Introduction to Trigonometry

Take a right triangle with the right angle at B, and focus on one acute angle A. With respect to angle A the three sides get special names:

• The side opposite to A is the perpendicular (here BC).
• The side next to A (other than the hypotenuse) is the base or adjacent side (here AB).
• The side opposite the right angle is always the hypotenuse (here AC), the longest side.

The three basic trigonometric ratios of angle A are defined as:

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

These ratios depend only on the size of the angle, not on how big the triangle is — a key idea that follows from similar triangles. Because each side is positive and the hypotenuse is the longest side, both sin A and cos A always lie between 0 and 1 for an acute angle.

Worked example. Suppose the triangle has BC = 3, AB = 4 and AC = 5 (note 3² + 4² = 5², so it is right-angled at B). For angle A:

• sin A = BC / AC = 3/5
• cos A = AB / AC = 4/5
• tan A = BC / AB = 3/4

Notice that tan A also equals sin A / cos A = (3/5) ÷ (4/5) = 3/4, which is true for every angle.

Every right triangle gives three more ratios, defined as reciprocals of the first three:

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

It also follows directly from the definitions that:

• cot A = cos A / sin A (the reciprocal of tan A = sin A / cos A)

A useful memory aid is to pair each ratio with its reciprocal: sin ↔ cosec, cos ↔ sec, tan ↔ cot. Since sin A and cos A are at most 1 for an acute angle, their reciprocals cosec A and sec A are always at least 1.

Worked example. Using the 3-4-5 triangle from before, where sin A = 3/5, cos A = 4/5 and tan A = 3/4, the reciprocal ratios are:

• cosec A = 5/3
• sec A = 5/4
• cot A = 4/3

Check: sin A × cosec A = (3/5) × (5/3) = 1, exactly as a ratio and its reciprocal should.

These are the standard angles. Their sine values follow an easy pattern: √0/2, √1/2, √2/2, √3/2, √4/2 for 0°, 30°, 45°, 60°, 90°, which simplify to 0, 1/2, 1/√2, √3/2, 1. The cosine values are the same list read backwards.

The key results are:

• sin: 0, 1/2, 1/√2, √3/2, 1
• cos: 1, √3/2, 1/√2, 1/2, 0
• tan: 0, 1/√3, 1, √3, and tan 90° is not defined

tan 90° is undefined because cos 90° = 0, and dividing by zero is not allowed. Similarly cosec 0°, cot 0° and sec 90° are not defined.

Worked example. Evaluate sin 30° cos 60° + cos 30° sin 60°.

1. Substitute values: (1/2)(1/2) + (√3/2)(√3/2).
2. Compute each term: 1/4 + 3/4.
3. Add: 1/4 + 3/4 = 1.

So the expression equals 1. A second quick check: tan 45° = sin 45° / cos 45° = (1/√2) ÷ (1/√2) = 1, matching the table.

An identity is an equation that is true for all values of the angle for which both sides are defined. The fundamental identity comes straight from the Pythagoras theorem applied to a right triangle:

• sin²θ + cos²θ = 1 (true for all θ)

Dividing this identity through by cos²θ gives a second identity, and dividing by sin²θ gives a third:

• 1 + tan²θ = sec²θ
• 1 + cot²θ = cosec²θ

Here 1 + tan²θ = sec²θ holds for 0° ≤ θ < 90°, and 1 + cot²θ = cosec²θ holds for 0° < θ ≤ 90° (where the ratios are defined).

Worked example 1 — finding a value. If sin A = 3/5, find cos A. Using sin²A + cos²A = 1:

1. cos²A = 1 − sin²A = 1 − 9/25 = 16/25.
2. cos A = √(16/25) = 4/5 (positive, since A is acute).

Worked example 2 — a proof. Prove that 1 − cos²θ = sin²θ. Starting from sin²θ + cos²θ = 1, subtract cos²θ from both sides to get sin²θ = 1 − cos²θ. Hence 1 − cos²θ equals sin²θ, as required.

Two acute angles are complementary if their sum is 90°. In a right triangle the two acute angles A and (90° − A) are always complementary. Comparing their ratios gives these results:

• sin(90° − A) = cos A
• cos(90° − A) = sin A
• tan(90° − A) = cot A
• cot(90° − A) = tan A
• sec(90° − A) = cosec A
• cosec(90° − A) = sec A

The pattern is simple: replace each ratio by its co-ratio (sine ↔ cosine, tangent ↔ cotangent, secant ↔ cosecant). This is exactly why cosine, cotangent and cosecant carry the prefix "co" — they are the ratios of the complementary angle.

Worked example 1. Evaluate sin 25° / cos 65°. Since 65° = 90° − 25°, we have cos 65° = cos(90° − 25°) = sin 25°. So sin 25° / cos 65° = sin 25° / sin 25° = 1.

Worked example 2. Evaluate tan 48° tan 23° tan 42° tan 67°. Pair the complementary angles: tan 48° = tan(90° − 42°) = cot 42°, so tan 48° × tan 42° = cot 42° × tan 42° = 1. Likewise tan 67° = cot 23°, so tan 23° × tan 67° = 1. The whole product is 1 × 1 = 1.