Q1. A lamp post casts a shadow 25 m long. The angle of elevation of the sun is 51°. What is the height of the lamp post?

Background

Topic: Right Triangle Trigonometry (Applications of Tangent)

This problem involves using trigonometric ratios to find the height of an object given the length of its shadow and the angle of elevation of the sun.

Key Terms and Formulas

• Angle of elevation: The angle formed by the line of sight above the horizontal.

• Tangent ratio:

Step-by-Step Guidance

1. Draw a right triangle where the lamp post is the vertical side (opposite), the shadow is the horizontal side (adjacent), and the angle of elevation is at the end of the shadow.

2. Label the height of the lamp post as , the length of the shadow as $25.

3. Set up the tangent ratio:

4. Rearrange the equation to solve for :

Try solving on your own before revealing the answer!

Q2. A hot air balloon (H) is hovering over a point on the ground (P) 95 ft. from where you are standing (G). The hot air balloon's angle of elevation with the ground is 12°. What is the altitude of the hot air balloon?

Background

Topic: Right Triangle Trigonometry (Applications of Tangent)

This question asks you to find the vertical height (altitude) of the balloon using the horizontal distance and the angle of elevation.

Key Terms and Formulas

• Angle of elevation: The angle above the horizontal from your eye to the object.

• Tangent ratio:

Step-by-Step Guidance

1. Draw a right triangle with the altitude of the balloon as the vertical side (opposite), the horizontal distance from you to the point below the balloon as the adjacent side (95 ft), and the angle of elevation as .

2. Let be the altitude of the balloon. Set up the tangent ratio:

3. Rearrange to solve for :

Try solving on your own before revealing the answer!

Q3. You are flying a kite and have let out 90 ft. of string. The kite's angle of elevation with the ground is 35°. If the string is stretched straight, how high is the kite above the ground?

Background

Topic: Right Triangle Trigonometry (Applications of Sine)

This problem involves finding the vertical height (opposite side) of a right triangle when the hypotenuse (string length) and the angle of elevation are known.

Key Terms and Formulas

• Sine ratio:

Step-by-Step Guidance

1. Draw a right triangle with the string as the hypotenuse (90 ft), the height as the opposite side, and the angle of elevation as .

2. Let be the height of the kite. Set up the sine ratio:

3. Rearrange to solve for :

Try solving on your own before revealing the answer!

Q4. A 25-foot ladder is leaning against a tree. The base of the ladder is 6 feet from the tree. Find the angle the ladder makes with the ground.

Background

Topic: Right Triangle Trigonometry (Applications of Cosine)

This question asks you to find an angle in a right triangle when the adjacent side and hypotenuse are known.

Key Terms and Formulas

• Cosine ratio:

• Inverse cosine:

Step-by-Step Guidance

1. Draw a right triangle with the ladder as the hypotenuse (25 ft), the distance from the base to the tree as the adjacent side (6 ft), and the angle at the ground as .

2. Set up the cosine ratio:

3. Take the inverse cosine to solve for :

Try solving on your own before revealing the answer!

Q5. A wire reaches from the top of a 130 ft. cell tower to the ground. The wire makes a 73° angle with the ground. Find the length of the wire.

Background

Topic: Right Triangle Trigonometry (Applications of Sine)

This problem involves finding the hypotenuse of a right triangle when the opposite side and the angle are known.

Key Terms and Formulas

• Sine ratio:

• Rearranged:

Step-by-Step Guidance

1. Draw a right triangle with the wire as the hypotenuse, the height of the tower as the opposite side (130 ft), and the angle with the ground as .

2. Set up the sine ratio: , where is the length of the wire.

3. Rearrange to solve for :

Try solving on your own before revealing the answer!

Q6. An airplane climbs at an angle of 16° with the ground. Find the ground distance it travels as it moves 3500 m through the air.

Background

Topic: Right Triangle Trigonometry (Applications of Cosine)

This question asks you to find the horizontal (ground) distance traveled, given the hypotenuse (air distance) and the angle of climb.

Key Terms and Formulas

• Cosine ratio:

• Rearranged:

Step-by-Step Guidance

1. Draw a right triangle with the air distance as the hypotenuse (3500 m), the ground distance as the adjacent side, and the angle of climb as .

2. Set up the cosine ratio: , where is the ground distance.

3. Rearrange to solve for :

Try solving on your own before revealing the answer!

Q7. A lighthouse operator at point P 45 ft. above sea level sights a sailboat at point S. The angle of depression of the sighting is 12°. How far is the boat from the base of the lighthouse?

Background

Topic: Right Triangle Trigonometry (Applications of Tangent and Angle of Depression)

This problem involves using the angle of depression (which equals the angle of elevation from the boat to the lighthouse) to find the horizontal distance from the lighthouse to the boat.

Key Terms and Formulas

• Angle of depression: The angle below the horizontal from the observer's eye to the object.

• Tangent ratio:

Step-by-Step Guidance

1. Draw a right triangle with the height of the lighthouse as the opposite side (45 ft), the horizontal distance to the boat as the adjacent side, and the angle of depression as .

2. Let be the distance from the base of the lighthouse to the boat. Set up the tangent ratio:

3. Rearrange to solve for :

Try solving on your own before revealing the answer!