📐Introduction to Trigonometry

Right triangles hide a remarkable fact: fix one acute angle, and the ratios among all three sides are fixed too. The triangle may grow or shrink, but the ratios never budge. Where the Pythagorean theorem manages the relationships among the three sides, trigonometry manages the relationship between sides and angles — it turns an angle into a number you can compute with.

Name the three sides first

Pick an acute angle. Relative to it, the three sides earn their names:

• The hypotenuse: the side facing the right angle, always the longest;
• The opposite side: the one directly facing your chosen angle;
• The adjacent side: the one touching your chosen angle that is not the hypotenuse.

Leg a3Leg b4

3² + 4² = 9 + 16 = 25 = c²

Hypotenuse c c = √25 ≈ 5.00整数勾股数 · Pythagorean triple!

Drag the two legs and watch the hypotenuse and the squares change. Keep one eye on this: similar right triangles may change every length, yet the ratios between sides stay put — that is what makes a table of trig values possible at all.

Three ratios: sin, cos, tan

Relative to the chosen angle, define three ratios:

• sine, sin = opposite ÷ hypotenuse;
• cosine, cos = adjacent ÷ hypotenuse;
• tangent, tan = opposite ÷ adjacent.

The mnemonic SOH-CAH-TOA squeezes all three definitions into nine letters: S-O-H, C-A-H, T-O-A. Each ratio depends only on the angle's size, so the values can be precomputed into a table (calculators store far more).

Values for three special angles

Angle	sin	cos	tan
30°	0.500	0.866	0.577
45°	0.707	0.707	1.000
60°	0.866	0.500	1.732

Two things hide in the table: at 45°, sin equals cos — that triangle's two legs are the same length; and the sin of 30° matches the cos of 60° — the same triangle, viewed from the other corner.

From angle to side, from side to angle

Angle known, side wanted. Hypotenuse 10, one acute angle 30° — how long is the side opposite the 30° angle?

Opposite = hypotenuse × sin 30° = $10 \times 0.5 = 5$.

In the same triangle, the side opposite the 60° angle is $10 \times \sin 60^\circ \approx 8.66$.

Side known, angle wanted. Measure a right triangle whose opposite side is 5 and hypotenuse 10: the ratio is $5 \div 10 = 0.5$. Look it up in the table (or press the $\sin^{-1}$ key on a calculator), and the angle reads 30°. Sides come from multiplying; angles come from reverse lookup — one calculator covers both.

Check yourself

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Quick quiz

↺ Try again

1. 1. sin is the ratio of which two sides?

   Opposite ÷ hypotenuseAdjacent ÷ hypotenuseOpposite ÷ adjacent

2. 2. Hypotenuse 10, one acute angle 30°. How long is the side opposite 30°?

   58.6620

3. 3. Which side of a right triangle is the longest?

   The opposite sideThe adjacent sideThe hypotenuse