1. Naming the sides of a right-angled triangle

Take a right-angled triangle with an angle θ (theta). Relative to θ, the three sides have fixed names:

• The hypotenuse — the longest side, always opposite the right angle (the Greek word hypoteinousa means "stretching under").
• The opposite side — the side across from θ, which does not touch θ.
• The adjacent side — the side next to θ, which, together with the hypotenuse, forms the angle.

2. The three basic ratios — SOH CAH TOA

The three trigonometric functions are defined as ratios of sides. The mnemonic SOH CAH TOA is used in every English-speaking classroom:

SOH:  sin θ = Opposite / Hypotenuse
CAH:  cos θ = Adjacent / Hypotenuse
TOA:  tan θ = Opposite / Adjacent

Read each line: "sine theta equals opposite over hypotenuse", "cosine theta equals adjacent over hypotenuse", "tangent theta equals opposite over adjacent". The word "over" means division.

3. Special angles worth memorising

Read them: "sine of thirty degrees is one half", "cosine of forty-five degrees is root two over two", "tangent of ninety degrees is undefined". Note: in English we always say the function name first — "sine of x", not "x's sine".

4. The unit circle — trigonometry beyond triangles

Angles bigger than 90° cannot fit in a right triangle, so we extend the definitions using the unit circle: the circle of radius 1 centred at the origin. For any angle θ, draw a radius making angle θ with the positive x-axis. Then:

cos θ = x-coordinate,   sin θ = y-coordinate,   tan θ = y / x

Read it: "On the unit circle, cosine theta is the x-coordinate and sine theta is the y-coordinate."

5. The graphs of sine and cosine

Plotting sin x and cos x against x reveals that both are periodic — they repeat every 2π (360°). Read it: "Sine and cosine are periodic functions with period two pi."

• The amplitude is the maximum distance from the middle line — for sin x and cos x it is 1.
• The period is the length of one full cycle — for sin x and cos x it is 2π.
• cos x is sin x shifted to the left by π/2: read it, "cosine x equals sine of x plus pi over two."

6. The sine rule and cosine rule

For triangles that are not right-angled, we need two powerful tools. In any triangle with sides a, b, c opposite angles A, B, C:

The sine rule (正弦定理):

a / sin A = b / sin B = c / sin C

Read it: "a over sine A equals b over sine B equals c over sine C."

The cosine rule (余弦定理):

c² = a² + b² − 2ab cos C

Read it: "c squared equals a squared plus b squared minus two a b cosine C." Note that when C = 90°, cos C = 0, and the cosine rule becomes Pythagoras' theorem — read it: "the cosine rule generalises Pythagoras' theorem."

7. Where trigonometry appears in real life

• Navigation — ships and aircraft use trigonometry to compute bearings and distances.
• Waves and sound — sound, light and radio waves are described by sine functions; read it: "A wave can be modelled by y = A sin(2πft + φ)."
• Physics — resolving a force into components uses sin and cos; read it: "The horizontal component is F cosine theta; the vertical component is F sine theta."
• Architecture — calculating roof slopes and support angles.

8. Worked example — full English solution

Problem. In a right-angled triangle, the hypotenuse is 10 and one angle is 30°. Find the opposite side.

Solution. Read it aloud:

sin 30° = opposite / hypotenuse
opposite = 10 · sin 30° = 10 · 1/2 = 5

"Sine of thirty degrees equals opposite over hypotenuse. Therefore the opposite side equals ten times sine thirty degrees, which is ten times one half, which is five." — the phrase "therefore" and the verb "equals" are the skeleton of every English maths solution.

9. Quick check — test yourself

1. In a right triangle, cos θ = 0.6. What is sin θ, given the identity sin²θ + cos²θ = 1?
2. What is the period and amplitude of y = 3 sin(2x)?
3. The sine rule relates which pairs of quantities?
4. Translate into English: "正切函数在九十度处无定义。"

Answers: 1. sin θ = √(1 − 0.36) = √0.64 = 0.8. 2. Amplitude 3, period π (because period = 2π/2). 3. Each side with the sine of its opposite angle. 4. "The tangent function is undefined at ninety degrees."

10. Vocabulary recap