Trigonometry · Ages 14–15
Finding an angle
The ratio, then sin⁻¹, cos⁻¹ or tan⁻¹.
How to do it
Method
The ratio, then sin⁻¹, cos⁻¹ or tan⁻¹
tan θ = 58, θ ≈ 32.0°
Another way
90° less the other angle
90° − 58.0° = 32.0°
In full
Given two sides, for an angle: decide sin, cos or tan by which two of opposite, adjacent and hypotenuse they are, work out the ratio, then the calculator's inverse (sin⁻¹, cos⁻¹, tan⁻¹). Opposite 5, adjacent 8: tan θ = 58 = 0.625, θ = tan⁻¹ 0.625 ≈ 32.0°. Opposite 6, hypotenuse 10: sin θ = 0.6, θ ≈ 36.9°. Put the fraction straight into the calculator: don't round the ratio first. Another way: find the other acute angle and take it from 90°: tan⁻¹(85) ≈ 58.0°, 90° − 58.0° = 32.0°. Check: the longer the opposite side, the bigger the angle; a sine or cosine is always between 0 and 1.
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 24 cm, and θ's opposite side is 16 cm. To the nearest 0.1°, what is θ?
Show the answer and working
Answer: 41.8°
sin θ = opposite ÷ hypotenuse = 1624, θ = sin⁻¹(1624) ≈ 41.8°.
Example 2
The right-angled triangle in the picture has shorter sides of 10 cm and 13 cm. To the nearest 0.1°, what is θ?
Show the answer and working
Answer: 37.6°
This θ is at the other corner: opposite 10, adjacent 13, so tan θ = 1013 and θ ≈ 37.6°.
Example 3
The right-angled triangle in the picture has shorter sides of 7 cm and 8 cm. To the nearest 0.1°, θ = 48.8°.
True or false?
Show the answer and working
Answer: True
tan θ = opp ÷ adj = 87, θ ≈ 48.8°; upside down gives the other acute angle.
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 22 cm, and θ's opposite side is 16 cm. To the nearest 0.1°, what is θ?
Answer:
2.The right-angled triangle in the picture has shorter sides of 8 cm and 6 cm. To the nearest 0.1°, θ = 53.1°.
True or false?
Answer:
3.The right-angled triangle in the picture has shorter sides of 17 cm and 15 cm. To the nearest 0.1°, what is θ?
Answer:
4.The right-angled triangle in the picture has a hypotenuse of 15 cm, and the shorter side beside θ is 12 cm. To the nearest 0.1°, what is θ?
Answer:
5.An isosceles triangle has two equal sides of 20 cm and a base of 19 cm. To the nearest 0.1°, what is each base angle?
Answer:
All the answers
- 1. 46.7°
- 2. False
- 3. 48.6°
- 4. 36.9°
- 5. 61.6°
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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