ACT Math : How to find a reference angle

Study concepts, example questions & explanations for ACT Math

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Example Questions

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Example Question #1 : Reference Angles

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

 

Possible Answers:

–sec(θ)

0

tan(θ)

1

cot(θ)

Correct answer:

1

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. 

Example Question #1 : How To Find A Reference Angle

Using trigonometry identities, simplify sinθcos2θ – sinθ

Possible Answers:

cos3θ

–sin3θ

sin2θcosθ

None of these answers are correct

cos2θsinθ

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.

Example Question #2 : How To Find A Reference Angle

Using trig identities, simplify sinθ + cotθcosθ

Possible Answers:

cscθ

secθ

sin2θ

tanθ

cos2θ

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.

Example Question #2 : How To Find A Reference Angle

Simplify sec4Θ – tan4Θ.

Possible Answers:

secΘ + sinΘ

tan2Θ – sin2Θ

sec2Θ + tan2Θ

cosΘ – sinΘ

sinΘ + cosΘ

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

Example Question #2 : How To Find A Reference Angle

Evaluate the expression below.

Possible Answers:

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

\sqrt{2}

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

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

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

Correct answer:

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

Explanation:

At , sine and cosine have the same value.

Cotangent is given by .

Now we can evaluate the expression.

Example Question #44 : Trigonometry

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

 

 

Possible Answers:

109

351

369

90

Correct answer:

90

Explanation:

3600 – 3510 = 90

 

 

 

 

 

Example Question #3 : How To Find A Reference Angle

Simplify the following expression:

 

 

Possible Answers:

tanΘ

sin2Θ

None of the answers are correct

cos2Θ

cscΘ

Correct answer:

sin2Θ

Explanation:

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

cotΘ =    cosΘ             secΘ = 1

              sinΘ               cosΘ

Example Question #2 : How To Find A Reference Angle

What is the reference angle for ?

Possible Answers:

Correct answer:

Explanation:

The reference angle is between  and , starting on the positive -axis and rotating in a counter-clockwise manor.

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

is equal to two complete revolutions, plus a  angle. Since is in Quadrant II, we subtract it from to get our reference angle:

Example Question #4 : How To Find A Reference Angle

Unit_circle

In the unit circle above, if , what are the coordinates of ?

Possible Answers:

Correct answer:

Explanation:

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

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

Example Question #4 : How To Find A Reference Angle

What is the reference angle for ?

Possible Answers:

Correct answer:

Explanation:

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

In this case, our angle  lies in Quadrant I, so the angle is its own reference angle.

Thus, the reference angle for  is .

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