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ACT Math : Cosine

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Example Question #1 : How To Find The Period Of The Cosine

A function with period P will repeat on intervals of length P, and these intervals are referred to as periods.

Find the period of the function

$cos\, (2x)$ .

Possible Answers:

$\pi$

$2\pi$

$\pi+2$

Correct answer:

$\pi$

Explanation:

For the function

cos(Ax)

the period is equal to

$\frac{2\pi}{A}$

or in this case

$\frac{2\pi}{2}$

which reduces to $\pi$ .

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Example Question #1 : How To Find The Period Of The Cosine

A function with period P will repeat on intervals of length P, and these intervals are referred to as periods.

Find the period of the function

$\cos \bigg(\frac{1}{4}\pi x\bigg)$ .

Possible Answers:

$4\pi$

$8\pi$

Correct answer:

Explanation:

For the function

cos(Ax)

the period is equal to

$\frac{2\pi}{A}$

or in this case

$\frac{2\pi}{\frac{1}{4}\pi}=2\pi \cdot \frac{4}{1\pi}$

which reduces to .

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Example Question #3 : How To Find The Period Of The Cosine

A function with period P will repeat on intervals of length P, and these intervals are referred to as periods.

Find the period of the function

$cos\, (\pi x)$ .

Possible Answers:

$2\pi$

$\pi$

Correct answer:

Explanation:

For the function

cos(Ax)

the period is equal to

$\frac{2\pi}{A}$

or in this case

$\frac{2\pi}{\pi}$

which reduces to .

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Example Question #4 : How To Find The Period Of The Cosine

A function with period will repeat its solutions in intervals of length .

What is the period of the function f(x) = cos(x) +3 ?

Possible Answers:

$3\pi$

$2\pi$

$\frac{2\pi}{3}$

Correct answer:

$2\pi$

Explanation:

For a trigonometric function $f(x) = \alpha cos(bx -c) +d$ , the period is equal to $\frac{2\pi}{b}$ . So, for f(x) = cos(x) +3 , $P = \frac{2\pi}{1} = 2\pi$ .

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Example Question #5 : How To Find The Period Of The Cosine

A function with period will repeat its solutions in intervals of length .

What is the period of the function $f(x) = 3cos(4\pi x-4) -2$ ?

Possible Answers:

$4\pi$

$\frac{\pi}{2}$

$8\pi^2$

$\frac{1}{2}$

$2\pi$

Correct answer:

$\frac{1}{2}$

Explanation:

For a trigonometric function $f(x) = \alpha cos(bx -c) +d$ , the period is equal to $\frac{2\pi}{b}$ . So, for $f(x) = 3cos(4\pi x-4) -2$ , $P = \frac{2\pi}{4\pi} = \frac{1}{2}$ .

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Example Question #6 : How To Find The Period Of The Cosine

A function with period will repeat its solutions in intervals of length .

What is the period of the function $f(x) = 2cos(\frac{1}{2}\pi x)$ ?

Possible Answers:

$\pi$

$4\pi$

$2\pi$

Correct answer:

Explanation:

For a trigonometric function $f(x) = \alpha cos(bx -c) +d$ , the period is equal to $\frac{2\pi}{b}$ . So, for $f(x) = 2cos(\frac{1}{2}\pi x)$ , $P = \frac{2\pi}{\frac{1}{2}\pi} = 4$ .

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Example Question #1 : How To Find The Range Of The Cosine

Simplify (cosΘ – sinΘ)²

Possible Answers:

cos2Θ – 1

1 – sin2Θ

sin2Θ – 1

1 + sin2Θ

1 + cos2Θ

Correct answer:

1 – sin2Θ

Explanation:

Multiply out the quadratic equation to get cosΘ² – 2cosΘsinΘ + sinΘ²

Then use the following trig identities to simplify the expression:

sin2Θ = 2sinΘcosΘ

sinΘ² + cosΘ² = 1

1 – sin2Θ is the correct answer for (cosΘ – sinΘ)²

1 + sin2Θ is the correct answer for (cosΘ + sinΘ)²

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Example Question #2 : How To Find The Range Of The Cosine

Which of the following represents a cosine function with a range of to ?

Possible Answers:

f(x)=9cos(22x)-21

f(x)=cos(18x)-30

f(x)=4cos(9x)-12

f(x)=cos(x)-30

f(x)=18cos(2x)-30

Correct answer:

f(x)=9cos(22x)-21

Explanation:

The range of a cosine wave is altered by the coefficient placed in front of the base equation. So, if you have f(x)=12cos(x) , this means that the highest point on the wave will be at and the lowest at ; however, if you then begin to shift the equation vertically by adding values, as in, f(x)=12cos(x)+1 , then you need to account for said shift. This would make the minimum value to be and the maximum value to be .

For our question, the range of values covers |-30-(-12)|=|-30+12|=18 . This range is accomplished by having either or as your coefficient. ( merely flips the equation over the -axis. The range "spread" remains the same.) We need to make the upper value to be instead of . To do this, you will need to subtract 9+12 , or , from . This requires an downward shift of . An example of performing a shift like this is:

f(x)=cos(x)-21

Among the possible answers, the one that works is:

f(x)=9cos(22x)-21

The 22x parameter does not matter, as it only alters the frequency of the function.

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Example Question #3 : How To Find The Range Of The Cosine

Which of the following represents a cosine function with a range from to ?

Possible Answers:

f(x)=cos(13x)+8

f(x)=-26cos(x)+8

f(x)=13cos(x)+21

f(x)=-13cos(x)+8

f(x)=12cos(x)+21

Correct answer:

f(x)=-13cos(x)+8

Explanation:

For our question, the range of values covers |-5-21|=|-26|=26 . This range is accomplished by having either or as your coefficient. ( merely flips the equation over the -axis. The range "spread" remains the same.) We need to make the upper value to be instead of . This requires an upward shift of . An example of performing a shift like this is:

f(x)=cos(x)+8

Among the possible answers, the one that works is:

f(x)=-13cos(x)+8

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Example Question #4 : How To Find The Range Of The Cosine

What is the range of the trigonometric function defined by $S(t) = -3\sin (1.5t)$ ?

Possible Answers:

[-1.5,1.5]

[1.5,3]

[-3,3]

[0,3]

[-3,1.5]

Correct answer:

[-3,3]

Explanation:

The range of a sine or cosine function spans from the negative amplitude to the positive amplitutde. The amplutide is given by $\left|a\right|$ in the equation $S(t)=a\sin(bt)$ . Thus the range for our function is [-3,3]

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ACT Math : Cosine

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #1 : How To Find The Period Of The Cosine

Example Question #1 : How To Find The Period Of The Cosine

Example Question #3 : How To Find The Period Of The Cosine

Example Question #4 : How To Find The Period Of The Cosine

Example Question #5 : How To Find The Period Of The Cosine

Example Question #6 : How To Find The Period Of The Cosine

Example Question #1 : How To Find The Range Of The Cosine

Example Question #2 : How To Find The Range Of The Cosine

Example Question #3 : How To Find The Range Of The Cosine

Example Question #4 : How To Find The Range Of The Cosine

All ACT Math Resources

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