Q1. A twenty-five foot ladder just reaches the top of a house and forms an angle of 41.5° with the wall of the house. How tall is the house? Round to the nearest 0.1 foot.

Background

Topic: Right Triangle Trigonometry

This question tests your ability to use trigonometric ratios (specifically sine, cosine, or tangent) to find the height of a right triangle given an angle and the length of the hypotenuse.

Key Terms and Formulas

• Hypotenuse: The side opposite the right angle (the ladder in this case).

• Opposite side: The side opposite the given angle (the height of the house).

• Sine function:

Step-by-Step Guidance

1. Draw a right triangle representing the situation, labeling the ladder as the hypotenuse (25 ft), the angle with the wall as , and the height of the house as the side opposite the angle.

2. Set up the sine equation: , where is the height of the house.

3. Rearrange the equation to solve for : .

Try solving on your own before revealing the answer!

Q2. To find the length of the span of a proposed ski lift from A to B, a surveyor measures the angle DAB to be 25°, then walks off a distance of L = 1950 feet to C and measures the angle ACB to be 15°. What is the distance from A to B?

Background

Topic: Law of Sines (Oblique Triangles)

This question tests your ability to use the Law of Sines to find an unknown side in a triangle when given two angles and one side.

Key Terms and Formulas

• Law of Sines:

• Given: , ft,

Step-by-Step Guidance

1. Draw triangle ABC, labeling the given angles and side. Let (the distance you want to find).

2. Find the third angle in the triangle: .

3. Set up the Law of Sines: .

4. Rearrange to solve for : .

Try solving on your own before revealing the answer!

Q3. The polar coordinates of a point are given: . Find the rectangular coordinates of the point.

Background

Topic: Polar to Rectangular Coordinate Conversion

This question tests your ability to convert from polar coordinates to rectangular coordinates .

Key Terms and Formulas

Step-by-Step Guidance

1. Identify and .

2. Calculate .

3. Calculate .

4. Recall that and .

Try solving on your own before revealing the answer!

Q4. The letters x and y represent rectangular coordinates. Write the equation using polar coordinates .

Background

Topic: Rectangular to Polar Equation Conversion

This question tests your ability to rewrite equations from rectangular form to polar form using the relationships between and .

Key Terms and Formulas

Step-by-Step Guidance

1. Substitute and into the equation: .

2. Expand and simplify: .

3. Factor out: .

4. Divide both sides by to solve for .

Try solving on your own before revealing the answer!

Q5. The letters r and θ represent polar coordinates. Write the equation using rectangular coordinates .

Background

Topic: Polar to Rectangular Equation Conversion

This question tests your ability to convert a polar equation to rectangular form using the relationships between and .

Key Terms and Formulas

Step-by-Step Guidance

1. Multiply both sides by : .

2. Expand: .

3. Recall that and .

4. Substitute for and for to write the equation in terms of $x$ and .

Try solving on your own before revealing the answer!

Q6. The rectangular coordinates of a point are given: (5, -5). Find polar coordinates for this point. Let and .

Background

Topic: Rectangular to Polar Coordinate Conversion

This question tests your ability to convert from rectangular coordinates to polar coordinates .

Key Terms and Formulas

• (adjust for quadrant)

Step-by-Step Guidance

1. Calculate .

2. Calculate .

3. Since the point is in the fourth quadrant, adjust to the correct value in .

Try solving on your own before revealing the answer!