Label the triangle, choose the correct ratio, then solve — three consistent steps every time. When the unknown is in the denominator, multiply across. When it is in the numerator, the answer falls straight out.
Use the PDF for classwork, homework or revision. It includes key ideas, activities, questions, an extend task and success-criteria proof.
A ramp rises 1.2 m vertically over a horizontal distance. You need to find the length of the ramp surface. What information would you need? What shape does this form? Which trigonometric ratio would connect the angle of the ramp to the length you need?
Type your initial response below — you will revisit this at the end of the lesson.
Write your initial response in your book. You will revisit it at the end of the lesson.
Come back to this at the end of the lesson.
Wrong: A percentage over 100% is impossible.
Right: Percentages over 100% are valid and common in contexts like percentage increase, profit margins, and scale factors.
Step 1 — Label and Choose
Always check your units before substituting into formulas. Converting to consistent units is a common source of errors in assessment tasks.
Before writing any formula, label the three sides of your right-angled triangle relative to the angle given. This one step prevents almost all SOHCAHTOA errors.
In a right-angled triangle, the hypotenuse is 18 m and one angle is 38°. Find the side opposite the 38° angle, correct to 2 decimal places.
A right-angled triangle has angle 52° and the adjacent side is 9.4 cm. Find the hypotenuse, correct to 2 decimal places.
From a point 24 m from the base of a vertical tree, the angle of elevation to the top is 35°. Find the height of the tree, correct to 1 decimal place.
A ladder makes an angle of 63°24' with the ground. If the ladder reaches 4.6 m up a vertical wall, find the length of the ladder, correct to 2 decimal places.
Section A — Standard Right-Angled Triangles
Section B — Degrees and Minutes
Section C — Practical Problems
$O = 15\sin42° \approx \mathbf{10.04 \text{ cm}}$
$A = 22\cos28° \approx \mathbf{19.43 \text{ m}}$
$O = 8.4\tan55° \approx \mathbf{12.00 \text{ cm}}$
$H = 7/\sin36° \approx \mathbf{11.90 \text{ m}}$
$H = 5.5/\cos71° \approx \mathbf{16.89 \text{ m}}$
$A = 12/\tan48° \approx \mathbf{10.80 \text{ cm}}$
$35°30' = 35.5°$; $O = 20\sin35.5° \approx \mathbf{11.60 \text{ m}}$
$47°12' = 47.2°$; $H = 6.3/\cos47.2° \approx \mathbf{9.27 \text{ m}}$
$O = 8\sin15° \approx \mathbf{2.07 \text{ m}}$
$h = 12\tan54° \approx \mathbf{16.5 \text{ m}}$
$H = 420/\sin28° \approx \mathbf{894 \text{ m}}$
Angle of depression $= 18°$ → angle inside triangle $= 18°$; $d = 25/\tan18° \approx \mathbf{76.9 \text{ m}}$
Look back at what you wrote in the Think First section. What has changed? What did you get right? What surprised you?
Multiple Choice
1 In a right-angled triangle, the angle is 34° and the hypotenuse is 20 cm. The side adjacent to the 34° angle is closest to:
? Regarding this topic, 1 In a right-angled triangle, the angle is 34° and the hypotenuse is 20 cm. The side adjacent to the 34° angle is closest to:
2 A vertical pole has a wire attached to its top. The wire makes an angle of 65° with the ground and is anchored 4.5 m from the base. The height of the pole is closest to:
? Regarding this topic, 2 A vertical pole has a wire attached to its top. The wire makes an angle of 65° with the ground and is anchored 4.5 m from the base. The height of the pole is closest to:
3 From a point on the ground, the angle of elevation to the top of a 60 m building is 41°. The distance from the point to the base of the building is closest to:
? Regarding this topic, 3 From a point on the ground, the angle of elevation to the top of a 60 m building is 41°. The distance from the point to the base of the building is closest to:
Short Answer
SA 4 3 marks A 6-metre ladder leans against a vertical wall. The ladder makes an angle of 72° with the ground.
(a) Find how high up the wall the ladder reaches (to 2 d.p.). (2 marks)
(b) Find the distance from the base of the wall to the foot of the ladder (to 2 d.p.). (1 mark)
$h = 6\sin72° \approx 6 \times 0.9511 \approx \mathbf{5.71 \text{ m}}$
$d = 6\cos72° \approx 6 \times 0.3090 \approx \mathbf{1.85 \text{ m}}$
SA 5 3 marks A surveyor stands 80 m from the base of a vertical cliff. She measures the angle of elevation to the top of the cliff as 38°42'.
(a) Convert 38°42' to decimal degrees. (1 mark)
(b) Find the height of the cliff, correct to the nearest metre. (2 marks)
$38 + 42/60 = \mathbf{38.7°}$
$h = 80\tan38.7° \approx 80 \times 0.8002 \approx \mathbf{64 \text{ m}}$
SA 6 4 marks A ship sails due east from port. After some time, the navigator observes a lighthouse at a bearing of N25°E, directly ahead of the ship's original direction. The lighthouse is known to be 3.2 km north of port.
Draw and label the right-angled triangle formed by port, the ship, and the lighthouse. (1 mark)
Find how far east the ship has sailed, correct to 2 decimal places. (2 marks)
Find the direct distance from the ship to the lighthouse, correct to 2 decimal places. (1 mark)
Right angle at the ship's position; north side = 3.2 km (O); east distance = A; angle at ship = 25° (bearing N25°E means 25° from north towards east); hypotenuse = direct distance to lighthouse
$\tan25° = \frac{A}{3.2}$... wait: the angle at the ship between north and the lighthouse direction is 25°; so O is the east distance, A is the north distance (3.2 km). $A_{\text{east}} = 3.2\tan25° \approx 3.2 \times 0.4663 \approx \mathbf{1.49 \text{ km}}$
$H = 3.2/\cos25° \approx 3.2/0.9063 \approx \mathbf{3.53 \text{ km}}$
Right-Angled Trigonometry: Finding Unknown Sides