Applications in 2D (Ex 6C)

Year 10 Mathematics Core · Cambridge Ch 6, §6C · Mr Wong

Name

Class

Date

Learning intentions.

Identify and measure angles of elevation (up) and depression (down) from the horizontal
Read a word problem and draw a clearly labelled right-angled triangle
Apply SOHCAHTOA and answer the question in a full sentence

Key results — elevation and depression

$\sin\theta=\dfrac{O}{H}\qquad \cos\theta=\dfrac{A}{H}\qquad \tan\theta=\dfrac{O}{A}$

Elevation is measured up from horizontal; depression is measured down. On the same diagram they are equal (alternate angles).

Warm-up

1. State the bearing direction and the trig ratio (sin/cos/tan) you would use:

a Plane 800 m up, angle of elevation 28°, find horizontal distance. Ratio =

b Cliff 50 m high, angle of depression to a boat 14°, find horizontal distance. Ratio =

c Cable 60 m, angle of elevation 35°, find vertical height the cable reaches. Ratio =

d Two poles 40 m apart, heights 12 m and 25 m, find angle of depression. Ratio =

2. The angle of elevation from point $A$ on the ground to the top of a tower at $B$ is $\mathbf{35°}$. Without solving, state the angle of depression from $B$ down to $A$. Answer: (Briefly explain why.)

Part A — Angle of elevation

Draw a labelled diagram, set up the trig equation, solve to 2 d.p., and answer in a sentence.

3. A helicopter hovers at $250$ m. The angle of elevation from the helipad to the helicopter is $35°$. Find the horizontal distance from the helipad to the helicopter (to the nearest centimetre).

4. A kite string is $80$ m long and makes an angle of elevation of $62°$ with the ground. How high is the kite above the ground?

5. A guy-wire is anchored from the ground to the top of a communications mast. The wire is $45$ m long and is at an elevation of $60°$. How tall is the mast?

Trig 6C — continued

Part B (depression) · Part C (mixed application)

Name

Part B — Angle of depression

6. Two vertical buildings $57$ m apart are $158$ m and $237$ m high. Find the angle of depression from the top of the taller to the top of the shorter (2 d.p.).

7. A lighthouse beacon is $112$ m above sea level. A boat is at a horizontal distance of $1.5$ km. Find the angle of depression from the beacon to the boat (2 d.p.). Watch units.

8. From the top of a $25$ m viewing tower, the angle of depression to a crocodile is $62°$. Find the direct (slant) distance from the top of the tower to the crocodile (2 d.p.).

Part C — Mixed application

9. A drone is $60$ m above a flat oval. The angle of depression from the drone to a player on the ground is $18°$. How far away (horizontally) is the player from a point directly below the drone? (2 d.p.)

10. A ramp $5.4$ m long rises to a loading dock at an elevation of $11°$ above the horizontal. (a) How high is the dock above the ground? (b) How far horizontally does the ramp cover? (2 d.p.)

11. Challenge. A surveyor stands $30$ m from the base of a tower. The angle of elevation to the top of the tower is $54°$; the angle of elevation to a flag pole on top of the tower is $58°$. How tall is the flag pole alone? (2 d.p.)