← Mathematics — Std 11
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Trigonometry — I

Ch. 2Std 11

Easy Overview

Trigonometry. The word itself sounds intimidating — but break it down and it's just 'trigon' (triangle) + 'metron' (measure). Triangle measuring. And at its core, that's exactly what it is. But somewhere along the way, trigonometry expands from measuring triangles to describing cycles, waves, and anything that repeats. Sound, light, seasons, heartbeats — they're all described by trig functions. Let's start with a right triangle. You've got three sides: opposite (across from the angle), adjacent (next to the angle), and hypotenuse (the long one). Every ratio of two sides gives you a trig function. The three primary ones are sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, and tan θ = opposite/adjacent. Then the reciprocals: cosec θ = 1/sin, sec θ = 1/cos, and cot θ = 1/tan. That's six functions from three ratios. Not so scary after all. But triangles only get you so far. What about an angle of 120°? You can't have a right triangle with a 120° angle (a triangle's angles sum to 180°, so a right triangle is at most 90°+something). That's where the unit circle comes in. Draw a circle of radius 1 centered at the origin. For any angle θ, draw a line from the origin at that angle — it hits the circle at a point (x, y). The x-coordinate is cos θ, and the y-coordinate is sin θ. This single idea extends trigonometry to all angles — positive, negative, any size. The unit circle also makes trig identities obvious. The Pythagorean identity sin²θ + cos²θ = 1 is just Pythagoras' theorem on a right triangle with hypotenuse 1. The ASTC rule (All, Sin, Tan, Cos) tells you which functions are positive in each quadrant, working anti-clockwise from Quadrant I where everything's positive. We'll also cover the trigonometric ratios of standard angles — the ones you should just know because they give nice exact values. You'll learn how trig functions behave with negative angles, with angles greater than 360°, and with allied angles like (180° − θ) and (90° + θ). By the end, you'll be able to find any trig ratio for any angle, simplify expressions using identities, and understand the graphs of sin, cos, and tan. This is the foundation for Trigonometry 2, where things get compound and exciting.

The six trigonometric ratios — the full family

In any right-angled triangle with angle θ: sin θ = opposite/hypotenuse (O/H), cos θ = adjacent/hypotenuse (A/H), tan θ = opposite/adjacent (O/A). Their reciprocals: cosec θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ. The mnemonic 'SOH CAH TOA' helps you remember the first three. For the reciprocals, just remember 'cosec is 1 over sin' and so on.

The unit circle — the single best tool in trig

Centre a circle of radius 1 at (0,0). For any angle θ, the point where the terminal ray meets the circle has coordinates (cos θ, sin θ). This means cos is the x-coordinate and sin is the y-coordinate. The unit circle lets you define trig functions for any angle — not just acute ones. It also shows you the range: cos and sin are always between −1 and 1.

ASTC rule — who's positive where?

The four quadrants of the coordinate plane each have a different sign pattern for trig functions. ASTC stands for All, Sin, Tan, Cos — reading anti-clockwise from Quadrant I. QI (0°−90°): all six functions are positive. QII (90°−180°): sin and cosec are positive. QIII (180°−270°): tan and cot are positive. QIV (270°−360°): cos and sec are positive. 'All Students Take Calculus' is a common mnemonic.

Trig ratios of standard angles — the exact values

Some angles give you clean, exact trig values. 0°: sin=0, cos=1, tan=0. 30° (π/6): sin=½, cos=√3/2, tan=1/√3. 45° (π/4): sin=1/√2, cos=1/√2, tan=1. 60° (π/3): sin=√3/2, cos=½, tan=√3. 90° (π/2): sin=1, cos=0, tan=undefined. These come from simple right triangles (30-60-90 and 45-45-90) and memorising them saves you time in exams.

Trigonometric ratios of complementary angles

Two angles that add up to 90° (π/2) are complementary. The key relationship: sin(90° − θ) = cos θ, cos(90° − θ) = sin θ, tan(90° − θ) = cot θ. For reciprocals: cosec(90° − θ) = sec θ, sec(90° − θ) = cosec θ, cot(90° − θ) = tan θ. These are useful for simplifying expressions and proving identities.

Allied angles — how trig functions transform

Angles like (90° ± θ), (180° ± θ), (270° ± θ), and (360° ± θ) are called allied angles. There are systematic rules: the function changes name when the multiple is 90° or 270° (sin ↔ cos, tan ↔ cot, sec ↔ cosec) and keeps the same name for 180° and 360°. The sign depends on the ASTC rule for the resulting quadrant.

The Pythagorean identities — the holy trinity

The fundamental identity: sin²θ + cos²θ = 1. Divide by cos²θ: tan²θ + 1 = sec²θ. Divide by sin²θ: 1 + cot²θ = cosec²θ. These three identities are the most used in the entire subject. Every trig simplification, every proof, every equation — you'll reach for these constantly. Memorise them like your name.

