Q: Which of the following is the correct formula for the area of a triangle given two sides and the included angle?

$$\text{Area} = \frac{1}{2}ab\sin C$$

Q: A triangle has sides $$a = 5$$, $$b = 7$$, and included angle $$C = 60^\circ. $$What is the area of the triangle?

$$\frac{1}{2} \times 5 \times 7 \times \sin 60^\circ$$

Question 1

Which of the following statements about oblique triangles is TRUE?

Accepted Answer

Oblique triangles do not contain a right angle.

Question 2

Given triangle $$ABC$$ with sides $$a$$, $$b$$, and $$c$$ opposite angles $$A$$, $$B$$, and $$C$$ respectively, which formula correctly represents the Law of Sines?

Accepted Answer

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

Question 3

A triangle has sides $$a = 7$$, $$b = 10$$, and angle $$A = 30^\circ. $$Using the Law of Sines, what is the value of $$\sin B$$?

Accepted Answer

$$\sin B = \frac{7 \cdot \sin 30^\circ}{10}$$

Question 4

Which of the following is NOT a valid step when solving for an unknown angle in an oblique triangle using the Law of Sines?

Accepted Answer

Apply the Pythagorean Theorem to find the missing angle.

Question 5

If a triangle has angles $$A = 40^\circ$$ and $$B = 65^\circ$$, what is the measure of angle $$C$$?

Accepted Answer

$$C = 75^\circ$$

Question 6

Which of the following is the correct formula for the area of a triangle given two sides and the included angle?

Accepted Answer

$$	ext{Area} = \frac{1}{2}ab\sin C$$

Question 7

A surveyor needs to find the distance across a river. She measures a $$60^\circ$$ angle from one bank and a $$45^\circ$$ angle from the other, with the distance between the two points on the banks being $$100$$ meters. What Law should she use to find the distance across the river?

Accepted Answer

Law of Sines

Question 8

Which of the following triangles is NOT an oblique triangle?

Accepted Answer

A triangle with angles $$90^\circ$$, $$45^\circ$$, and $$45^\circ$$

Question 9

Given triangle $$ABC$$ with $$a = 8$$, $$A = 50^\circ$$, and $$B = 70^\circ$$, what is the value of $$b$$ using the Law of Sines?

Accepted Answer

$$b = \frac{8 \cdot \sin 70^\circ}{\sin 50^\circ}$$

Question 10

A triangle has sides $$a = 5$$, $$b = 7$$, and included angle $$C = 60^\circ. $$What is the area of the triangle?

Accepted Answer

$$\frac{1}{2} 	imes 5 	imes 7 	imes \sin 60^\circ$$

Question 11

Which of the following is a necessary condition for using the Law of Sines to solve a triangle?

Accepted Answer

You must know at least one angle-side opposite pair.

Question 12

If $$\sin B = 1.2 is $$obtained while solving for angle $$B in a $$triangle, what does this indicate?

Accepted Answer

There is no such triangle with the given measurements.

Question 13

A triangle has sides $$a = 9$$, $$b = 12$$, and angle $$A = 40^\circ. $$Which equation would you use to find angle $$B$$?

Accepted Answer

$$\frac{9}{\sin 40^\circ} = \frac{12}{\sin B}$$

Question 14

Which of the following best describes an acute triangle?

Accepted Answer

A triangle with all angles less than $$90^\circ$$

Question 15

A triangle has sides $$a = 6$$, $$b = 8$$, and included angle $$C = 45^\circ. $$What is the area of the triangle to the nearest tenth?

Accepted Answer

$$\frac{1}{2} 	imes 6 	imes 8 	imes \sin 45^\circ \approx 17.0$$

Question 16

Which of the following is a correct application of the Law of Sines in a real-world context?

Accepted Answer

Finding the height of a mountain by measuring angles from two different points and the distance between them.

The Law of Sines and Area of Oblique Triangles

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