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Trigonometry : Trigonometric Functions

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Example Questions

Example Question #16 : Simplifying Trigonometric Functions

Simplify.

$\frac{\sin x\tan x}{\cos x}$

Possible Answers:

$2\sin x$

$\sin x\cos x$

$\tan^2 x$

$\tan x$

Correct answer:

$\tan^2 x$

Explanation:

First, put everything in terms of sine and cosine:

$\frac{\sin x\tan x}{\cos x} \rightarrow \frac{\sin x\tfrac{\sin x}{\cos x}}{\cos x}\rightarrow \frac{\sin^2 x}{\cos^2 x}$

This simplifies to $\tan^2 x$ .

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Example Question #17 : Simplifying Trigonometric Functions

$\small \sec x*\sin x* \cot x$ If you simplify this equation. What might the new expression be?

Possible Answers:

$\small \sec x$

$\small \cos x$

$\small \tan x$

$\small \sin x$

Correct answer:

Explanation:

You should know two things. $\small \sec x = \frac{1}{\cos x}$ and $\small \cot x = \frac{\cos x}{\sin x}$ . If you place these to the equation, you will get $\small \frac{1}{\cos x}*\sin x*\frac{\cos x}{\sin x}$ . After you get rid of the same values, which are in the denominator and numerator, you will get 1.

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Example Question #21 : Simplifying Trigonometric Functions

Which of the following is equivalent to

$\tan\theta\csc\theta$ ?

Possible Answers:

$\sin\theta$

$\tan\theta$

$\csc\theta$

$\sec\theta$

$\cos\theta$

Correct answer:

$\sec\theta$

Explanation:

In order to evaluate this expression, rewrite the trigonometric identity in terms of sines and cosines. The tangent is equal to the sine over the cosine and the cosecant is the reciprocal of the sine; thus, we can write the following:

$\tan(\theta)\csc(\theta)= \frac{\sin(\theta)}{\cos(\theta)}\times\frac{1}{\sin(\theta)}$

Now, can simplify. Notice that the sine terms cancel each other out.

$\frac{\sin(\theta)}{\cos(\theta)}\times\frac{1}{\sin(\theta)}= \frac{1}{\cos(\theta)}$

Remember, that the reciprocal of the cosine is the secant.

$\frac{1}{\cos(\theta)}= \sec(\theta)$

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Example Question #22 : Simplifying Trigonometric Functions

Change a $-240^{\circ}$ angle to radians.

Possible Answers:

2/3

$\frac{4\pi }{3}$

$4\pi$

$\frac{-4\pi }{3}$

Correct answer:

$\frac{-4\pi }{3}$

Explanation:

In order to change an angle into radians, you must multiply the angle by $\frac{\pi }{180^{\circ}}$ .

Therefore, to solve:

$-240^{\circ} \cdot \frac{\pi }{180^{\circ}} = \frac{-4\pi }{3}$

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Example Question #23 : Simplifying Trigonometric Functions

$\frac{\cos ^{2}x}{1+\sin x}$ The simple way to express this equation is:

Possible Answers:

$\small 1+\sin x$

$\small \tan x$

$\small 1-\sin x$

$\small 1$

$\small \sin x + \cos x$

Correct answer:

$\small 1-\sin x$

Explanation:

If $\sin ^{2}x+\cos ^{2}x=1$ , then $\cos ^{2}x=1-\sin ^{2}x$ . Place $\sin ^{2}x-1$ to $\cos ^{2}x$ . Then turn it to $\frac{(\sin x-1)*(\sin x+1)}{(\sin x-1)}$ . Get rid of $(\sin x-1)$ , and you will get $\sin x+1$ .

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Example Question #1 : Graphs Of Inverse Trigonometric Functions

True or False: The inverse of the function $\sin (x)$ is also a function.

Possible Answers:

True

False

Correct answer:

False

Explanation:

Consider the graph of the function sin(x) . It passes the vertical line test, that is if a vertical line is drawn anywhere on the graph it only passes through a single point of the function. This means that sin(x) is a function.

Screen shot 2020 08 27 at 11.59.46 am

Now, for its inverse to also be a function it must pass the horizontal line test. This means that if a horizontal line is drawn anywhere on the graph it will only pass through one point.

Screen shot 2020 08 27 at 12.00.18 pm

This is not true, and we can also see that if we graph the inverse of sin(x) ( $sin^{-1}(x)$ ) that this does not pass the vertical line test and therefore is not a function. If you wish to graph the inverse of sin(x) , then you must restrict the domain so that your graph will pass the vertical line test.

Screen shot 2020 08 27 at 12.00.40 pm

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Example Question #1 : Graphs Of Inverse Trigonometric Functions

Which of the following is the graph of the inverse of sin(x) with $D: \frac{-\pi}{2} \leq x \leq \frac{\pi}{2}$ ?

