Trigonometry · Ages 14–15
Trigonometry in plane figures
Find the right-angled triangles; the shared side first.
How to do it
Method
Find the right-angled triangles; the shared side first
BC = htan B + htan C
Another way
The shared side first
AD = AB sin B, then DC = AD ÷ tan C
In full
In a composite figure find the right-angled triangles first: a rectangle's diagonal, an isosceles triangle's height and a perpendicular from a corner to the base each make some. In triangle ABC with AD ⊥ BC and AD = h: BD = h ÷ tan B, DC = h ÷ tan C, BC = BD + DC; AB = h ÷ sin B. An isosceles triangle with equal sides L and top angle 2θ: its height splits it into two right-angled triangles, so the base = 2 × L sin θ. A rhombus's diagonals bisect each other at right angles, making four. Writing a length with ratios, write the expression first (htan B, say) and put the numbers in at the end. Another way: find a shared side first (AD, say) by one triangle's ratio, then use it in the other triangle.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
In triangle ABC in the picture the height AD = 18 cm, ∠B = 65° and ∠C = 29°. What is BD?
- A. 18 ÷ sin 65°
- B. 18 × tan 65°
- C. 18 ÷ tan 65°
- D. 18 × sin 65°
Choose the right answer
Show the answer and working
Answer: C. 18 ÷ tan 65°
In the right-angled triangle ABD, tan 65° = ADBD = 18BD, so BD = 18 ÷ tan 65°.
Example 2
In triangle ABC in the picture the height AD = 18 cm, ∠B = 58° and ∠C = 47°. Correct to 1 decimal place, what is BC, in cm?
Show the answer and working
Answer: 28 cm
BD = 18 ÷ tan 58°, DC = 18 ÷ tan 47°, BC = BD + DC ≈ 28 cm (rounded at the end).
Example 3
In triangle ABC in the picture the height AD = 14 cm, ∠B = 64° and ∠C = 30°. Correct to 1 decimal place: x = BD = cm, y = DC = cm.
Fill in each blank
Show the answer and working
Answer: 6.8, 24.2
The two right-angled triangles share AD: BD = 14 ÷ tan 64° ≈ 6.8, DC = 14 ÷ tan 30° ≈ 24.2.
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.In triangle ABC in the picture the height AD = 15 cm, ∠B = 37° and ∠C = 49°. What is BD?
- A. 15 × tan 37°
- B. 15 ÷ sin 37°
- C. 15 ÷ tan 37°
- D. 15 × sin 37°
Choose the right answer
Answer:
2.In triangle ABC in the picture the height AD = 9 cm, ∠B = 37° and ∠C = 50°. Correct to 1 decimal place, what is BC, in cm?
Answer:
3.An isosceles triangle has two equal sides of 9 cm and an angle of 46° at the top. Correct to 1 decimal place, how long is its base, in cm?
Answer:
4.In triangle ABC in the picture the height AD = 13 cm, ∠B = 59° and ∠C = 53°. Correct to 1 decimal place, AB = 15.2 cm.
True or false?
Answer:
5.A rhombus has sides of 30 cm and an acute angle of 62°. Correct to 1 decimal place, how long is its longer diagonal, in cm?
Answer:
All the answers
- 1. C. 15 ÷ tan 37°
- 2. 19.5 cm
- 3. 7 cm
- 4. True
- 5. 51.4 cm
VIP practice notes and papers
Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.
- Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
- Practice paper: Set out in parts like a school paper, marked, with answers and full working
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