Mathematics Standard • Year 11 • Module 2 • Lesson 4

Introduction to Trigonometry

Practise HSC Mathematics Standard 2-style writing on SOHCAHTOA — multi-mark short answers and one structured extended response.

Master · Past-Paper Style

1. Short-answer questions

1.1 In a right-angled triangle the reference angle is 55° and the adjacent side is 10 m. Find the opposite side correct to 2 d.p. 2 marks Band 3

1.2 Find the hypotenuse of a right-angled triangle in which the reference angle is 36° and the adjacent side is 12 cm. Give your answer correct to 2 d.p. 3 marks Band 3-4

1.3 A right-angled triangle has opposite side 11 cm and adjacent side 8 cm. 4 marks Band 4
(a) Find the reference angle θ correct to the nearest degree.
(b) Convert your answer in (a) to degrees and minutes (use the unrounded θ from your calculator).

Stuck on 1.3(b)? Find the decimal degree first, then take the decimal part of that, then × 60 for minutes.

2. Extended response

2.1 A surveyor needs to find the height of a flagpole that stands on top of a building. From a point on flat ground 60 m horizontally from the base of the building, she measures:

• Angle of elevation to the TOP of the BUILDING (where the flagpole begins): 28°.

• Angle of elevation to the TOP of the FLAGPOLE: 35°.

(a) Find the height of the building to 2 d.p.
(b) Find the height from the ground to the top of the flagpole to 2 d.p.
(c) Calculate the height of the flagpole itself (top minus base), and state the answer clearly in a one-sentence conclusion. 7 marks Band 5-6

Explicit marking criteria

Part (a) — 2 marks

• 1 mark — correctly identifies tan as the ratio (O = building height unknown, A = 60 m known).

• 1 mark — correct building height using h = 60 × tan 28°.

Part (b) — 2 marks

• 1 mark — applies tan with the larger 35° angle.

• 1 mark — correct top-of-flagpole height to 2 d.p.

Part (c) — 3 marks

• 1 mark — subtracts (b) − (a) (NOT adding).

• 1 mark — correct numerical flagpole height to 2 d.p.

• 1 mark — explicit conclusion sentence stating the flagpole height with units.

Your response:

Stuck on (c)? Flagpole height = (height of top of flagpole) − (height of top of building). Both heights share the same 60 m base, so the tan ratio works for both.

How did this worksheet feel?

Got it Partly Lost

What I'll revisit before next class:

Answers — sample responses + marking notes

1.1 — θ = 55°, A = 10, find O (2 marks)

Sample response.
O and A → use tan. tan 55° = O / 10 → O = 10 × tan 55° = 14.281...
O = 14.28 m (to 2 d.p.).

Marking notes. 1 mark — correct ratio choice and equation setup. 1 mark — correct rounded answer with unit.

1.2 — θ = 36°, A = 12, find H (3 marks)

Sample response.
A and H → use cos. cos 36° = 12 / H.
Rearrange: H × cos 36° = 12 → H = 12 / cos 36° = 14.832...
H = 14.83 cm (to 2 d.p.).

Marking notes. 1 mark — correct ratio. 1 mark — correct rearrangement showing division by cos 36° (NOT multiplication). 1 mark — correct rounded answer with unit. Common error: writing H = 12 × cos 36° = 9.71 cm — wrong (answer is smaller than the adjacent, which is impossible for the hypotenuse).

1.3 — O = 11, A = 8, find θ (4 marks)

Sample response.
(a) O and A → use tan. tan θ = 11 / 8 = 1.375. θ = tan⁻¹(1.375) = 53.97...° ≈ 54°.
(b) Unrounded: θ = 53.9726...°. Whole degrees: 53°. Decimal part: 0.9726 × 60 = 58.36 ≈ 58 min. θ = 53°58'.

Marking notes. (a) 1 mark — correct ratio and inverse. 1 mark — correct nearest degree. (b) 1 mark — uses the unrounded θ from the calculator (NOT 54.0 from part a). 1 mark — correct degrees-and-minutes form.

2.1 — Flagpole on a building (7 marks): sample Band-6 response with annotations

Sample Band-6 response.

(a) Building height.

O and A → use tan. [1 mark — ratio choice.]
tan 28° = h_building / 60 → h_building = 60 × tan 28° = 60 × 0.5317... = 31.904...
h_building = 31.90 m (to 2 d.p.). [1 mark — value.]

(b) Top-of-flagpole height.

tan 35° = h_top / 60. [1 mark — uses 35° with same adjacent 60.]
h_top = 60 × tan 35° = 60 × 0.7002... = 42.012...
h_top = 42.01 m (to 2 d.p.). [1 mark — value.]

(c) Flagpole height.

Flagpole = h_top − h_building. [1 mark — subtracts, does not add.]
Flagpole = 42.01 − 31.90 = 10.11 m (using rounded values; using unrounded: 42.012 − 31.904 = 10.108 ≈ 10.11 m). [1 mark — correct value.]

Conclusion: the flagpole standing on top of the building is approximately 10.11 m tall. [1 mark — explicit conclusion sentence with units.]

Total: 7/7.

Band descriptors for marker.

Band 3: Calculates only the building height (a), or chooses cos instead of tan. ≈ 2-3 marks.

Band 4: Both heights correct, but adds them (42.01 + 31.90 = 73.91 m) instead of subtracting — misinterprets the geometry. ≈ 4-5 marks.

Band 5: Both heights correct, correct subtraction, but conclusion is a bare number ("10.11") without units or naming what was found. ≈ 6 marks.

Band 6: Complete, ratios correct, subtraction correct, conclusion sentence with units and clear identification of the flagpole. 7/7.