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High School Math : Trigonometric Functions and Graphs

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Example Questions

Example Question #11 : Graphing The Sine And Cosine Functions

Which of the following is not in the range of the function $y = \sin{x}$ ?

Possible Answers:

$\frac{2}{3}$

$\frac{3}{4}$

$\frac{3}{2}$

$-\frac{1}{2}$

Correct answer:

$\frac{3}{2}$

Explanation:

The range of the function $y= \sin x$ is all numbers between and (the sine wave never goes above or below this).

Of the choices given, $\frac{3}{2}$ is greater than and thus not in this range.

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Example Question #11 : Functions

Simplify the function below:

$f(x)=cos^2(\frac{1}{2}cos^{-1}x)$

Possible Answers:

$\frac{x}{2}$

cosx

$\frac{1-x}{2}$

$\frac{1+x}{2}$

Correct answer:

$\frac{1+x}{2}$

Explanation:

We need to use the following formulas:

a) $cos^2x=\frac{1+cos2x}{2}$

and

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

We can simplify f(x) as follows:

$f(x)=cos^2(\frac{1}{2}cos^{-1}x)=\frac{1+cos(cos^{-1}x)}{2}=\frac{1+x}{2}$

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Example Question #1 : Understanding Period And Amplitude

What is the period of

$\tan \left ( \frac{x}{\pi } \right )$ ?

Possible Answers:

$\pi$

$2\pi$

$\frac{1}{\pi }$

$\pi^{2}$

Correct answer:

$\pi^{2}$

Explanation:

The period for $\tan(x)$ is $\pi$ . However, if a number is multiplied by , you divide the period $\pi$ by what is being multiplied by . Here, $(1/\pi )$ is being multiplied by . $\pi/(1/\pi )$ equals $\pi^{2}$ .

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Example Question #1 : Understanding Period And Amplitude

What is the amplitude of $y=3\sin(2x+5)$ ?

Possible Answers:

$-\frac{5}{2}$

$-\frac{5}{3}$

Correct answer:

Explanation:

The amplitude of a wave function like $y=3\sin(2x+5)$ is always going to be the coefficient of the function. In this case, that is .

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Example Question #3 : Understanding Period And Amplitude

What is the local maximum of $\sin(2x)$ between $-\pi$ and $\pi$ ?

Possible Answers:

0.05

Correct answer:

Explanation:

The fastest way to solve this problem is to graph it and observe the answer. However, the other option is to think of this equation in terms of period.

When the coefficient of the variable increases, the frequency increases and the period decreases by that rate.

Since our equation is $\sin(2x)$ , our period will be $\frac{1}{2}$ the normal period of a $\sin(x)$ wave. Since only the period is changing, the amplitude is not. Therefore the amplitude (the highest and lowest points) of $\sin(2x)$ will be the same as that of $\sin(x)$ . The amplitude of a sine wave is , so the amplitude of $\sin(2x)$ will also be .

Therefore, our maximum will be .

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Example Question #1 : Find The Amplitude Of A Sine Or Cosine Function

Which of the given functions has the greatest amplitude?

Possible Answers:

$\sin(x+3)$

$2\sin(x)$

$\sin(4x)$

$\sin\frac{2\pi }{3}(x+2)$

$\frac{1}{2}\sin(4x)$

Correct answer:

$2\sin(x)$

Explanation:

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.

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High School Math : Trigonometric Functions and Graphs

Study concepts, example questions & explanations for High School Math

All High School Math Resources

Example Questions

Example Question #11 : Graphing The Sine And Cosine Functions

Example Question #11 : Functions

Example Question #1 : Understanding Period And Amplitude

Example Question #1 : Understanding Period And Amplitude

Example Question #3 : Understanding Period And Amplitude

Example Question #1 : Find The Amplitude Of A Sine Or Cosine Function

All High School Math Resources

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