Periodicity of trigonometric functions

Trig functions repeat. sin(θ + 360°) = sin θ, and the same for cos. So sin and cos have a period of 360° (2π). Tan has a smaller period: tan(θ + 180°) = tan θ, so its period is 180° (π). This periodic nature reflects rotational symmetry — going around the circle again brings you back to the same point.

Even and odd trig functions

Cos is an even function: cos(−θ) = cos θ (symmetric about the y-axis). Sin and tan are odd: sin(−θ) = −sin θ and tan(−θ) = −tan θ (symmetric about the origin). These properties help simplify expressions with negative angles and are useful in integration later.

Domain and range of trig functions

Sin and cos have domain ℝ (all real numbers) and range [−1, 1]. Tan has domain ℝ except odd multiples of 90° (where cos = 0) and range ℝ. Cosec and sec have range (−∞, −1] ∪ [1, ∞). Cot has domain ℝ except multiples of 180° and range ℝ. Understanding these helps avoid undefined operations.

Graphs of sin, cos, and tan

The graph of sin x is a smooth wave starting at 0, peaking at 1 at 90°, back to 0 at 180°, trough at −1 at 270°, back to 0 at 360°. Cos x is the same wave shifted left by 90°. Tan x has vertical asymptotes at odd multiples of 90° and rises from −∞ to +∞ between them. These graphs visualise everything you need to know about amplitude, period, and frequency.

Maximum and minimum values of trig expressions

Since sin and cos are bounded between −1 and 1, expressions like a sin θ + b cos θ have max/min of ±√(a² + b²). This is a standard technique: write it as R sin(θ + α) or R cos(θ + α). For example, 3 sin θ + 4 cos θ has max value 5 (since √(3²+4²) = 5). This is useful in optimisation problems in physics and engineering.

Solving simple trigonometric equations

Basic trig equations like sin θ = ½, cos θ = 0, or tan θ = 1 can be solved using standard angles and ASTC. Sin θ = ½ gives θ = 30° and 150° (in 0°−360°). Cos θ = 0 gives θ = 90° and 270°. Tan θ = 1 gives θ = 45° and 225°. Multiple solutions exist because trig functions are periodic.

Trigonometric ratios in terms of sides of a triangle

Given the sides of a right triangle, you can find all six trig ratios for either acute angle. If the opposite side is 3, adjacent is 4, and hypotenuse is 5, then sin = 3/5, cos = 4/5, tan = 3/4, and so on. This is the most direct application of trig — given a triangle, find the ratios.

Finding all ratios from one given ratio

If sin θ = 3/5 and θ is in QII, you can find cos θ using the Pythagorean identity: cos²θ = 1 − sin²θ = 1 − 9/25 = 16/25, so cos θ = ±4/5. In QII, cos is negative, so cos θ = −4/5. Then tan θ = sin θ/cos θ = −3/4. This 'given one ratio, find the rest' is a classic problem type.

Areas of triangles using trig

The area of any triangle can be found using two sides and the included angle: Area = ½ab sin C. When C = 90°, sin C = 1, and this reduces to the familiar ½ × base × height. This formula connects trigonometry back to geometry and is essential for solving non-right triangles.

Key Points

  • •SOH CAH TOA: sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent
  • •Reciprocal ratios: cosec = 1/sin, sec = 1/cos, cot = 1/tan
  • •Unit circle: (cos θ, sin θ) for any angle θ on a circle of radius 1
  • •ASTC rule (anti-clockwise from Q1): All positive, Sin positive, Tan positive, Cos positive
  • •sin²θ + cos²θ = 1 — the master Pythagorean identity
  • •1 + tan²θ = sec²θ and 1 + cot²θ = cosec²θ — the derived forms
  • •sin(90° − θ) = cos θ, cos(90° − θ) = sin θ — complementary angle identities
  • •Sin and cos have period 360° (2π); tan has period 180° (π)
  • •cos(−θ) = cos θ (even); sin(−θ) = −sin θ, tan(−θ) = −tan θ (odd)
  • •Sin and cos range in [−1, 1]; tan ranges over all ℝ
  • •Standard angle values: 0°, 30°, 45°, 60°, 90° — memorise the exact values
  • •Allied angle rules: function changes for 90° and 270°, stays same for 180° and 360°
  • •Maximum of a sin θ + b cos θ = √(a² + b²)
  • •Area of triangle = ½ab sin C

Practice Questions

  • If sin θ = 5/13 and θ lies in Quadrant II, find cos θ, tan θ, and the remaining three trigonometric ratios.
  • Prove that sin⁴θ − cos⁴θ = sin²θ − cos²θ using the Pythagorean identity.
  • Prove that (1 + cot θ − cosec θ)(1 + tan θ + sec θ) = 2.
  • Find the values of sin 150°, cos 240°, and tan 315° using allied angle formulas.
  • If tan θ = 4/3 and θ is in QIII, find all six trigonometric ratios.
  • Express sin 75° and cos 15° in terms of standard angles and find their exact values.
  • Prove that sec²θ + cosec²θ = sec²θ·cosec²θ.
  • Find the maximum value of 5 cos θ + 12 sin θ and the angle at which it occurs.