MISSION · Geometry
Sine, Cosine, and Tangent
Three ratios that depend only on the angle
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The idea in one sentence
In a right triangle, the ratio between any two sides is fixed by the angle alone — so knowing one angle and one side is enough to find every other side.
Why the ratios are constant
Draw a right triangle with a \(30°\) angle. Draw a bigger one, also with a \(30°\) angle. The two are similar — same shape, different size — so every pair of matching sides is in the same proportion.
That means the ratio \(\frac{\text{opposite}}{\text{hypotenuse}}\) comes out the same for both, and for every \(30°\) right triangle anywhere. It is exactly 0.5, forever. The ratio depends on the angle and nothing else, which is what makes it worth naming and tabulating.
The three names
Label the sides relative to the angle you care about. The hypotenuse is always the long side opposite the right angle. The opposite is across from your angle. The adjacent is the remaining one, touching your angle.
\[\sin \theta = \frac{\text{opp}}{\text{hyp}} \qquad \cos \theta = \frac{\text{adj}}{\text{hyp}} \qquad \tan \theta = \frac{\text{opp}}{\text{adj}}\]The usual mnemonic is SOH-CAH-TOA, one syllable per ratio.
The labels are relative, and that’s the point people miss. Switch your attention to the other acute angle and opposite and adjacent swap places — which is why \(\sin \theta = \cos(90° - \theta)\), and why cosine is named for the complementary angle’s sine.
Using them
Finding a side. A ladder 6 m long leans at \(70°\). How high does it reach? The height is opposite the angle and the ladder is the hypotenuse, so sine is the one: \(\sin 70° = \frac{h}{6}\), giving \(h = 6 \sin 70° \approx 5.6\) m.
Finding an angle. If the opposite is 3 and the adjacent is 4, then \(\tan \theta = 0.75\), so \(\theta = \tan^{-1}(0.75) \approx 36.9°\). The inverse functions — \(\sin^{-1}\), \(\cos^{-1}\), \(\tan^{-1}\) — take a ratio and hand back the angle.
The method is always the same: label the sides relative to the known angle, see which two you’re dealing with, pick the ratio that uses exactly those two.
Some values worth knowing
| \(\theta\) | \(\sin\) | \(\cos\) | \(\tan\) |
|---|---|---|---|
| \(0°\) | 0 | 1 | 0 |
| \(30°\) | 0.5 | 0.866 | 0.577 |
| \(45°\) | 0.707 | 0.707 | 1 |
| \(60°\) | 0.866 | 0.5 | 1.732 |
| \(90°\) | 1 | 0 | undefined |
Tangent is undefined at \(90°\) because the adjacent side has shrunk to nothing, and dividing by it is dividing by zero.
Common mix-ups
- Labelling the sides before choosing the angle. Opposite and adjacent only mean something once you’ve fixed which angle you’re working from.
- Calculator in radians. If \(\sin 30°\) doesn’t give 0.5, your calculator is in the wrong mode.
- Using these on a triangle with no right angle. They need the right angle; other triangles need the sine and cosine rules.
- Confusing \(\sin^{-1}\) with \(\frac{1}{\sin}\). The first is the inverse function; the second is a reciprocal.
Check yourself
A right triangle has a \(40°\) angle and a hypotenuse of 10. Find the side opposite the angle, and check the third angle.
Opposite \(= 10 \sin 40° \approx 6.43\). The angles must total \(180°\), so the third is \(180 - 90 - 40 = 50°\).
LEVEL 3 · REFLECT
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