Possible Answers:

Screen shot 2020 08 27 at 10.47.13 am

Screen shot 2020 08 27 at 10.47.06 am

Screen shot 2020 08 27 at 10.46.15 am

Screen shot 2020 08 27 at 10.45.10 am

Correct answer:

Screen shot 2020 08 27 at 10.46.15 am

Explanation:

Note that the inverse of sin(x) is not $\frac{1}{sin(x)}$ , that is the reciprocal. The inverse of sin(x) is $sin^{-1}(x)$ also written as arcsin(x) . The graph of sin(x) with $D: \frac{-\pi}{2} \leq x \leq \frac{\pi}{2}$ is as follows.

Screen shot 2020 08 27 at 10.45.10 am

And so the inverse of this graph must be the following with $D: -1 \leq x \leq 1$ and $R: \frac{-\pi}{2} \leq y \leq \frac{\pi}{2}$

Screen shot 2020 08 27 at 10.46.15 am

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Example Question #3 : Graphs Of Inverse Trigonometric Functions

Which best describes the easiest method to graph an inverse trigonometric function (or any function) based on the parent function?

Possible Answers:

The values are swapped with $\frac{1}{y}$ so where y = cos(x) for the parent function, $x = cos(\frac{1}{y})$ for the inverse function.

Let $y = \frac{1}{y}$ so where y = cos(x) for the parent function $y = \frac{1}{cos(x)}$ for the inverse function.

The and values are switched so where y = cos(x) for the parent function, x = cos(y) for the inverse function.

Let $x = \frac{1}{x}$ so where y = cos(x) for the parent function, $y = cos(\frac{1}{x})$ for the inverse function.

Correct answer:

The and values are switched so where y = cos(x) for the parent function, x = cos(y) for the inverse function.

Explanation:

To find an inverse function you swap the and values. Take cos(x) for example, to find the inverse we use the following method.

y = cos(x)

x = cos(y) (swap the and values)

$cos^{-1}(x) = y$ (solving for )

$y = cos^{-1}(x)$

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Example Question #4 : Graphs Of Inverse Trigonometric Functions

Which of the following represents the graph of $tan^{-1}(x)$ with $R: \frac{-\pi}{2} \leq y \leq \frac{\pi}{2}$ ?

Possible Answers:

Screen shot 2020 08 28 at 9.02.30 am

Screen shot 2020 08 28 at 9.00.48 am

Screen shot 2020 08 28 at 9.02.37 am

Screen shot 2020 08 28 at 9.01.00 am

Correct answer:

Screen shot 2020 08 28 at 9.01.00 am

Explanation:

If we are looking for the graph of $tan^{-1}(x)$ with $R: \frac{-\pi}{2} \leq x \leq \frac{\pi}{2}$ , that means this is the inverse of tan(x) with $D:\frac{-\pi}{2} \leq x \leq \frac{\pi}{2}$ . The graph of tan(x) with $D:\frac{-\pi}{2} \leq x \leq \frac{\pi}{2}$ is

Screen shot 2020 08 28 at 9.00.48 am

Switching the and values to graph the inverse we get the graph

Screen shot 2020 08 28 at 9.01.00 am

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Example Question #1 : Graphs Of Inverse Trigonometric Functions

Which of the following is the graph of $cos^{-1}(x) + 2$ with $D: -1 \leq x \leq 1$ ?

Possible Answers:

Screen shot 2020 08 27 at 2.14.48 pm

Screen shot 2020 08 27 at 2.14.55 pm

Screen shot 2020 08 27 at 2.13.29 pm

Screen shot 2020 08 27 at 2.14.41 pm

Correct answer:

Screen shot 2020 08 27 at 2.13.29 pm

Explanation:

We first need to think about the graph of the function $cos^{-1}(x)$ .

Screen shot 2020 08 27 at 2.13.25 pm

Using the formula y = D + Acos(B(x - C)) where is the vertical shift, we have to perform a transformation of moving the function $cos^{-1}(x)$ up two units on the graph.

Screen shot 2020 08 27 at 2.13.29 pm

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Trigonometry : Trigonometric Functions

Study concepts, example questions & explanations for Trigonometry

All Trigonometry Resources

Example Questions

Example Question #16 : Simplifying Trigonometric Functions

Example Question #17 : Simplifying Trigonometric Functions

Example Question #21 : Simplifying Trigonometric Functions

Example Question #22 : Simplifying Trigonometric Functions

Example Question #23 : Simplifying Trigonometric Functions

Example Question #1 : Graphs Of Inverse Trigonometric Functions

Example Question #1 : Graphs Of Inverse Trigonometric Functions

Example Question #3 : Graphs Of Inverse Trigonometric Functions

Example Question #4 : Graphs Of Inverse Trigonometric Functions

Example Question #1 : Graphs Of Inverse Trigonometric Functions

All Trigonometry Resources

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