Finding Unknown Angles
Every question with a fully worked solution.
- Finding Unknown Angles - Video - finding angles in right-angled triangles Watch
Theory
To find an unknown angle in a right-angled triangle, use the two known sides to write \(\sin\theta\), \(\cos\theta\) or \(\tan\theta\), then apply the inverse key. This Year 10 maths page revises Year 9: choosing the ratio, rounding to the nearest degree or minute, finding the other acute angle, and practical problems with rectangles, isosceles triangles and rhombuses.
The inverse trigonometric functions undo the ratios. If \(\sin\theta = 0.5\), then \(\theta = \sin^{-1}(0.5) = 30^\circ\). The \(-1\) is not a power: \(\sin^{-1}\) means “the angle whose sine is”.
To find an angle you need two sides. Label them from the angle: opposite and hypotenuse give sine, adjacent and hypotenuse give cosine, opposite and adjacent give tangent.
Angles are often rounded to the nearest minute. There are \(60\) minutes in a degree, so the decimal part of the degrees is multiplied by \(60\): \(32.5^\circ = 32^\circ 30'\).
The two acute angles of a right-angled triangle add to \(90^\circ\), so once you know one, the other is \(90^\circ\) minus it.
| You know | Use |
|---|---|
| opposite and hypotenuse | \(\sin^{-1}\) |
| adjacent and hypotenuse | \(\cos^{-1}\) |
| opposite and adjacent | \(\tan^{-1}\) |
How to find an unknown angle
- Label the two known sides from the angle you want (O, A or H).
- Choose the ratio that uses those two sides and write it as a fraction.
- Apply the inverse key in degree mode.
- Round as asked: to the nearest degree, or convert the decimal part to minutes.
Harder problems
- Other acute angle: subtract from \(90^\circ\).
- Isosceles triangle: the height halves the base, giving a right-angled triangle for the base angle; use the angle sum for the top angle with the unrounded value.
- Rhombus: the diagonals cross at right angles and halve each other and the angles. Find half an angle, then double it; the angles next to each other add to \(180^\circ\).
- Rectangle: a diagonal and two sides make a right-angled triangle.
- Two triangles: find each angle in its own triangle, then add or subtract them (an exterior angle equals the sum of the two opposite interior angles).
- Units: write both lengths in the same unit before dividing.
From \(\theta\): \(7.5\) cm is opposite and \(11\) cm is the hypotenuse. O and H, so use sine.
| \(\sin\theta\) | \(=\) | \(\dfrac{7.5}{11}\) |
| \(\theta\) | \(=\) | \(\sin^{-1}\left(\dfrac{7.5}{11}\right)\) |
| \(\) | \(=\) | \(42.98\ldots^\circ\) |
| \(\) | \(\approx\) | \(\mathbf{43^\circ}\) |
Type the whole fraction inside the bracket, or work out \(7.5 \div 11\) first and keep every digit.
Use inverse cos, then change the decimal part of the degrees into minutes (multiply by \(60\)).
| \(\theta\) | \(=\) | \(\cos^{-1}(0.62)\) |
| \(\) | \(=\) | \(51.6838\ldots^\circ\) |
| \(0.6838\ldots \times 60\) | \(=\) | \(41.03\ldots'\) |
| \(\theta\) | \(\approx\) | \(\mathbf{51^\circ 41'}\) |
The two acute angles add to \(90^\circ\):
| \(90^\circ - 51^\circ 41'\) | \(=\) | \(\mathbf{38^\circ 19'}\) |
The degrees-minutes-seconds key does this conversion for you. \(51.68^\circ\) is not \(51^\circ 68'\).
The string is the hypotenuse; the height is opposite \(\theta\). Use sine, not tangent: the \(60\) m is not measured along the ground.
| \(\sin\theta\) | \(=\) | \(\dfrac{41}{60}\) |
| \(\theta\) | \(=\) | \(43.1046\ldots^\circ\) |
| \(\) | \(\approx\) | \(\mathbf{43^\circ 6'}\) |
Lengths in different units (km and m) must be changed to the same unit before you divide.
(i) each base angle \(b\)
(ii) the angle \(t\) at the top, each correct to the nearest degree.
The height from the top cuts the base in half, making two right-angled triangles with hypotenuse \(9\) and base \(7\).
(i)
| \(\cos b\) | \(=\) | \(\dfrac{7}{9}\) |
| \(b\) | \(=\) | \(38.9424\ldots^\circ\) |
| \(\) | \(\approx\) | \(\mathbf{39^\circ}\) |
(ii) Use the angle sum with the unrounded value.
| \(t\) | \(=\) | \(180^\circ - 2 \times 38.94\ldots^\circ\) |
| \(\) | \(=\) | \(102.11\ldots^\circ\) |
| \(\) | \(\approx\) | \(\mathbf{102^\circ}\) |
Common pitfalls
Frequently asked questions
How do you find an angle using trigonometry?
Label the two sides you know from the angle, choose sine, cosine or tangent to match them, write the ratio as a fraction, then use the inverse key on your calculator in degree mode.
What does sin to the power of minus 1 mean?
\(\sin^{-1}\) is the inverse sine, not \(\dfrac{1}{\sin}\). It gives the angle that has that sine: \(\sin^{-1}(0.5) = 30^\circ\).
How do you round an angle to the nearest minute?
Keep the whole degrees, multiply the decimal part by \(60\) and round to a whole number of minutes. Or use the degrees-minutes-seconds key.
Why does my calculator give a strange angle?
It is probably in radian mode. Look for D or DEG on the screen and change the mode to degrees, then work the angle out again.
How do you find the other angle in a right-angled triangle?
The two acute angles add to \(90^\circ\), so subtract the angle you found from \(90^\circ\). You can check it with a second inverse ratio from the other angle.