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ACT Math : How to find a reference angle

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Example Questions

Example Question #1 : How To Find A Reference Angle

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

Possible Answers:

cot(θ)

–sec(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 : How To Find A Reference Angle

Using trigonometry identities, simplify sinθcos²θ – sinθ

Possible Answers:

None of these answers are correct

sin²θcosθ

cos³θ

cos²θsinθ

–sin³θ

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 #1 : How To Find A Reference Angle

Using trig identities, simplify sinθ + cotθcosθ

Possible Answers:

secθ

tanθ

sin²θ

cos²θ

cscθ

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 #2 : How To Find A Reference Angle

Simplify sec⁴Θ – tan⁴Θ.

Possible Answers:

tan²Θ – sin²Θ

secΘ + sinΘ

sinΘ + cosΘ

cosΘ – sinΘ

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 #2 : How To Find A Reference Angle

Evaluate the expression below.

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

Possible Answers:

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

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

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

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

$\sqrt{2}$

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 : Reference Angles

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

Possible Answers:

351

109

369

Correct answer:

Explanation:

3600 – 3510 = 90

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Example Question #2 : Reference Angles

Simplify the following expression:

$\frac{sin\theta}{cot\theta sec\theta}$

Possible Answers:

tanΘ

sin²Θ

None of the answers are correct

cscΘ

cos²Θ

Correct answer:

sin²Θ

Explanation:

Convert cotΘ and secΘ to sinΘ and cosΘ and simplify the resulting complex fraction.

cotΘ = cosΘ secΘ = 1

sinΘ cosΘ

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Example Question #3 : How To Find A Reference Angle

What is the reference angle for $855^{\circ}$ ?

Possible Answers:

$720^{\circ}$

$55^{\circ}$

$45^{\circ}$

$495^{\circ}$

$360^{\circ}$

Correct answer:

$45^{\circ}$

Explanation:

The reference angle is between 0^o and $90^{\circ}$ , starting on the positive -axis and rotating in a counter-clockwise manor.

To find the reference angle, we subtract 360^o for each complete revolution until we get a positive number less than 360^o .

855 - 360 = 495

495 - 360 = 135

855^o is equal to two complete revolutions, plus a $135^{\circ}$ angle. Since $135^{\circ}$ is in Quadrant II, we subtract it from $180^{\circ}$ to get our reference angle:

$180-135=45^{\circ}$

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Example Question #4 : How To Find A Reference Angle

Unit_circle

In the unit circle above, if $\Theta =30^{\circ}$ , what are the coordinates of (X,Y) ?

Possible Answers:

$\left (\frac{1}{2},\frac{1}{2} \right )$

$\left (\frac{1}{2},\frac{\sqrt{3}}{2} \right )$

$\left (\frac{\sqrt{3}}{2},\frac{1}{2} \right )$

$\left (1,\frac{1}{2} \right )$

$\left (\frac{\sqrt{3}}{2},\frac{\sqrt{3}}{2} \right )$

Correct answer:

$\left (\frac{\sqrt{3}}{2},\frac{1}{2} \right )$

Explanation:

On the unit circle, (X,Y) = (cos Θ, sin Θ).

(cos Θ,sin Θ) = (cos 30º, sin 30º) = (√3 / 2 , 1 / 2)

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Example Question #10 : How To Find A Reference Angle

What is the reference angle for $45^{\circ}$ ?

Possible Answers:

$315^{\circ}$

$135^{\circ}$

$45^{\circ}$

$90^{\circ}$

$180^{\circ}$

Correct answer:

$45^{\circ}$

Explanation:

A reference angle is the smallest possible angle between a given angle measurement and the x-axis.

In this case, our angle $45^{\circ}$ lies in Quadrant I, so the angle is its own reference angle.

Thus, the reference angle for $45^{\circ}$ is $45^{\circ}$ .

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ACT Math : How to find a reference angle

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #1 : How To Find A Reference Angle

Example Question #1 : How To Find A Reference Angle

Example Question #1 : How To Find A Reference Angle

Example Question #2 : How To Find A Reference Angle

Example Question #2 : How To Find A Reference Angle

Example Question #3 : Reference Angles

Example Question #2 : Reference Angles

Example Question #3 : How To Find A Reference Angle

Example Question #4 : How To Find A Reference Angle

Example Question #10 : How To Find A Reference Angle

All ACT Math Resources

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