Trigonometry — I
Easy Overview
Imagine you're on a Ferris wheel. As you go up, come down, and go around, your height above the ground keeps changing in a smooth, wavy pattern. That's trigonometry — it's the maths of cycles, angles, and triangles, but really it's about how things go up and down and around.
The six trigonometric functions
Meet sin, cos, tan, cosec, sec, and cot. At their heart, they're just ratios of sides of a right-angled triangle. For an angle θ: sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = sin/cos. The other three are just reciprocals — cosec = 1/sin, sec = 1/cos, cot = 1/tan. That's it. Nothing more.
The unit circle — your best friend
Draw a circle of radius 1 at the center of a graph. Pick any point on it. The x-coordinate is cos θ and the y-coordinate is sin θ. That's the whole idea. As you walk around the circle, sin and cos trace out smooth waves. It connects triangles to circles, and circles to waves. This one picture explains pretty much everything in trig.
ASTC rule — sign of trig functions
In which quadrants are sin, cos, and tan positive? There's a stupid-simple trick. Draw four quadrants and label 'em: A (all positive), S (sin positive), T (tan positive), C (cos positive). Start from the first quadrant and go anti-clockwise — A, S, T, C. That's the ASTC rule. In Q1 everything's happy. In Q2 only sin is. In Q3 tan survives. In Q4 cos is the lone positive one.
Trigonometric identities you can't escape
The big daddy: sin²θ + cos²θ = 1. It's just Pythagoras on the unit circle. Divide it by cos²θ and you get 1 + tan²θ = sec²θ. Divide by sin²θ and you get cot²θ + 1 = cosec²θ. These three show up everywhere — in proving stuff, in solving equations, in integration later. Just memorise them.
Key Points
- •sin = opposite/hypotenuse, cos = adjacent/hypotenuse
- •tan = sin/cos, cot = cos/sin
- •Unit circle: (cos θ, sin θ) for any angle θ
- •ASTC rule — All, Sin, Tan, Cos (anti-clockwise from Q1)
- •sin²θ + cos²θ = 1 — the master identity
- •1 + tan²θ = sec²θ and 1 + cot²θ = cosec²θ
- •Trig functions are periodic — sin and cos repeat every 360°
Practice Questions
- Prove that sin⁴θ − cos⁴θ = sin²θ − cos²θ.
- If sin θ = 3/5 and θ lies in Q2, find cos θ and tan θ.
- Prove that (1 + cot θ − cosec θ)(1 + tan θ + sec θ) = 2.
- Express sin 150° and cos 240° in terms of acute angles and find their values.
- If tan θ = 4/3 and θ is in Q3, find all six trigonometric ratios.