Finding an Unknown Angle
Find an unknown angle in a right-angled triangle for NSW Year 12 Mathematics Standard 1 (NESA outcome MST-12-S1-04). Once you know two sides, the trigonometric ratios sine, cosine and tangent take you straight to the missing angle.
You will name the hypotenuse, opposite and adjacent sides, pick the right ratio with SOH CAH TOA, use the inverse key on your calculator, and give the angle in degrees or in degrees and minutes — handy for the slope of a ramp, a ladder or a roof.
Theory
To find an unknown angle in a right-angled triangle, use the two sides you know to choose sine, cosine or tangent, then press the inverse key \(\sin^{-1}\), \(\cos^{-1}\) or \(\tan^{-1}\). This Year 12 Standard 1 (NSW) guide also shows how to give the angle in degrees and minutes.
To find an unknown angle in a right-angled triangle you need two known sides. Name the sides relative to the angle \(\theta\) you want: the hypotenuse is opposite the right angle and is the longest side, the opposite side faces \(\theta\), and the adjacent side touches \(\theta\) but is not the hypotenuse.
Pick the ratio from SOH CAH TOA that uses the two sides you know, for example \(\tan\theta=\dfrac{O}{A}\). Then undo the ratio with the inverse key: \(\theta=\tan^{-1}\!\left(\dfrac{O}{A}\right)\). Your calculator must be in degree mode (it shows D).
Angles are often given in degrees and minutes, where \(1^\circ=60'\). Keep the whole degrees and multiply only the decimal part by \(60\): \(41.25^\circ\) is \(41^\circ\) and \(0.25\times60'=15'\), so \(41^\circ 15'\). The degrees-minutes-seconds key on your calculator does this for you.
Look at which two sides you know, then use the matching inverse ratio:
| The two sides you know | Use |
|---|---|
| opposite and hypotenuse | \(\theta=\sin^{-1}\!\left(\dfrac{O}{H}\right)\) |
| adjacent and hypotenuse | \(\theta=\cos^{-1}\!\left(\dfrac{A}{H}\right)\) |
| opposite and adjacent | \(\theta=\tan^{-1}\!\left(\dfrac{O}{A}\right)\) |
Converting to degrees and minutes:
How to find an unknown angle
- Label the sides as hypotenuse, opposite and adjacent, measured from the angle you want.
- Choose the ratio that uses the two sides you know: \(\sin\) (O and H), \(\cos\) (A and H) or \(\tan\) (O and A).
- Write the equation, for example \(\sin\theta=\dfrac{4}{9}\).
- Apply the inverse key in degree mode, then round to the nearest degree, or convert the decimal part to minutes.
Sketch the triangle. The opposite side and the hypotenuse are known, so use sine and its inverse.
| \(\sin\theta\) | \(=\) | \(\dfrac{4}{9}\) |
| \(\theta\) | \(=\) | \(\sin^{-1}\!\left(\dfrac{4}{9}\right)\) |
| \(\theta\) | \(=\) | \(26.38\ldots^\circ\) |
| \(\theta\) | \(\approx\) | \(26^\circ\) |
So \(\sin\theta=\dfrac{4}{9}\) and \(\theta\approx 26^\circ\).
Sketch the triangle. The rise is opposite the angle and the horizontal distance is adjacent, so use tangent.
| \(\tan\theta\) | \(=\) | \(\dfrac{2.4}{5.5}\) |
| \(\theta\) | \(=\) | \(\tan^{-1}\!\left(\dfrac{2.4}{5.5}\right)\) |
| \(\theta\) | \(=\) | \(23.57\ldots^\circ\) |
| \(\theta\) | \(\approx\) | \(24^\circ\) |
The roof is at about \(24^\circ\) to the horizontal.
Sketch the triangle. The ladder is the hypotenuse and the base distance is adjacent to \(\theta\), so use cosine.
| \(\cos\theta\) | \(=\) | \(\dfrac{1.8}{6}\) |
| \(\theta\) | \(=\) | \(\cos^{-1}\!\left(\dfrac{1.8}{6}\right)\) |
| \(\theta\) | \(=\) | \(72.5424\ldots^\circ\) |
| \(0.5424\ldots^\circ\) | \(=\) | \(0.5424\ldots\times 60' = 32.5\ldots'\approx 33'\) |
| \(\theta\) | \(\approx\) | \(72^\circ 33'\) |
The ladder makes an angle of about \(72^\circ 33'\) with the ground.
Sketch the triangle. Use tangent for the angle, then Pythagoras for the third side.
| \(\tan\theta\) | \(=\) | \(\dfrac{9}{14}\) |
| \(\theta\) | \(=\) | \(\tan^{-1}\!\left(\dfrac{9}{14}\right)=32.73\ldots^\circ\) |
| \(0.73\ldots^\circ\) | \(=\) | \(0.73\ldots\times 60'\approx 44'\) |
| \(\text{hyp}\) | \(=\) | \(\sqrt{9^2+14^2}=\sqrt{277}\) |
| \(\text{hyp}\) | \(=\) | \(16.64\ldots\) |
(i) \(33^\circ\) (ii) \(32^\circ 44'\) (iii) \(16.6\text{ cm}\)
Common pitfalls
Frequently asked questions
How do you find an unknown angle in a right-angled triangle?
Label the sides from the angle as opposite, adjacent and hypotenuse. Choose sine, cosine or tangent using the two sides you know, write the equation, then use the inverse key (sin inverse, cos inverse or tan inverse) on your calculator in degree mode.
When do I use sin inverse, cos inverse or tan inverse?
Use sin inverse when you know the opposite side and the hypotenuse, cos inverse when you know the adjacent side and the hypotenuse, and tan inverse when you know the opposite and adjacent sides.
Why is my calculator giving the wrong angle?
The most common reason is that the calculator is in radian or gradian mode. Change it to degree mode so the screen shows D. Also check you pressed the inverse key (shift then sin, cos or tan).
How do I change a decimal angle into degrees and minutes?
Keep the whole number of degrees, then multiply the decimal part by 60 to get the minutes. For example, 41.25 degrees is 41 degrees plus 0.25 times 60, which is 15 minutes, so the angle is 41 degrees 15 minutes.
Is 26.39 degrees the same as 26 degrees 39 minutes?
No. There are 60 minutes in a degree, not 100. 26.39 degrees is 26 degrees plus 0.39 times 60, which is about 23 minutes, so it is 26 degrees 23 minutes.