Trigonometry · Ages 14–15
Finding a side
Unknown on top: multiply; underneath: divide.
How to do it
Method
Unknown on top: multiply; underneath: divide
x = 10 × sin 35° ≈ 5.7 cm
Another way
Use the other acute angle
10 × cos 55° ≈ 5.7
In full
Given an angle and a side, for another side: say which two of opposite, adjacent and hypotenuse the known side and the one wanted are, pick the ratio with those two, and solve. Hypotenuse 10 cm, θ = 35°, the opposite side x: sin 35° = x10, so x = 10 × sin 35° ≈ 5.7 cm. Adjacent 8 cm, θ = 40°, the hypotenuse y: cos 40° = 8y, so y = 8 ÷ cos 40° ≈ 10.4 cm — with the unknown underneath, divide, don't multiply. Don't round sin 35° on the way: round only the last step, to the accuracy asked. Another way: use the other acute angle, 90° − θ, and its ratio — the same answer: x = 10 × cos 55°.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
The right-angled triangle in the picture has a hypotenuse of 32 cm, and the angle marked is 45°. Correct to 1 decimal place, what is x, in cm?
Show the answer and working
Answer: 22.6 cm
x is opposite the 45° angle, the hypotenuse known: sin 45° = x32, so x = 32 × sin 45° ≈ 22.6 cm.
Example 2
In the right-angled triangle in the picture the angle marked is 36°, and the side beside it is 25 cm. Correct to 1 decimal place, what is x, in cm?
Show the answer and working
Answer: 18.2 cm
x is opposite and 25 adjacent: tan 36° = x25, so x = 25 × tan 36° ≈ 18.2 cm.
Example 3
The right-angled triangle in the picture has a hypotenuse of 5 cm, and the angle marked is 37°. Correct to 1 decimal place, x = 4 cm.
True or false?
Show the answer and working
Answer: False
x is the opposite side: 5 × sin 37° ≈ 3 cm; cos gives the adjacent side.
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.The right-angled triangle in the picture has a hypotenuse of 7 cm, and the angle marked is 14°. Which working finds x?
- A. x = 7 ÷ sin 14°
- B. x = 7 × tan 14°
- C. x = 7 × sin 14°
- D. x = 7 × cos 14°
Choose the right answer
Answer:
2.The right-angled triangle in the picture has a hypotenuse of 26 cm, and the angle marked is 66°. Correct to 1 decimal place, what is x, in cm?
Answer:
3.In the right-angled triangle in the picture the angle marked is 41°, and the side beside it is 21 cm. Correct to 1 decimal place, what is x, in cm?
Answer:
4.In the right-angled triangle in the picture the angle marked is 75°, and the side opposite it is 20 cm. Correct to 1 decimal place, what is the hypotenuse x, in cm?
Answer:
5.The right-angled triangle in the picture has a hypotenuse of 16 cm, and the angle marked is 56°. Correct to 1 decimal place, x = 8.9 cm.
True or false?
Answer:
All the answers
- 1. C. x = 7 × sin 14°
- 2. 10.6 cm
- 3. 18.3 cm
- 4. 20.7 cm
- 5. False
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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