Mathematics • Year 9 • Unit 4 • Lesson 1
Trigonometry in the Real World
Use SOH CAH TOA in everyday contexts — slides at the park, a skateboard ramp, a ladder against a wall, a flagpole shadow and a beach umbrella. Then explain your method in your own words.
1. Word problems
Each problem uses sin, cos or tan from Lesson 1. Draw a quick triangle, label opposite / adjacent / hypotenuse, then choose the matching ratio. Show your working — a single number with no working only earns half marks. Calculator in degree mode. Round to 2 dp unless told otherwise.
1.1 — Playground slide. A straight playground slide is 3.5 m long (the slope, i.e. the hypotenuse) and meets the ground at an angle of 38°.
(a) How high is the top of the slide above the ground? 3 marks
1.2 — Skateboard ramp. A skateboard ramp is built with a horizontal base of 1.2 m and a vertical height of 0.5 m. Skaters want to know the angle the ramp's surface makes with the ground.
(a) Which trig ratio links the opposite (0.5 m) and the adjacent (1.2 m)?
(b) Find the ramp angle to the nearest degree. 3 marks
1.3 — Ladder against the wall. A 6 m ladder leans against a vertical wall, making a 70° angle with the ground (this is a safe ladder angle in NSW guidelines).
(a) How high up the wall does the ladder reach?
(b) How far is the foot of the ladder from the wall? 3 marks
1.4 — Flagpole shadow. At 9 am the sun shines on a flagpole at the local high school. The flagpole's shadow on the flat ground is 8 m long, and the angle from the tip of the shadow up to the top of the flagpole is 55°.
(a) Sketch the triangle, labelling the flagpole (height = ?), the shadow (8 m) and the 55° angle.
(b) Find the flagpole height. 3 marks
1.5 — Beach umbrella. A beach umbrella pole is stuck vertically in the sand. A guy-rope of length 2.4 m runs from the top of the pole to a peg in the sand, making a 40° angle with the pole (not with the ground).
(a) How tall is the pole above the sand? (The pole is ADJACENT to the 40° angle, and the rope is the hypotenuse.)
(b) How far is the peg from the base of the pole? 3 marks
2. Explain your thinking
This question is about communication, not just answers. Use full sentences. 4 marks
2.1 A classmate is working on Question 1.4 (the flagpole shadow). They say: "I always use sin because sin is the easy one." In your own words, explain (i) why "always use sin" is wrong, (ii) how you choose between sin, cos and tan, and (iii) which ratio was actually needed for the flagpole question, and why. Refer to the words opposite, adjacent and hypotenuse somewhere in your explanation.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Playground slide
Height = opposite the 38° angle, slide = hypotenuse = 3.5 m.
sin 38° = height / 3.5 → height = 3.5 × sin 38° = 3.5 × 0.6157 ≈ 2.16 m.
Sanity check: less than 3.5 m, makes sense as the slope is longer than its height.
1.2 — Skateboard ramp
(a) Opposite + adjacent → tan (TOA).
(b) tan θ = 0.5 / 1.2 = 0.4167 → θ = tan⁻¹(0.4167) ≈ 23° (to the nearest degree).
That's a fairly gentle ramp — competition ramps are often 30°+.
1.3 — Ladder against the wall
Hypotenuse = 6 m (the ladder), angle with ground = 70°.
(a) Height = 6 × sin 70° = 6 × 0.9397 ≈ 5.64 m.
(b) Distance from wall = 6 × cos 70° = 6 × 0.3420 ≈ 2.05 m.
Check with Pythagoras: 5.64² + 2.05² ≈ 31.8 + 4.2 = 36 = 6². ✓
1.4 — Flagpole shadow
Shadow = adjacent = 8 m, flagpole = opposite (unknown), angle at the tip of shadow = 55°.
tan 55° = h / 8 → h = 8 × tan 55° = 8 × 1.4281 ≈ 11.43 m.
Realistic — most school flagpoles are around 10-12 m tall.
1.5 — Beach umbrella
Tricky: the 40° angle is at the TOP, between the pole and the rope. So the pole is ADJACENT and the horizontal distance to the peg is OPPOSITE. Rope = hypotenuse = 2.4 m.
(a) Pole height = adjacent = 2.4 × cos 40° = 2.4 × 0.7660 ≈ 1.84 m.
(b) Peg distance = opposite = 2.4 × sin 40° = 2.4 × 0.6428 ≈ 1.54 m.
The names "opposite" and "adjacent" depend on WHICH angle you're looking at — not on which side is horizontal.
2.1 — Explain your thinking (sample response)
"Always use sin" is wrong because each trig ratio links a specific pair of sides. Sin links opposite and hypotenuse, cos links adjacent and hypotenuse, and tan links opposite and adjacent. To choose the right one, I look at which two sides are involved in the question (one I know, one I want), then I pick the ratio that connects exactly those two. In the flagpole question (1.4), I knew the shadow (adjacent to the 55° angle) and I wanted the flagpole height (opposite the angle). Opposite + adjacent → tan, so tan 55° = height / 8 was the correct choice. Using sin would have given the wrong equation because the flagpole shadow is not the hypotenuse.
Marking: 1 mark for "each ratio links a specific pair"; 1 for naming opposite/adjacent/hypotenuse; 1 for identifying tan in 1.4; 1 for a clear, full-sentence explanation.