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Trigonometry : Trigonometric Functions and Graphs

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Example Questions

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Example Question #41 : Trigonometric Functions

Simplify.

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

Possible Answers:

$\tan x$

$2\sin x$

$\sin x\cos x$

$\tan^2 x$

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 #111 : Trigonometry

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

Possible Answers:

$\small \cos x$

$\small \tan x$

$\small \sec x$

$\small \sin x$

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 #42 : Trigonometric Functions

Which of the following is equivalent to

$\tan\theta\csc\theta$ ?

Possible Answers:

$\sec\theta$

$\cos\theta$

$\sin\theta$

$\tan\theta$

$\csc\theta$

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 #121 : Trigonometry

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

Possible Answers:

2/3

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

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

$4\pi$

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 #41 : Trigonometric Functions And Graphs

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

Possible Answers:

$\small 1+\sin x$

$\small 1$

$\small 1-\sin x$

$\small \sin x + \cos x$

$\small \tan x$

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 : Period And Amplitude

What is the amplitude in the graph of the following equation:

$-2y=6sin(2\Theta )$

Possible Answers:

Correct answer:

Explanation:

The general form for a sine equation is:

$y=Asin(B(\Theta -C))+D$

The amplitude of a sine equation is the absolute value of .

Since our equation begins with , we would simplify the equation:

$y=-3(\frac{sin(2\Theta )}{2})$

The absolute value of would be .

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Example Question #1 : Trigonometric Graphs

What is the amplitude of y = sinx ?

Possible Answers:

Correct answer:

Explanation:

Amplitude describes the distance from the middle of a periodic function to its local maximum. y = sinx covers the range from -1 to 1. Thus, it covers a distance of 2 vertically. Half of this, or 1, gives us the amplitude of the function. It is often helpful to think of the amplitude of a periodic function as its "height".

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Example Question #3 : Period And Amplitude

What is the amplitude of y = 4cosx ?

Possible Answers:

Correct answer:

Explanation:

The amplitude of a function describes its height from the midline to the maximum. The amplitude of the parent function, y = cosx , is 1, since it goes from -1 to 1. In this case our function has been multiplied by 4. Think of the effects this multiplication has on the outputs. In , we get our maximum at x = 0 , and cosx = 1 . Here, we will get 4. The same thing happens for our minimum, at $\pi$ , cosx = -1 . Here, we get -4. Thus, by this analysis, it is clear that the amplitude is 4. In the future, remember that the number preceding the cosine function will always be its amplitude.

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Example Question #4 : Period And Amplitude

What is the period of the function y= sinx ?

Possible Answers:

$3\pi$

$\pi$

$2\pi$

Correct answer:

$2\pi$

Explanation:

By definition, the period of a function is the length of for which it repeats. y = sinx starts at 0, continues to 1, goes back to 0, goes to -1, and then back to 0.

This complete cycle goes from x = 0 to x = $2\pi$ .

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Example Question #5 : Period And Amplitude

What is the period and amplitude of the following trigonometric function?

$y = 9\ cos(6\pi x \ + 7) \ + 12$

Possible Answers:

$Amplitude = 9, \ period = \frac{1}{3}$

$Amplitude = 7, \ period = 2\pi$

$Amplitude = 12, \ period = 3$

$Amplitude = 3, \ period = \frac{\pi }{3}$

$Amplitude = 9, \ period = 3$

Correct answer:

$Amplitude = 9, \ period = \frac{1}{3}$

Explanation:

Recall the form of a sinusoid:

$y = A\sin (Bx + C) + D$ or y = Acos(Bx + C) + D

The important quantities for this question are the amplitude, given by , and period given by $\frac{2\pi }{B}{}$ .

For this problem, amplitude is equal to $A = \mathbf{9}$ and period is $\frac{2\pi }{6\pi } = \mathbf{\frac{1}{3}}{}$ .

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Trigonometry : Trigonometric Functions and Graphs

Study concepts, example questions & explanations for Trigonometry

All Trigonometry Resources

Example Questions

Example Question #41 : Trigonometric Functions

Example Question #111 : Trigonometry

Example Question #42 : Trigonometric Functions

Example Question #121 : Trigonometry

Example Question #41 : Trigonometric Functions And Graphs

Example Question #1 : Period And Amplitude

Example Question #1 : Trigonometric Graphs

Example Question #3 : Period And Amplitude

Example Question #4 : Period And Amplitude

Example Question #5 : Period And Amplitude

All Trigonometry Resources

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