Introduction to Trigonometry: Sine, Cosine, Tangent, Cotangent

Trigonometry studies the relationships between the angles and sides of triangles, and its basic tools are the ratios sine, cosine, tangent and cotangent of an acute angle in a right triangle. In brief: in a right triangle, sin A = opposite/hypotenuse, cos A = adjacent/hypotenuse, tan A = opposite/adjacent and cot A = adjacent/opposite. In a 3-4-5 triangle, the angle opposite the side 3 has sin A = 3/5, cos A = 4/5, tan A = 3/4 and cot A = 4/3.

These ratios depend only on the angle, not on the size of the triangle, because triangles with the same angles are similar. This is why a table or a calculator can give the ratio for any angle, and why trigonometry can be used to find distances that cannot be measured directly. This guide defines the four ratios, shows how to name the sides, how to find sides and angles in right triangles, the use of the calculator and a table of values.

How do you name the sides of a right triangle?

In a right triangle ABC with the right angle at C, the hypotenuse c is opposite the right angle. For an acute angle A, the leg a across from A is the opposite side, and the leg b that touches A is the adjacent side. The names depend on the chosen angle: for the other acute angle B, the roles of the legs are exchanged.

What are sine, cosine, tangent and cotangent?

• sin A = opposite / hypotenuse
• cos A = adjacent / hypotenuse
• tan A = opposite / adjacent = sin A / cos A
• cot A = adjacent / opposite = cos A / sin A = 1 / tan A

A memory aid: SOH-CAH-TOA. Since the legs are shorter than the hypotenuse, sine and cosine of an acute angle are between 0 and 1. Tangent and cotangent can be any positive number. For the two acute angles of a right triangle, sin A = cos B, because the opposite side of A is the adjacent side of B. This gives the cofunction rule sin A = cos(90° − A).

Example 1

Right triangle with legs 5 and 12 and hypotenuse 13. For the angle A opposite the leg 5: sin A = 5/13 ≈ 0.385, cos A = 12/13 ≈ 0.923, tan A = 5/12 ≈ 0.417 and cot A = 12/5 = 2.4. For the other angle B: sin B = 12/13 and cos B = 5/13.

How do you find a side?

1. Mark the known angle and the sides involved: which side is given, and which is wanted.
2. Choose the ratio that connects these two sides (SOH-CAH-TOA).
3. Write the equation and solve for the unknown side.

Example 2

A ladder of 6 m leans against a wall at an angle of 70° to the ground. The height reached is opposite the angle, and the ladder is the hypotenuse: sin 70° = h/6, so h = 6 · 0.9397 ≈ 5.64 m. The distance of the foot from the wall is adjacent: cos 70° = d/6, so d = 6 · 0.3420 ≈ 2.05 m.

Example 3

In a right triangle, the angle A = 35° and the adjacent leg b = 8. The opposite leg: tan 35° = a/8, so a = 8 · 0.7002 ≈ 5.60. The hypotenuse: cos 35° = 8/c, so c = 8/0.8192 ≈ 9.77. Check with Pythagoras: √(64 + 31.4) ≈ 9.77 ✓.

How do you find an angle?

If two sides are known, use the inverse function: A = sin⁻¹(opposite/hypotenuse), A = cos⁻¹(adjacent/hypotenuse) or A = tan⁻¹(opposite/adjacent). Make sure that the calculator is in degree mode.

Example 4

A ramp rises 1.2 m over a horizontal distance of 10 m. The angle of inclination: tan A = 1.2/10 = 0.12, so A = tan⁻¹(0.12) ≈ 6.84°.

Example 5

A right triangle has the hypotenuse 20 and one leg 12. Then cos A = 12/20 = 0.6, so A ≈ 53.13°, and the other angle is 36.87°. The third side is 16.

Table of trigonometric values for 10° to 80°

Notice the symmetry: sin 10° = cos 80° and tan 10° = cot 80°. Exact values for 30°, 45° and 60° come from the special triangles.

Where does trigonometry come from?

The word comes from Greek words for “triangle” and “measure”. Greek astronomers, especially Hipparchus and later Ptolemy, produced tables of chords of circles to compute positions of stars. Indian mathematicians introduced the sine function (jya) in the Gupta period, and the name “sine” came into European languages through a chain of translations from Sanskrit through Arabic to Latin. Arab mathematicians added tangent and cotangent and developed the subject into its modern form.

Where is trigonometry used today?

Surveyors measure inaccessible heights and distances. Engineers resolve forces into components, and architects compute the slope of roofs. GPS, computer graphics, music synthesis and medical imaging depend on trigonometric functions.

Which mistakes are common?

Wrong calculator mode

sin 30° = 0.5 in degree mode, but in radian mode sin 30 ≈ −0.988. Check the mode before computing.

Mixing up opposite and adjacent

The names refer to the chosen angle. Mark the angle and the sides first.

Using sin instead of sin⁻¹

To find an angle from a ratio, use the inverse function, not sin of the ratio.

Applying the ratios to a non-right triangle

SOH-CAH-TOA is for right triangles. For other triangles, use the law of sines and the law of cosines.