Naming the sides
Trigonometry starts with a right-angled triangle. Pick one of the two acute angles and call it θ (theta). Relative to θ:
the hypotenuse is the side opposite the right angle (always the longest side),
the opposite side is the side facing θ,
the adjacent side is the side next to θ that is not the hypotenuse.
If you switch to the other acute angle, the opposite and adjacent sides swap. The hypotenuse stays the same.
The six trigonometric ratios
Ratio | Definition |
|---|---|
sin θ | opposite / hypotenuse |
cos θ | adjacent / hypotenuse |
tan θ | opposite / adjacent |
cosec θ | hypotenuse / opposite = 1 / sin θ |
sec θ | hypotenuse / adjacent = 1 / cos θ |
cot θ | adjacent / opposite = 1 / tan θ |
A useful consequence: tan θ = sin θ / cos θ.
These ratios depend only on the angle, not on the size of the triangle. Two right triangles with the same acute angle are similar, so their side ratios are equal.
Example. In a right triangle ABC, right-angled at B, AB = 5 cm and BC = 12 cm. Find sin A and cos A.
First find the hypotenuse: AC = √(5² + 12²) = √169 = 13 cm. Relative to angle A, the opposite side is BC = 12 and the adjacent side is AB = 5. So sin A = 12/13 and cos A = 5/13.
Values you should know
θ | 0° | 30° | 45° | 60° | 90° |
|---|---|---|---|---|---|
sin θ | 0 | 1/2 | 1/√2 | √3/2 | 1 |
cos θ | 1 | √3/2 | 1/√2 | 1/2 | 0 |
tan θ | 0 | 1/√3 | 1 | √3 | not defined |
A quick way to remember the sine row: write √0/2, √1/2, √2/2, √3/2, √4/2 and simplify. The cosine row is the same list in reverse order.
tan 90° is not defined because cos 90° = 0, and you cannot divide by zero.
Example. Evaluate 2 sin 30° cos 60° + tan 45°.
= 2 × (1/2) × (1/2) + 1 = 1/2 + 1 = 3/2.
Ratios of complementary angles
In a right triangle the two acute angles add up to 90°, so they are complementary. The side that is opposite one angle is adjacent to the other, which gives:
sin(90° − θ) = cos θ, cos(90° − θ) = sin θ, tan(90° − θ) = cot θ
Example. Evaluate sin 18° / cos 72°.
Since 18° + 72° = 90°, cos 72° = sin 18°. The value is 1.
The fundamental identity
Apply Pythagoras to the triangle: opposite² + adjacent² = hypotenuse². Divide every term by hypotenuse² and you get
sin²θ + cos²θ = 1
Dividing the same Pythagoras relation by adjacent² or by opposite² instead gives two more identities:
1 + tan²θ = sec²θ and 1 + cot²θ = cosec²θ
Example. If sin θ = 3/5 and θ is acute, find cos θ and tan θ.
cos²θ = 1 − 9/25 = 16/25, so cos θ = 4/5 (positive, since θ is acute). Then tan θ = (3/5) ÷ (4/5) = 3/4.
How to prove an identity
Start from the more complicated side.
Rewrite everything in terms of sin and cos if you are stuck.
Use sin²θ + cos²θ = 1 to simplify.
Never move terms across the equals sign as if you were solving an equation. Work on one side until it matches the other.
Quick summary
sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent.
Learn the 0°, 30°, 45°, 60°, 90° table by heart.
sin(90° − θ) = cos θ for complementary angles.
sin²θ + cos²θ = 1 is the identity behind most proofs.