Finding Unknown Angle
To find an angle, write the ratio of two known sides and use the inverse function on a calculator, such as \(\tan^{-1}\). This NSW Year 9 Advanced maths subtopic finds angles in right-angled triangles to the nearest degree or minute (NESA outcome MA5‑TRG‑C‑01).
You will find angles from diagrams and from side lengths, work out the pitch of a roof and the angle of a guy wire, and explain why \(\sin\theta = \frac{9}{7}\) is impossible. You will find the angles of an isosceles triangle and a rhombus, the change in a kite string's angle, an angle made by two triangles, and the angle between a TV screen's diagonal and its base. It is part of Year 9 Advanced Trigonometry.
Theory
To find an unknown angle, write a trig ratio from two known sides, then use the inverse key: \(\sin^{-1}\), \(\cos^{-1}\) or \(\tan^{-1}\). This Year 9 Advanced maths page covers answers to the nearest minute, exact angles, angles inside isosceles triangles and rhombuses, and angles that are the difference of two angles.
When two sides of a right-angled triangle are known, their ratio fixes the angle. The inverse trig functions undo the ratios: if \(\tan\theta = 0.7\), then \(\theta\) \(= \tan^{-1} 0.7\) \(\approx 35^\circ\).
\(\sin^{-1}\) does not mean \(\dfrac{1}{\sin}\). It means "the angle whose sine is". On most calculators it is the shift (or 2nd) function of the sin key.
Angles can be given to the nearest degree or to the nearest minute, where \(1^\circ = 60'\).
For an acute angle, \(\sin\theta\) and \(\cos\theta\) lie between \(0\) and \(1\). A ratio such as \(\sin\theta = \dfrac{11}{8}\) is impossible: the hypotenuse is the longest side. The calculator shows an error.
| \(\theta\) | \(\sin\theta\) | \(\cos\theta\) | \(\tan\theta\) |
|---|---|---|---|
| \(30^\circ\) | \(\dfrac{1}{2}\) | \(\dfrac{\sqrt{3}}{2}\) | \(\dfrac{1}{\sqrt{3}}\) |
| \(45^\circ\) | \(\dfrac{1}{\sqrt{2}}\) | \(\dfrac{1}{\sqrt{2}}\) | \(1\) |
| \(60^\circ\) | \(\dfrac{\sqrt{3}}{2}\) | \(\dfrac{1}{2}\) | \(\sqrt{3}\) |
How to find an unknown angle
- Label H, O and A from the angle you want.
- Choose the ratio that uses the two sides you know.
- Invert: use \(\sin^{-1}\), \(\cos^{-1}\) or \(\tan^{-1}\) on the whole fraction, in degree mode.
- Round as asked: to the nearest degree, or convert to degrees and minutes.
- Check: the larger angle is opposite the longer of the two shorter sides.
Harder problems
- Find a side first with Pythagoras' theorem if the two sides you need are not both given.
- Isosceles triangle: drop the perpendicular from the apex; use half the base. The apex angle is \(180^\circ\) minus twice the base angle.
- Rhombus: the diagonals cross at right angles and halve each other. Find half an angle from the half-diagonals, then double it. Neighbouring angles add to \(180^\circ\).
- Angle between two lines from one point: find each angle in its own right-angled triangle, then subtract (Example 3).
- Ratios and gradients: a ratio of sides such as \(4 : 3\) or a gradient \(1 : 12\) is a tangent; use \(\tan^{-1}\).
- Change of position: when one length changes (a kite rising, a ladder slipping), find the angle in each position and subtract.
From \(K\): \(KL\) is adjacent and \(KM\) is the hypotenuse, so use cosine.
| \(\cos\angle LKM\) | \(=\) | \(\dfrac{5.3}{8.6}\) |
| \(\angle LKM\) | \(=\) | \(\cos^{-1}\left(\dfrac{5.3}{8.6}\right)\) |
| \(\) | \(=\) | \(51.9550\ldots^\circ\) |
| \(\) | \(\approx\) | \(\mathbf{51^\circ 57'}\) |
| \(\angle KML\) | \(=\) | \(90^\circ - 51^\circ 57'\) |
| \(\) | \(=\) | \(\mathbf{38^\circ 3'}\) |
The two acute angles of a right-angled triangle add to \(90^\circ\), so the second angle needs no calculator.
The line from the apex to the midpoint of the base is perpendicular to it. Each half is a right-angled triangle with base \(4.8\) m and hypotenuse \(5.8\) m.
| \(\cos\alpha\) | \(=\) | \(\dfrac{4.8}{5.8}\) |
| \(\alpha\) | \(=\) | \(34.1484\ldots^\circ\) |
| \(\) | \(\approx\) | \(\mathbf{34^\circ 9'}\) |
| \(\beta\) | \(=\) | \(180^\circ - 2 \times 34^\circ 9'\) |
| \(\) | \(=\) | \(\mathbf{111^\circ 42'}\) |
Use half the span, not the whole span. The angle sum of the triangle gives \(\beta\).
\(\theta\) is not in a right-angled triangle. It is the difference of two angles that are: one to the top of the pole, one to the roof.
| \(\text{to the top}\) | \(=\) | \(\tan^{-1}\left(\dfrac{38}{20}\right) = 62.2414\ldots^\circ\) |
| \(\text{to the roof}\) | \(=\) | \(\tan^{-1}\left(\dfrac{32}{20}\right) = 57.9946\ldots^\circ\) |
| \(\theta\) | \(=\) | \(4.2468\ldots^\circ\) |
| \(\) | \(\approx\) | \(\mathbf{4^\circ 15'}\) |
Subtract the full calculator values, then round. Rounding each angle first can change the minutes.
Turn both into angles with the horizontal and compare. The rule gives rise and run, so it uses tangent; the ramp gives rise and slope length, so it uses sine.
| \(\text{Steepest allowed}\) | \(=\) | \(\tan^{-1}\left(\dfrac{1}{14}\right) = 4.0856\ldots^\circ \approx 4^\circ 5'\) |
| \(\text{This ramp}\) | \(=\) | \(\sin^{-1}\left(\dfrac{0.45}{6.0}\right) = 4.3012\ldots^\circ \approx 4^\circ 18'\) |
\(4^\circ 18' > 4^\circ 5'\), so the ramp is too steep and does not meet the rule.
Common pitfalls
Frequently asked questions
How do you find an angle using trigonometry?
Label the sides from the unknown angle, choose sin, cos or tan so the ratio uses the two sides you know, write the ratio as a fraction, then apply the inverse function on your calculator.
What does sin to the power of -1 mean?
Inverse sine means the angle whose sine is a given number. It is not 1 divided by sin. For example the inverse sine of 0.5 is 30 degrees.
How do you give an angle to the nearest minute?
Multiply the decimal part of the degrees by 60 to get minutes and round, or press the degrees-minutes-seconds key. For example 41.6 degrees is 41 degrees 36 minutes.
Why does my calculator say error when I find an angle?
Usually because the ratio for sine or cosine is bigger than 1, which happens when the hypotenuse is not on the bottom of the fraction. Check your labelling. Also check the calculator is in degree mode.
How do you find the angle of a slope from its gradient?
A gradient is rise over run, which is the tangent of the angle with the horizontal. Use inverse tan of the gradient. A gradient of 1 in 10 is inverse tan of 0.1, about 5.7 degrees.