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ACT Math : ACT Math

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Example Questions

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

Find the domain of $y=tan(x-\frac{3\pi}{4})$ . Assume is for all real numbers.

Possible Answers:

$\textup{Everywhere except: }x=\frac{3\pi}{4}\pm \pi k$

$x=\frac{5\pi}{4}\pm \pi k$

$x=\frac{3\pi}{4}\pm \pi k$

$\textup{Everywhere except: }x=\frac{5\pi}{4}\pm \pi k$

$\textup{Everywhere except: }x=\frac{5\pi}{4}\pm \frac{\pi}{2} k$

Correct answer:

$\textup{Everywhere except: }x=\frac{5\pi}{4}\pm \pi k$

Explanation:

The domain of tan(x) does not exist at $\frac{\pi}{2}\pm \pi k$ , for is an integer.

The ends of every period approaches to either positive or negative infinity. Notice that for this problem, the entire graph shifts to the right $\frac{3\pi}{4}$ units. This means that the asymptotes would also shift right by the same distance.

The asymptotes will exist at:

$x=\frac{\pi}{2}\pm \pi k + \frac{3\pi}{4} = \frac{5\pi}{4}\pm \pi k$

Therefore, the domain of $y=tan(x-\frac{3\pi}{4})$ will exist anywhere EXCEPT:

$\frac{5\pi}{4}\pm \pi k$

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

Find the range of: $y= 2\tan(\theta)+15$

Possible Answers:

$[7.5,\infty)$

$[15,\infty)$

$(15,\infty)$

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

$(7.5,\infty)$

Correct answer:

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

Explanation:

The function $y= 2\tan(\theta)+15$ is related to $y=\tan(\theta)$ . The range of the tangent function is ${}(-\infty,\infty)$ .

The range of $y= 2\tan(\theta)+15$ is unaffected by the amplitude and the y-intercept. Therefore, the answer is ${}(-\infty,\infty)$ .

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Example Question #2951 : Act Math

What is the range of the trigonometric fuction defined by:
$f(t) = -2 \tan(3t)$ ?

Possible Answers:

$\mathbb{R}$

$\mathbb{N}$

[-1,1]

[-2,2]

$\mathbb{Z}$

Correct answer:

$\mathbb{R}$

Explanation:

For tangent and cotangent functions, the range is always all real numbers.

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Example Question #2952 : Act Math

What is the range of the given trigonometric function:

$n(t) = 4\cot(2t)$

Possible Answers:

[-4,4]

[-4,0]

[-2,2]

[0,2]

$(-\infty, \infty)$

Correct answer:

$(-\infty, \infty)$

Explanation:

The range of a function is every value that the funciton's results take. For tangent and cotangent, the function spans from $(-\infty, \infty)$ and so the range is:

$(-\infty, \infty)$

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Example Question #1 : How To Find A Reference Angle

Which of the following is equivalent to cot(θ)sec(θ)sin(θ)

Possible Answers:

–sec(θ)

cot(θ)

tan(θ)

Correct answer:

Explanation:

The first thing to do is to breakdown the meaning of each trig function, cot = cos/sin, sec = 1/cos, and sin = sin. Then put these back into the function and simplify if possible, so then (cos (Θ)/sin (Θ))*(1/cos (Θ))*(sin (Θ)) = (cos (Θ)*sin(Θ))/(sin (Θ)*cos(Θ)) = 1, since they all cancel out.

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Example Question #1 : Reference Angles

Using trigonometry identities, simplify sinθcos²θ – sinθ

Possible Answers:

cos²θsinθ

None of these answers are correct

–sin³θ

cos³θ

sin²θcosθ

Correct answer:

–sin³θ

Explanation:

Factor the expression to get sinθ(cos²θ – 1).

The trig identity cos²θ + sin²θ = 1 can be reworked to becomes cos²θ – 1 = –sinθ resulting in the simplification of –sin³θ.

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Example Question #2 : How To Find A Reference Angle

Using trig identities, simplify sinθ + cotθcosθ

Possible Answers:

cscθ

tanθ

cos²θ

secθ

sin²θ

Correct answer:

cscθ

Explanation:

Cotθ can be written as cosθ/sinθ, which results in sinθ + cos²θ/sinθ.

Combining to get a single fraction results in (sin²θ + cos²θ)/sinθ.

Knowing that sin²θ + cos²θ = 1, we get 1/sinθ, which can be written as cscθ.

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Example Question #1 : Reference Angles

Simplify sec⁴Θ – tan⁴Θ.

Possible Answers:

secΘ + sinΘ

sinΘ + cosΘ

cosΘ – sinΘ

tan²Θ – sin²Θ

sec²Θ + tan²Θ

Correct answer:

sec²Θ + tan²Θ

Explanation:

Factor using the difference of two squares: a² – b² = (a + b)(a – b)

The identity 1 + tan²Θ = sec²Θ which can be rewritten as 1 = sec²Θ – tan²Θ

So sec⁴Θ – tan⁴Θ = (sec²Θ + tan²Θ)(sec²Θ – tan²Θ) = (sec²Θ + tan²Θ)(1) = sec²Θ + tan²Θ

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Example Question #2953 : Act Math

Evaluate the expression below.

sin(45^o) + cot(45^o)

Possible Answers:

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

$\frac{1 + \sqrt{3}}{2}$

$\frac{2 + \sqrt{2}}{2}$

$\sqrt{2}$

$\frac{2 + \sqrt{3}}{2}$

Correct answer:

$\frac{2 + \sqrt{2}}{2}$

Explanation:

At 45^o , sine and cosine have the same value.

$sin(45^o)=cos(45^o)=\frac{\sqrt{2}}{2}$

Cotangent is given by $\frac{cos}{sin}$ .

$cot(45^o=\frac{cos(45^o)}{sin(45^o)})=\frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}}=1$

Now we can evaluate the expression.

$sin(45^o)+cot(45^o)=(\frac{\sqrt{2}}{2})+1=(\frac{\sqrt{2}}{2})+\frac{2}{2}=\frac{\sqrt{2}+2}{2}$

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Example Question #3 : How To Find A Reference Angle

What is the reference angle of an angle that measures 3510 in standard position?

Possible Answers:

369

351

109

Correct answer:

Explanation:

3600 – 3510 = 90

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ACT Math : ACT Math

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

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

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

Example Question #2951 : Act Math

Example Question #2952 : Act Math

Example Question #1 : How To Find A Reference Angle

Example Question #1 : Reference Angles

Example Question #2 : How To Find A Reference Angle

Example Question #1 : Reference Angles

Example Question #2953 : Act Math

Example Question #3 : How To Find A Reference Angle

All ACT Math Resources

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