=\frac{1}{5}$ , so our amplitude is .

Compare your answer with the correct one above

Tap the card to reveal the answer