Trigonometric Ratios: Sine, Cosine, and Tangent

In any right triangle, the three sides have special names relative to an acute angle: the opposite side (the side facing the angle), the adjacent side (the side that touches the angle but is not the hypotenuse), and the hypotenuse (the longest side, always opposite the 90° right angle).

Trigonometric ratios are simply divisions between these sides. The sine of an angle θ is the quotient of the opposite side and the hypotenuse. The cosine is the quotient of the adjacent side and the hypotenuse. The tangent is the quotient of the opposite side and the adjacent side.

These relationships are constant for a given angle, regardless of the size of the triangle. A triangle with a 30° angle and a hypotenuse of 10 cm has an opposite side of 5 cm. Another with a hypotenuse of 20 cm has an opposite side of 10 cm. The ratio — the sine — is always 0.5.

SOH-CAH-TOA is the most widely used mnemonic in the world for memorizing the three ratios:

SOH: Sine = Opposite / Hypotenuse. CAH: Cosine = Adjacent / Hypotenuse. TOA: Tangent = Opposite / Adjacent.

A good way to remember it is to say it aloud while pointing to the corresponding sides on a drawn triangle. After a few repetitions, the relationship becomes automatic.

It is worth noting that tangent can be derived from the other two: tan θ = sin θ / cos θ. So ultimately, sine and cosine are the fundamental ratios; tangent is a direct consequence of them.

When angle θ approaches 0°, the opposite side shrinks and the adjacent side almost equals the hypotenuse. That is why sin 0° = 0 and cos 0° = 1. As θ grows toward 90°, the opposite side grows and the adjacent shrinks: sin 90° = 1 and cos 90° = 0.

The tangent grows rapidly: tan 0° = 0, tan 45° = 1, and tan 89° already exceeds 57. At exactly 90° the tangent is undefined, because the adjacent side becomes zero and we cannot divide by zero.

Complementary angles share a beautiful relationship: sin θ = cos(90° − θ). The sine of 30° equals the cosine of 60°, both equal to 0.5. This makes geometric sense: swapping the roles of opposite and adjacent is equivalent to looking at the complementary angle.

When you know an acute angle and one side, you can calculate any other side. Example: a roof makes a 35° angle with the horizontal and has a length (hypotenuse) of 8 m. The vertical height is: height = 8 × sin 35° ≈ 8 × 0.574 ≈ 4.59 m.

When you know two sides and want the angle, use the inverse function. If the opposite side is 3 and the hypotenuse is 5, then sin θ = 3/5 = 0.6, so θ = arcsin(0.6) ≈ 36.87°.

A scientific calculator has the sin, cos, tan keys and their inverses sin⁻¹ (or arcsin), cos⁻¹, and tan⁻¹. Make sure to set it to degrees (DEG) or radians (RAD) as the problem requires.

The right triangle works well for angles between 0° and 90°, but what about larger or negative angles? This is where the unit circle comes in — a circle of radius 1 centered at the origin of the Cartesian plane.

For any angle θ, we draw a ray forming that angle with the positive x-axis. The point where the ray meets the circle has coordinates (cos θ, sin θ). This allows the trigonometric ratios to be extended to all real angles.

For now, it is enough to know that the unit circle is the natural generalization of the right-triangle ratios. When we study the trigonometric cycle, we will explore this concept in depth.