Q: A surveyor stands 35 feet from the center of a circular hole with a radius of 10 feet and wants to direct a laser beam through the center of the hole. What angle of elevation $$\theta$$ should the laser be aimed at to hit the center of the hole? (Assume the laser is at the same height as the center of the hole.)

$$\theta = \arctan\left(\frac{10}{35}\right)$$

Question 1

A surveyor stands 35 feet from the center of a circular hole with a radius of 10 feet and wants to direct a laser beam through the center of the hole. What angle of elevation $$	heta$$ should the laser be aimed at to hit the center of the hole? (Assume the laser is at the same height as the center of the hole.)

Accepted Answer

$$	heta = \arctan\left(\frac{10}{35}ight)$$

Question 2

Given a right triangle with legs $$a$$ and $$b$$, and hypotenuse $$c$$, which equation correctly relates the sides?

Accepted Answer

$$a^2$$ + $$b^2$$ = $$c^2$$

Question 3

A ship leaves port with a bearing of S80°E and travels 15 knots for 1 hour, then turns 90° toward the south and continues for 2 hours at the same speed. What is the best method to find the bearing from the port to the ship's final position?

Accepted Answer

Use the Law of Cosines to find the distance, then use the Law of Sines to find the bearing angle.

Question 4

Which of the following is the Law of Sines for any triangle with sides $$a$$, $$b$$, $$c$$ and opposite angles $$A$$, $$B$$, $$C$$?

Accepted Answer

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

Question 5

A triangle has sides $$a = 4$$, $$b = 5$$, and $$c = 3. $$Which law should be used to solve for the angles of the triangle?

Accepted Answer

Law of Cosines

Question 6

Given triangle $$ABC$$ with $$a = 3$$, $$b = 7$$, and $$A = 70°$$, how many possible triangles can be formed?

Accepted Answer

Two triangles (ambiguous case)

Question 7

Which formula gives the area $$K of a $$triangle with sides $$a$$, $$b$$, and included angle $$C$$?

Accepted Answer

$$K = \frac{1}{2}ab\sin C$$

Question 8

A forest ranger walks on a path inclined at $$5°$$ toward a $$100-$$foot-tall fire tower. The angle of elevation from the path to the top of the tower is $$40°. $$Which law should be used to find the distance from the ranger to the tower?

Accepted Answer

Law of Sines

Question 9

Which of the following is Heron's Formula for the area $$K of a $$triangle with sides $$a$$, $$b$$, and $$c$$?

Accepted Answer

$$K = \sqrt{s(s-a)(s-b)(s-c)}$$ where $$s = \frac{1}{2}(a+b+c)$$

Question 10

In navigation, the bearing from point $$O to $$point $$P is $$defined as:

Accepted Answer

The acute angle between the ray $$OP$$ and the north-south line through $$O$$

Question 11

Given triangle $$ABC$$ with $$B = 40°$$, $$b = 4$$, and $$c = 3$$, which law should be used to determine if a triangle can be formed and to solve for the unknowns?

Accepted Answer

Law of Sines (SSA ambiguous case)

Question 12

A triangle has sides $$a = 4$$, $$b = 3$$, $$c = 6. $$What is the area of the triangle using Heron's Formula? (Round to two decimals)

Accepted Answer

$$K = 5.33$$

Question 13

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

Accepted Answer

SSS (three sides known)

Question 14

The Law of Cosines can be used to solve for a side $$c in a $$triangle when:

Accepted Answer

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

Question 15

A triangle has $$a = 2$$, $$c = 1$$, and $$B = 10°. $$What is the area of the triangle?

Accepted Answer

$$K = \frac{1}{2} \cdot 2 \cdot 1 \cdot \sin 10°$$

Question 16

Which of the following is true about the sum of the angles in any triangle?

Accepted Answer

$$A + B + C = 180°$$

Question 17

If you are given two sides and the included angle (SAS) of a triangle, which law should you use to solve for the third side?

Accepted Answer

Law of Cosines

Question 18

A triangle has sides $$a = 3$$, $$b = 4$$, and included angle $$C = 30°. $$What is the area of the triangle?

Accepted Answer

$$K = \frac{1}{2} \cdot 3 \cdot 4 \cdot \sin 30°$$

Question 19

Which of the following is a correct interpretation of the ambiguous case (SSA) when solving triangles using the Law of Sines?

Accepted Answer

There may be one, two, or no triangle possible depending on the given values.

Applications and Laws of Triangles in Precalculus: Right, Oblique, and Area Calculations

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