# ACT Math : Reference Angles

## Example Questions

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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)

0

cot(θ)

–sec(θ)

1

tan(θ)

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 : Reference Angles

Using trigonometry identities, simplify sinθcos2θ – sinθ

cos3θ

sin2θcosθ

cos2θsinθ

None of these answers are correct

–sin3θ

–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 #41 : Trigonometry

Using trig identities, simplify sinθ + cotθcosθ

secθ

sin2θ

tanθ

cscθ

cos2θ

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 #1 : How To Find A Reference Angle

Simplify sec4Θ – tan4Θ.

sec2Θ + tan2Θ

secΘ + sinΘ

sinΘ + cosΘ

cosΘ – sinΘ

tan2Θ – sin2Θ

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 #2951 : Act Math

Evaluate the expression below.

Explanation:

At , sine and cosine have the same value.

Cotangent is given by .

Now we can evaluate the expression.

### Example Question #3 : Reference Angles

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

351

369

109

90

90

Explanation:

3600 – 3510 = 90

### Example Question #1 : How To Find A Reference Angle

Simplify the following expression:

tanΘ

None of the answers are correct

cos2Θ

cscΘ

sin2Θ

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 : Reference Angles

What is the reference angle for ?

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 #1 : How To Find A Reference Angle

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

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 #2 : Reference Angles

What is the reference angle for ?