Q: A surveyor measures two sides of a triangular plot of land as 120 m and 150 m, with the angle between them as $$60^\circ. $$What is the length of the third side?

$$\sqrt{120^2$ + 150^2$ - 2 \times 120 \times 150 \times \cos 60^\circ}$$

Q: A triangle has sides $$a = 8$$, $$b = 15$$, and $$c = 17. $$What is the measure of the largest angle?

$$\cos C = \frac{8^2$ + 15^2$ - 17^2$}{2 \times 8 \times 15}$$

Q: A civil engineer needs to determine the angle between two roads that meet at a junction. The roads are 200 m and 250 m long, and the distance between their endpoints is 300 m. Which formula should be used to find the angle between the roads?

$$\cos A = \frac{200^2$ + 250^2$ - 300^2$}{2 \times 200 \times 250}$$

Q: A triangle has sides $$a = 6$$, $$b = 8$$, and $$c = 10. $$What is the value of $$\cos A$$?

$$\cos A = \frac{8^2$ + 10^2$ - 6^2$}{2 \times 8 \times 10}$$

Question 1

Which of the following is the correct Law of Cosines formula for finding side $$a in a $$triangle with sides $$a$$, $$b$$, $$c$$ and angle $$A$$ opposite side $$a$$?

Accepted Answer

$$a^2$$ = $$b^2$$ + $$c^2 - 2bc$$ \cos A$$

Question 2

When solving an oblique triangle, which case requires the use of the Law of Cosines rather than the Law of Sines?

Accepted Answer

When you know two sides and the included angle (SAS case)

Question 3

Given triangle $$ABC$$ with $$a = 7$$, $$b = 5$$, $$c = 9$$, which formula would you use to find angle $$A$$?

Accepted Answer

$$\cos A = \frac{b^2$ + c^2$ - a^2$}{2bc}$$

Question 4

A surveyor measures two sides of a triangular plot of land as 120 m and 150 m, with the angle between them as $$60^\circ. $$What is the length of the third side?

Accepted Answer

$$\sqrt{120^2$ + 150^2$ - 2 	imes 120 	imes 150 	imes \cos 60^\circ}$$

Question 5

Which of the following is the alternate form of the Law of Cosines for finding angle $$C$$?

Accepted Answer

$$\cos C = \frac{a^2$ + b^2$ - c^2$}{2ab}$$

Question 6

A triangle has sides $$a = 8$$, $$b = 15$$, and $$c = 17. $$What is the measure of the largest angle?

Accepted Answer

$$\cos C = \frac{8^2$ + 15^2$ - 17^2$}{2 	imes 8 	imes 15}$$

Question 7

In solving a triangle with sides $$a = 10$$, $$b = 7$$, $$c = 5$$, which angle is the largest?

Accepted Answer

Angle $$A$$ opposite side $$a$$

Question 8

A civil engineer needs to determine the angle between two roads that meet at a junction. The roads are 200 m and 250 m long, and the distance between their endpoints is 300 m. Which formula should be used to find the angle between the roads?

Accepted Answer

$$\cos A = \frac{200^2$ + 250^2$ - 300^2$}{2 	imes 200 	imes 250}$$

Question 9

If you are given all three sides of a triangle, which sequence of steps is correct for finding all the angles?

Accepted Answer

Use the Law of Cosines to find the largest angle, then use the Law of Sines or Cosines to find another angle, and finally use the triangle sum to find the last angle.

Question 10

A triangle has sides $$a = 6$$, $$b = 8$$, and $$c = 10. $$What is the value of $$\cos A$$?

Accepted Answer

$$\cos A = \frac{8^2$ + 10^2$ - 6^2$}{2 	imes 8 	imes 10}$$

Question 11

Which of the following scenarios best illustrates the use of the Law of Cosines in a real-world application?

Accepted Answer

Calculating the distance between two ships at sea given the distance from each ship to a lighthouse and the angle between their paths.

Question 12

A triangle has sides $$a = 9$$, $$b = 12$$, and $$c = 15. $$What is the measure of angle $$B$$?

Accepted Answer

$$\cos B = \frac{9^2$ + 15^2$ - 12^2$}{2 	imes 9 	imes 15}$$

Question 13

Which of the following is NOT a valid use of the Law of Cosines?

Accepted Answer

Finding the area of a triangle given only its three angles.

Question 14

A triangle has sides $$a = 5$$, $$b = 7$$, and $$c = 10. $$What is the measure of angle $$C$$?

Accepted Answer

$$\cos C = \frac{5^2$ + 7^2$ - 10^2$}{2 	imes 5 	imes 7}$$

Question 15

In a triangle, if $$a = 13$$, $$b = 14$$, and $$C = 60^\circ$$, what is the length of side $$c$$?

Accepted Answer

$$\sqrt{13^2$ + 14^2$ - 2 	imes 13 	imes 14 	imes \cos 60^\circ}$$

Question 16

A triangle has sides $$a = 11$$, $$b = 13$$, and $$c = 17. $$Which angle is the smallest?

Accepted Answer

Angle $$A$$ opposite side $$a$$

Question 17

Which of the following steps is NOT part of solving a triangle using the SSS case with the Law of Cosines?

Accepted Answer

Using the Law of Sines to find the first angle.

Question 18

A triangle has sides $$a = 4$$, $$b = 6$$, and $$c = 8. $$What is the value of $$\cos B$$?

Accepted Answer

$$\cos B = \frac{4^2$ + 8^2$ - 6^2$}{2 	imes 4 	imes 8}$$

The Law of Cosines: Solving Oblique Triangles

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