Sketch, Label and Solve: Right-Angled Triangles

Name:

Date:

Right-angled triangle with trigonometry ratios

📐 Quick Reference & Triangle Labelling

Formula Box:
SOHCAHTOA: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent
Inverse functions: θ = sin⁻¹(opp/hyp), θ = cos⁻¹(adj/hyp), θ = tan⁻¹(opp/adj)
Angle of elevation: looking UP from horizontal | Angle of depression: looking DOWN from horizontal
Reminder: Check your calculator is in DEGREE mode!

1. In the right-angled triangle below, angle C = 90°, angle A = 35°. Label the sides as opposite (O), adjacent (A), or hypotenuse (H) relative to angle A.

        B
        |\
        | \
     12m|  \
        |   \
        |____\
        C  8m  A

2. For the same triangle, what trigonometric ratio would you use to find the length of side AB? Circle your answer.

sine

cosine

tangent

🔢 Finding Missing Sides and Angles

3. A ladder leans against a wall at an angle of 65° to the ground. The ladder is 4.2m long. How high up the wall does the ladder reach? Sketch the situation, label known information, show your working.

4. From a lookout point 45m above sea level, a ranger spots a boat. The angle of depression to the boat is 28°. How far is the boat from the base of the lookout (horizontal distance)? Sketch and solve.

5. A right-angled triangle has sides of 7.3cm and 11.8cm. Find the acute angle between these two sides. Round to the nearest degree.

6. A surveyor measures that a building casts a 15.6m shadow when the angle of elevation to the sun is 42°. Calculate the height of the building.

🏗️ Multi-Step Problem Solving

7. A wheelchair ramp must have a gradient no steeper than 8° for safety. If the ramp needs to reach a height of 1.2m, what is the minimum horizontal distance required? If the available space is only 7m horizontally, will this meet safety requirements? Show all working and explain your conclusion.

8. A drone flies 200m horizontally from its launch point, then climbs at an angle of elevation of 35° for 150m (along the slope). From this final position, what is the drone's total horizontal distance from the launch point and its height above the launch point? Sketch the path and calculate both distances.

9. Reflection: Describe one common error students make when solving trigonometry problems and explain how to avoid it.