Q: A pilot is flying from point $$A to $$point $$B$$ and then to point $$C. $$The angle at $$A is$$ $$40^\circ$$, at $$B is$$ $$65^\circ$$, and the distance from $$A to$$ $$B is$$ $$65$$ miles. What is the distance from $$A to$$ $$C$$?

$$\dfrac{65 \times \sin 40^\circ}{\sin 75^\circ}$$

Q: A triangle has $$a = 65$$, $$A = 20^\circ$$, and $$C = 114^\circ. $$What is the value of $$c$$?

$$c = \dfrac{65 \times \sin 114^\circ}{\sin 20^\circ}$$

Question 1

What is the sum of the interior angles of any triangle?

Accepted Answer

$$180^\circ$$

Question 2

A surveyor needs to find the distance across a river. She measures two angles from one side: $$A = 40^\circ$$, $$B = 65^\circ$$, and the side between them is $$c = 120$$ meters. Which method should she use to solve for the unknown sides?

Accepted Answer

Law of Sines

Question 3

Given triangle $$ABC$$ with $$a = 75$$, $$b = 100$$, and $$A = 50^\circ$$, which equation correctly sets up the Law of Sines to solve for angle $$B$$?

Accepted Answer

$$\dfrac{\sin 50^\circ}{75} = \dfrac{\sin B}{100}$$

Question 4

A right triangle has one leg of length $$15.8 ft $$and an angle of $$34^\circ. $$What is the length of the hypotenuse?

Accepted Answer

$$12.44 ft$$

Question 5

Which of the following is the correct Law of Sines formula for any triangle $$ABC$$?

Accepted Answer

$$\dfrac{\sin A}{a} = \dfrac{\sin B}{b} = \dfrac{\sin C}{c}$$

Question 6

In triangle solving, which case is known as the 'ambiguous case' where there may be one, two, or no solutions?

Accepted Answer

Side-Side-Angle (SSA)

Question 7

A construction engineer needs to find the height of a building. She stands $$50 ft $$from the base and measures the angle of elevation to the top as $$29^\circ. $$What is the height of the building?

Accepted Answer

$$50 	imes 	an 29^\circ$$

Question 8

Which of the following sets of information is sufficient to solve a triangle using the Law of Sines?

Accepted Answer

Two angles and one side (AAS or ASA)

Question 9

If you are given $$a = 65$$, $$A = 20^\circ$$, and $$C = 114^\circ in $$triangle $$ABC$$, which formula would you use to find side $$c$$?

Accepted Answer

$$\dfrac{a}{\sin A} = \dfrac{c}{\sin C}$$

Question 10

A pilot is flying from point $$A to $$point $$B$$ and then to point $$C. $$The angle at $$A is$$ $$40^\circ$$, at $$B is$$ $$65^\circ$$, and the distance from $$A to$$ $$B is$$ $$65$$ miles. What is the distance from $$A to$$ $$C$$?

Accepted Answer

$$\dfrac{65 	imes \sin 40^\circ}{\sin 75^\circ}$$

Question 11

Which scenario best describes when you should use right triangle trigonometry instead of the Law of Sines?

Accepted Answer

When the triangle contains a $$90^\circ$$ angle

Question 12

A triangle has sides $$a = 8$$, $$b = 12$$, and angle $$A = 30^\circ. $$What is the first step to determine if the triangle can be solved using the Law of Sines?

Accepted Answer

Check if you have an angle and its opposite side

Question 13

In triangle $$ABC$$, if $$A = 114^\circ$$, $$B = 14^\circ$$, and $$C = 52^\circ$$, what is the sum of the angles?

Accepted Answer

$$180^\circ$$

Question 14

A triangle has $$a = 65$$, $$A = 20^\circ$$, and $$C = 114^\circ. $$What is the value of $$c$$?

Accepted Answer

$$c = \dfrac{65 	imes \sin 114^\circ}{\sin 20^\circ}$$

Question 15

Which of the following is NOT a valid case for using the Law of Sines?

Accepted Answer

Three sides known (SSS)

Question 16

A triangle has $$a = 75$$, $$b = 100$$, and $$A = 50^\circ. $$After setting up the Law of Sines, what is the next step to find angle $$B$$?

Accepted Answer

Take the inverse sine of the result

Question 17

A triangle has $$A = 29^\circ$$, $$B = 114^\circ$$, and $$C = 37^\circ. $$What is the measure of angle $$C$$?

Accepted Answer

$$37^\circ$$

Question 18

A triangle has $$a = 65$$, $$A = 20^\circ$$, and $$C = 114^\circ. $$What is the correct setup to solve for $$c$$ using the Law of Sines?

Accepted Answer

$$\dfrac{65}{\sin 20^\circ} = \dfrac{c}{\sin 114^\circ}$$

Triangle Geometry and the Law of Sines

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