Why Trigonometry matters in everyday life
Ever wondered how engineers calculate the height of a skyscraper without climbing it? They use trigonometric functions – the same tools you’ll meet in ISC Maths.
💡 In Simple Words: Trigonometric functions tell us how the sides of a right‑angled triangle change as the angle grows. Think of the angle as the dial on a water tap and the side lengths as the water flow – turn the dial, and the flow changes predictably.
Basic Trigonometric Functions
In a right‑angled triangle, the three primary ratios are:
- sin θ (sine) = opposite side ÷ hypotenuse
- cos θ (cosine) = adjacent side ÷ hypotenuse
- tan θ (tangent) = opposite side ÷ adjacent side
Imagine the triangle as a slide in a playground. The hypotenuse is the slide itself, the opposite side is the vertical drop, and the adjacent side is the horizontal run.
Reciprocal Functions
The other three functions are just the “flipped” versions of the above:
| Function | Definition |
|---|---|
| csc θ (cosecant) | 1 ÷ sin θ |
| sec θ (secant) | 1 ÷ cos θ |
| cot θ (cotangent) | 1 ÷ tan θ |
Fundamental Trigonometric Identities
Identities are equations that are always true, no matter what angle you plug in. They let you swap one function for another, which is handy in exams.
- Pythagorean identities: derived from the Pythagoras theorem (a² + b² = c²). The most used one is sin²θ + cos²θ = 1.
- Reciprocal identities: directly from the definitions, e.g., 1 + tan²θ = sec²θ and 1 + cot²θ = csc²θ.
- Quotient identities: relate sine and cosine to tangent and cotangent, e.g., tan θ = sin θ ÷ cos θ, cot θ = cos θ ÷ sin θ.
How to Prove a Trig Identity – A Step‑by‑Step Checklist
- Start with the side that looks more complicated.
- Replace every function using basic definitions or known identities.
- Simplify algebraically until both sides match.
- Check your work by plugging a simple angle like 0° or 30°.
Worked Example: Prove sin θ + cos θ = √2 · sin(θ + 45°)
Step 1: Write the right‑hand side using the sine addition formula (sin A + B = sin A cos B + cos A sin B).
sin(θ + 45°) = sin θ cos 45° + cos θ sin 45°. Since cos 45° = sin 45° = √2/2, the expression becomes (√2/2)(sin θ + cos θ).
Step 2: Multiply by √2:
√2 · sin(θ + 45°) = √2 · (√2/2)(sin θ + cos θ) = (sin θ + cos θ).
Both sides are identical, so the identity holds for every θ.
Quick Reference Table
| Identity | What it tells you |
|---|---|
| sin²θ + cos²θ = 1 | Links sine and cosine directly. |
| 1 + tan²θ = sec²θ | Turns a tangent into a secant. |
| 1 + cot²θ = csc²θ | Turns a cotangent into a cosecant. |
| tan θ = sin θ ÷ cos θ | Defines tangent as a ratio of sine and cosine. |
| cot θ = cos θ ÷ sin θ | Defines cotangent as a ratio of cosine and sine. |
📝 Likely Exam Questions
- Find the value of sin 30° and cos 60° without a calculator.
Answer: Both equal ½ because the 30°‑60°‑90° triangle has sides in the ratio 1 : √3 : 2. - Prove the identity 1 + tan²θ = sec²θ.
Answer: Start with tan θ = sin θ/ cos θ, square it, add 1, and use sin²θ + cos²θ = 1 to obtain sec²θ = 1/ cos²θ. - Simplify ( sin θ · csc θ ) + ( cos θ · sec θ ).
Answer: sin θ · csc θ = 1 and cos θ · sec θ = 1, so the expression equals 2. - If sin θ = 3/5 and θ is acute, find cos θ and tan θ.
Answer: cos θ = √(1 – (3/5)²) = 4/5, tan θ = (3/5)/(4/5) = 3/4. - Show that sin θ + cos θ = √2 · sin(θ + 45°).
Answer: Use the sine addition formula as demonstrated in the worked example.