### All ACT Math Resources

## Example Questions

### Example Question #1 : Reference Angles

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

**Possible Answers:**

cot(θ)

–sec(θ)

tan(θ)

0

1

**Correct answer:**

1

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 #2 : Reference Angles

Using trigonometry identities, simplify sinθcos^{2}θ – sinθ

**Possible Answers:**

cos^{3}θ

None of these answers are correct

sin^{2}θcosθ

cos^{2}θsinθ

–sin^{3}θ

**Correct answer:**

–sin^{3}θ

Factor the expression to get sinθ(cos^{2}θ – 1).

The trig identity cos^{2}θ + sin^{2}θ = 1 can be reworked to becomes cos^{2}θ – 1 = –sinθ resulting in the simplification of –sin^{3}θ.

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

Using trig identities, simplify sinθ + cotθcosθ

**Possible Answers:**

cscθ

tanθ

secθ

cos^{2}θ

sin^{2}θ

**Correct answer:**

cscθ

Cotθ can be written as cosθ/sinθ, which results in sinθ + cos^{2}θ/sinθ.

Combining to get a single fraction results in (sin^{2}θ + cos^{2}θ)/sinθ.

Knowing that sin^{2}θ + cos^{2}θ = 1, we get 1/sinθ, which can be written as cscθ.

### Example Question #3 : Reference Angles

Simplify sec^{4}*Θ* – tan^{4}*Θ*.

**Possible Answers:**

tan^{2}*Θ* – sin^{2}*Θ*

sin*Θ* + cos*Θ*

sec*Θ* + sin*Θ*

cos*Θ* – sin*Θ*

sec^{2}*Θ* + tan^{2}*Θ*

**Correct answer:**

sec^{2}*Θ* + tan^{2}*Θ*

Factor using the difference of two squares: *a*^{2} – *b*^{2} = (*a* + *b*)(*a* – *b*)

The identity 1 + tan^{2}*Θ* = sec^{2}*Θ* which can be rewritten as 1 = sec^{2}*Θ* – tan^{2}*Θ*

So sec^{4}*Θ* – tan^{4}*Θ* = (sec^{2}*Θ* + tan^{2}*Θ*)(sec^{2}*Θ* – tan^{2}*Θ*) = (sec^{2}*Θ* + tan^{2}*Θ*)(1) = sec^{2}*Θ* + tan^{2}*Θ*

### Example Question #4 : Reference Angles

Evaluate the expression below.

**Possible Answers:**

**Correct answer:**

At , sine and cosine have the same value.

Cotangent is given by .

Now we can evaluate the expression.

### Example Question #2 : How To Find A Reference Angle

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

**Possible Answers:**

90

109

369

351

**Correct answer:**

90

3600 – 3510 = 90

### Example Question #6 : Reference Angles

Simplify the following expression:

**Possible Answers:**

cscΘ

tanΘ

None of the answers are correct

sin^{2}Θ

cos^{2}Θ

**Correct answer:**

sin^{2}Θ

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

cotΘ = cosΘ secΘ = 1

sinΘ cosΘ

### Example Question #7 : Reference Angles

What is the reference angle for ?

**Possible Answers:**

**Correct answer:**

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 #8 : Reference Angles

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

**Possible Answers:**

**Correct answer:**

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

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

### Example Question #3 : How To Find A Reference Angle

What is the reference angle for ?

**Possible Answers:**

**Correct answer:**

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 .