Right Triangles & Trigonometry

Sine, Cosine & Tangent

Defining and applying the three basic trigonometric ratios in a right triangle.

Advanced20 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

The Pythagorean Theorem

Why a Ratio Can Depend on an Angle Alone

Draw two different right triangles, both containing a 30° angle, but at completely different sizes. Before reading on, think back to AA similarity: since both triangles share the right angle and the 30° angle, what must be true about the ratio of any two corresponding sides between them?

By AA similarity, every right triangle containing a 30° angle is similar to every other one — meaning their corresponding sides all share the same ratio, regardless of the triangle's actual size. This is the entire reason a ratio like 'opposite side over hypotenuse' can be treated as a fixed number attached to an angle alone, rather than something that depends on which specific triangle you happened to draw.

Definition — Sine, Cosine, and Tangent

For an acute angle θ in a right triangle: sine (sin θ) = opposite/hypotenuse, cosine (cos θ) = adjacent/hypotenuse, and tangent (tan θ) = opposite/adjacent — where 'opposite' and 'adjacent' are measured relative to angle θ specifically, not fixed to any particular side of the triangle.

Worked Example — Finding Trig Ratios from a Right Triangle

A right triangle has a leg of 3 (opposite angle θ), a leg of 4 (adjacent to θ), and a hypotenuse of 5. Find sin θ, cos θ, and tan θ. sin θ = 3/5, cos θ = 4/5, tan θ = 3/4.

Worked Example — Finding a Missing Side Using a Trig Ratio

A right triangle has a 35° angle and a hypotenuse of 20. Find the side opposite the 35° angle. Since sine relates the opposite side and hypotenuse: sin 35° = opposite/20. Solving: opposite = 20 × sin 35° ≈ 20 × 0.574 ≈ 11.5.

Tip

Relabel 'opposite' and 'adjacent' fresh for each angle you're working with in a triangle — the same leg is 'opposite' relative to one acute angle but 'adjacent' relative to the other, since these labels depend entirely on which angle is currently in focus.

Common Mistakes

Mixing up which leg is 'opposite' and which is 'adjacent' relative to the angle actually being used in a calculation.
Look specifically at the angle in question — the side that touches the angle (other than the hypotenuse) is adjacent to it; the side that doesn't touch it at all is opposite it.
Applying a trig ratio to a triangle's hypotenuse as if it were one of the two legs in the ratio.
The hypotenuse is always the side opposite the right angle, never labeled 'opposite' or 'adjacent' relative to an acute angle — sine and cosine both use it specifically as the denominator.

Key Takeaways

Sine, cosine, and tangent are ratios between a right triangle's sides, depending only on a chosen acute angle, not the triangle's size.
This works because every right triangle sharing that angle is similar by AA, guaranteeing the same side ratios throughout.
'Opposite' and 'adjacent' are defined relative to whichever specific angle is currently being used.

Summary

Trigonometric ratios connect an angle to a fixed relationship between a right triangle's sides. The next lesson uses these ratios, and their inverses, to fully solve a right triangle from partial information.

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