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ACT Math : Sine

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Example Questions

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

What is the period of 2sin(4Θ)?

Possible Answers:

None of the answers are correct

$2\pi$

$\frac{\pi}{2}$

Correct answer:

$\frac{\pi}{2}$

Explanation:

The period of sinΘ is 2Π, so we set the new angle equal to the base period of 2Π and solve for Θ.

So 4Θ = 2Π and Θ = Π/2.

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

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

Find the period of

$sin\, (\pi x)$ .

Possible Answers:

$4\pi$

$2\pi$

$\pi$

Correct answer:

Explanation:

For the function

$sin\, (Ax)$

the period is equal to

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

in this case

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

which reduces to .

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Example Question #1 : Sine

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

$\sin \bigg(\frac{1}{2}x\bigg)$ .

Possible Answers:

$\frac{1}{2}$

$4\pi$

$\pi$

Correct answer:

$4\pi$

Explanation:

For the function

$sin\, (Ax)$

the period is equal to

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

in this case

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

which reduces to $4\pi$ .

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

What is the period of the function $f(t) = -2\sin(4t)$ ?

Possible Answers:

$2\pi$

$-\pi$

$\frac{\pi}{2}$

Correct answer:

$\frac{\pi}{2}$

Explanation:

To find the period of Sine and Cosine functions you use the formula:
$\frac{2\pi}{\left | b\right |}$ where comes from $f(t) = a\sin(bt)$ . Looking at our formula you see b is 4 so
$\frac{2\pi}{\left | 4\right |}=\frac{\pi}{2}$

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Example Question #52 : Trigonometry

What is the period of the given trigonometric function:

$c(t) = 3\sin(-2t)$ . Leave your answer in terms of $\pi$ , simplify all fractions.

Possible Answers:

$-\pi$

$\frac{\pi}{2}$

$\frac{\pi}{3}$

$\pi$

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

Correct answer:

$\pi$

Explanation:

To find the period of a sine, cosine, cosecant, or secant funciton use the formula:

$\omega = \frac{2\pi}{\left|b\right|}$ where comes from the general formula: $c(t)=a\sin(bt)$ . We see that for our equation $\left|b\right| = 2$ and so the period is $\pi$ when you reduce the fraction.

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Example Question #1 : Sine

Find the period of the following formula:

$f(x)=2\sin(4\theta)$

Possible Answers:

$\frac{\pi}{2}$

$\frac{1}{\pi}$

$2\pi$

$\pi$

$\frac{2}{\pi}$

Correct answer:

$\frac{\pi}{2}$

Explanation:

To find period, simply remember the following formula:

$\textup{period}=\frac{2\pi}{B}$

where B is the coefficient in front of x. Thus,

$\textup{period}=\frac{2\pi}{4}=\frac{\pi}{2}$

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Example Question #1 : How To Find The Domain Of The Sine

Find the domain of the function: $y=-9\sin(5\theta-3)-3$

Possible Answers:

$\left(-\frac{2\pi}{5},\infty\right)$

$\left(-\frac{2\pi}{5},\frac{2\pi}{5}\right)$

${}(-\infty,\infty)$

$\left[-\frac{2\pi}{5},\frac{2\pi}{5}\right]$

$\left(-\frac{5}{3},\frac{2\pi}{5}\right)$

Correct answer:

${}(-\infty,\infty)$

Explanation:

The function $y=-9\sin(5\theta-3)-3$ is related to the parent function $y=\sin(\theta)$ , which has a domain of $(-\infty.\infty)$ .

The value of theta for $y=-9\sin(5\theta-3)-3$ has no restriction and is valid for all real numbers.

The answer is $(-\infty,\infty)$ .

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Example Question #2 : Sine

What is the domain of the given trigonometric function:

$z(t) = 2\sin(-2t)$

Possible Answers:

$[0,\infty]$

$[-\pi,\pi]$

[-2,2]

$(-\infty,\infty)$

[-2,0]

Correct answer:

$(-\infty,\infty)$

Explanation:

For both Sine and Cosine, since there are no asymptotes like Tangent and Cotangent functions, the function can take in any value for . Thus the domain is:

$(-\infty,\infty)$

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

Which of the following statements is (are) true:

I. The domain of the tangent function is all real numbers

II. The range of the sine function is all real numbers

III. The periods of the tangent, sine, and cosine functions are the same.

Possible Answers:

II and III only

I and II only

II only

I only

none of the above

Correct answer:

II only

Explanation:

The domain of the tangent function does not include any values of x that are odd multiples of π/2 .

The range of the sine function is from [-1, 1].

The period of the tangent function is π, whereas the period for both sine and cosine is 2π.

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

Which of the following represents a sine wave with a range of $-5 \leq y \leq 5$ ?

Possible Answers:

f(x)=sin(5x)

f(x)=5sin(x)+5

f(x)=5sin(3x)

f(x)=sin(x)+5

f(x)=5sin(x) + 3

Correct answer:

f(x)=5sin(3x)

Explanation:

The range of a sine wave is altered by the coefficient placed in front of the base equation. So, if you have f(x)=5sin(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)=5sin(x)+5 , 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, then, it is fine to use f(x)=5sin(3x) . The for the function parameter only alters the period of the equation, making its waves "thinner."

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ACT Math : Sine

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

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

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

Example Question #1 : Sine

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

Example Question #52 : Trigonometry

Example Question #1 : Sine

Example Question #1 : How To Find The Domain Of The Sine

Example Question #2 : Sine

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

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

All ACT Math Resources

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