How Sine, Cosine and Tangent Work in Right Triangles

Trigonometry studies the relationship between angles and lengths. Its starting point is three ratios, sine, cosine, and tangent, which compare two sides of a right triangle relative to one of its acute angles. These ratios let you find unknown sides and angles, and they later become the functions that describe waves, rotation, and oscillation. This guide defines the ratios, shows exact values for common angles, and explains how the same ideas extend beyond triangles.

The three sides relative to an angle

Take a right triangle and choose one of the two acute angles, call it θ. Name the sides from the point of view of that angle. The hypotenuse is the longest side, opposite the right angle. The opposite side is the one across from θ. The adjacent side is the one that touches θ and is not the hypotenuse.

The labels depend on which angle you choose. The same side can be opposite for one angle and adjacent for the other. Always name the sides from the angle you are working with, so the ratios come out correctly.

Definitions of the three ratios

The three basic ratios are defined as follows.

Sine of θ is the opposite side divided by the hypotenuse: sin θ = opposite / hypotenuse.

Cosine of θ is the adjacent side divided by the hypotenuse: cos θ = adjacent / hypotenuse.

Tangent of θ is the opposite side divided by the adjacent side: tan θ = opposite / adjacent.

A memory aid is SOH-CAH-TOA. Sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, and tangent is opposite over adjacent. The first letters of each ratio spell the phrase.

Because each ratio divides two lengths, the answer does not depend on the size of the triangle. Two right triangles with the same angle are similar, so their side ratios match exactly.

A worked example

Suppose a right triangle has hypotenuse 10 and an angle θ with opposite side 6. Then sin θ = 6 / 10 = 0.6. To find the adjacent side, use the Pythagorean theorem: adjacent² = 10² - 6² = 64, so the adjacent side is 8. Then cos θ = 8 / 10 = 0.8 and tan θ = 6 / 8 = 0.75.

Notice that sin² θ + cos² θ = 0.36 + 0.64 = 1. This identity holds for every angle and follows directly from the Pythagorean theorem.

Exact values for common angles

Some angles produce exact values built from simple roots. The table below lists the standard angles, all measured in degrees, and their sine, cosine, and tangent. These values come from a 45-45-90 triangle and a 30-60-90 triangle.

Angle	sin	cos	tan
0°	0	1	0
30°	1/2	√3/2	√3/3
45°	√2/2	√2/2	1
60°	√3/2	1/2	√3
90°	1	0	undefined

The 45-45-90 triangle has two equal legs, so its sine and cosine at 45 degrees are equal. The 30-60-90 triangle has sides in the ratio 1 : √3 : 2, which gives the values at 30 and 60 degrees. Memorizing these values pays off quickly in exams and in quick estimates.

Using the ratios to find unknowns

Trigonometry solves two types of problem. In the first, you know an angle and one side and need another side. In the second, you know two sides and need an angle.

If a ramp rises 2 metres over a horizontal distance of 10 metres, the angle of the ramp satisfies tan θ = 2 / 10 = 0.2. Then θ = arctan(0.2), which is approximately 11.3 degrees. The inverse function undoes the ratio and returns the angle.

If you know the angle is 30 degrees and the hypotenuse is 20 metres, the opposite side is 20 × sin 30° = 20 × 0.5 = 10 metres. Multiplying by the ratio gives the side length directly.

Extending beyond acute angles

The definitions above work for acute angles in a triangle. To handle any angle, including those larger than 90 degrees and negative angles, place the angle at the centre of a circle of radius 1, called the unit circle. Start at the point (1, 0) and rotate counterclockwise by θ.

The endpoint of the rotation has coordinates (cos θ, sin θ). This makes the cosine the horizontal coordinate and the sine the vertical coordinate. The tangent is the sine divided by the cosine, which equals the slope of the ray from the origin.

This view explains why sine and cosine are periodic. Rotating by a full turn of 360 degrees returns to the same point, so the values repeat. It also shows why cosine is positive in the first and fourth quadrants and sine is positive in the first and second.

Why the functions matter

Sine and cosine describe many repeating motions. A pendulum, a vibrating string, an alternating current, and the sound wave from a piano note all follow sinusoidal patterns. Engineers use these functions to analyse signals, design bridges, and model seasonal change in data.

The shift from triangles to the unit circle is the step that turns trigonometry from a tool for measuring into a language for periodic phenomena. Later courses use radians instead of degrees, because the calculus of these functions is simplest in that unit.

Common mistakes

The most frequent error is using a calculator in the wrong mode. Many calculators have a setting for degrees and another for radians. If your answer for sin 30 is not 0.5, check the mode before doing anything else.

Another common error is mixing up opposite and adjacent. Always identify the angle first, then label the sides from that angle. Writing the labels on the diagram prevents confusion.

A third mistake is treating sin² θ as sin(θ²). The notation sin² θ means (sin θ) squared, not the sine of the squared angle.

A fourth error is rounding too early. Keep full precision through the calculation and round only the final answer. Rounding the sine of 30 degrees to 0.5 is exact, but rounding an intermediate value such as tan 38° can shift the final answer noticeably once the sides are scaled up to real measurements.

Key takeaways

Sine, cosine, and tangent are ratios of two sides of a right triangle relative to an acute angle. Sine uses opposite over hypotenuse, cosine uses adjacent over hypotenuse, and tangent uses opposite over adjacent.

Learn the exact values at 0, 30, 45, 60, and 90 degrees. They come from two special triangles and make many problems fast to solve.

On the unit circle, cosine and sine give the coordinates of a rotating point. This extension explains periodicity and connects trigonometry to waves and rotation.