Reference Angles

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ACT Math › Reference Angles

Questions 1 - 10

1

Using trig identities, simplify sinθ + cotθcosθ

tanθ

secθ

sin2θ

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.

2

Using trig identities, simplify sinθ + cotθcosθ

tanθ

secθ

sin2θ

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.

3

What is the reference angle for $125^{\circ}$ ?

$55^{\circ}$

$75^{\circ}$

$35^{\circ}$

$125^{\circ}$

$235^{\circ}$

Explanation

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

In this case, our angle $125^{\circ}$ lies in Quadrant II, so we can find our reference angle using the formula

$\angle A_r = 180^{\circ}- \angle A$ .

$\angle A_r = 180^{\circ}- \angle A$ $180^{\circ} - 125^{\circ} = 55^{\circ}$

Thus, the reference angle for $125^{\circ}$ is $55^{\circ}$ .

4

What is the reference angle for $125^{\circ}$ ?

$55^{\circ}$

$75^{\circ}$

$35^{\circ}$

$125^{\circ}$

$235^{\circ}$

Explanation

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

In this case, our angle $125^{\circ}$ lies in Quadrant II, so we can find our reference angle using the formula

$\angle A_r = 180^{\circ}- \angle A$ .

$\angle A_r = 180^{\circ}- \angle A$ $180^{\circ} - 125^{\circ} = 55^{\circ}$

Thus, the reference angle for $125^{\circ}$ is $55^{\circ}$ .

5

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

$45^{\circ}$

$135^{\circ}$

$90^{\circ}$

$180^{\circ}$

$315^{\circ}$

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 $45^{\circ}$ .

6

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

$45^{\circ}$

$135^{\circ}$

$90^{\circ}$

$180^{\circ}$

$315^{\circ}$

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 $45^{\circ}$ .

7

What is the reference angle for $257^{\circ}$ ?

$77^{\circ}$

$13^{\circ}$

$103^{\circ}$

$257^{\circ}$

$93^{\circ}$

Explanation

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

In this case, our angle $257^{\circ}$ lies in Quadrant III, so the angle is found by the formula $\angle A_r = \angle A - 180^{\circ}$ .

$\angle A_r = \angle A - 180^{\circ}$ $257^{\circ} - 180^{\circ} = 77^{\circ}$

Thus, the reference angle for $257^{\circ}$ is $77^{\circ}$ .

8

What is the reference angle for $257^{\circ}$ ?

$77^{\circ}$

$13^{\circ}$

$103^{\circ}$

$257^{\circ}$

$93^{\circ}$

Explanation

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

In this case, our angle $257^{\circ}$ lies in Quadrant III, so the angle is found by the formula $\angle A_r = \angle A - 180^{\circ}$ .

$\angle A_r = \angle A - 180^{\circ}$ $257^{\circ} - 180^{\circ} = 77^{\circ}$

Thus, the reference angle for $257^{\circ}$ is $77^{\circ}$ .

9

Evaluate the expression below.

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

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

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

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

$\sqrt{2}$

Explanation

At , 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.

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

10

Evaluate the expression below.

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

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

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

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

$\sqrt{2}$

Explanation

At , 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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