Trigonometric Functions: Radians, the Unit Circle & Identities Made Simple

Stop memorising blindly — understand angles as radians, meet sine and cosine on the unit circle, master the sign of each ratio in every quadrant, and see exactly where the key identities come from.

By the PadhoDost Team · 📖 7 min read · Updated 4 August 2026

Part of Class 11 (CBSE) prep

🧠 The running-track secret

Imagine you are jogging on a circular stadium track, Dost. Instead of asking "how many degrees have I turned?", ask "how many radius-lengths of track have I run?" That second question is exactly what a radian measures. Angles were always secretly about the distance you travel around a circle — degrees just hid it from you. Once you see angles this way, all of trigonometry starts to click.

Radians: A Smarter Way to Measure Angles

You already know degrees: a full circle is 360°. But 360 is a leftover from ancient Babylonian astronomy — there is nothing natural about it. A radian is built from the circle itself. Draw an arc whose length equals the radius; the angle that arc makes at the centre is exactly 1 radian. In general, angle (in radians) = arc length ÷ radius, or θ = l/r. Because it is a ratio of two lengths, a radian has no unit at all — it is a pure number. Since the full circumference is 2πr, going all the way around is 2π radians, which must equal 360° (so π radians = 180°).

π radians = 180° → degrees × (π/180) = radians, radians × (180/π) = degrees. Taking π ≈ 22/7: 1 radian ≈ 57°16′ and 1° ≈ 0.01746 rad. Arc length: l = r·θ (θ in radians).

📝 Worked example: converting and using radians

Part A — Convert 150° into radians.

Multiply by π/180: 150 × (π/180) = 150π/180.

Simplify the fraction: = 5π/6 radians ≈ 2.62 rad.

Part B — A circular park has radius r = 5 m. Find the arc length for a central angle of 60°.

First convert the angle: 60° = 60 × (π/180) = π/3 rad.

Use l = r·θ (θ must be in radians): l = 5 × (π/3) = 5π/3.

So l ≈ 5.24 m.

Note the units: radian is unit-less (arc ÷ radius), so l keeps the unit of r — metres.

The Unit Circle: Where Sine and Cosine Live

In earlier classes, sin and cos came from right triangles — but a triangle can't handle a 210° angle. The unit circle fixes this. Draw a circle of radius 1 centred at the origin. Start from the positive x-axis and rotate anticlockwise by angle θ (clockwise is negative). Wherever you land is a point P with coordinates (x, y). We simply define cos θ = x and sin θ = y. That is it. This definition works for any angle — big, small, or negative. And because P lies on the circle, the Pythagoras theorem gives x² + y² = 1, which is the same as sin²θ + cos²θ = 1. The identity is not a magic formula to cram; it is just Pythagoras in disguise.

On the unit circle at angle θ: cos θ = x, sin θ = y, tan θ = y/x = sin θ/cos θ (x ≠ 0). Reciprocals: cosec θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ.
💡 Since x and y are coordinates on a circle of radius 1, they can never be bigger than 1 or smaller than −1. So sin θ and cos θ always stay between −1 and +1. If a calculation ever gives you sin θ = 1.5, stop — a mistake has crept in somewhere.

Signs of the Ratios in the Four Quadrants

As θ grows, point P moves through four quadrants, and the signs of x and y change — so the signs of the ratios change too. The famous memory hook is "Add Sugar To Coffee" (or "All Students Take Coffee"): in Quadrant I All ratios are positive, in Quadrant II only Sine (and its reciprocal cosec), in Quadrant III only Tangent (and cot), and in Quadrant IV only Cosine (and sec). A reciprocal ratio always carries the same sign as its parent (sec follows cos, cosec follows sin, cot follows tan).

QuadrantAngle rangePositive ratiosNegative ratios
I0° to 90°All (sin, cos, tan)None
II90° to 180°sin, coseccos, tan, sec, cot
III180° to 270°tan, cotsin, cos, sec, cosec
IV270° to 360°cos, secsin, tan, cosec, cot
⚠️ Three classic slips to avoid: (1) sin²θ means (sin θ)², NOT sin(θ²). (2) sin(A + B) is NOT sin A + sin B — sine does not split across a sum. (3) Before pressing buttons, check whether your calculator is in DEGREE or RADIAN mode; the wrong mode gives wildly wrong answers.

The Three Fundamental Identities

From x² + y² = 1 on the unit circle: sin²θ + cos²θ = 1. Divide that by cos²θ → 1 + tan²θ = sec²θ. Divide it by sin²θ → 1 + cot²θ = cosec²θ. (All three come from one idea — Pythagoras.)

📝 Worked example: using an identity with quadrant signs

Given: cos θ = −3/5 and θ lies in the second quadrant. Find sin θ and tan θ.

Use the identity: sin²θ + cos²θ = 1.

sin²θ = 1 − cos²θ = 1 − (−3/5)² = 1 − 9/25 = 16/25.

Take the square root: sin θ = ±4/5.

Choose the sign using the quadrant: in Quadrant II sine is positive, so sin θ = +4/5.

Now tan θ = sin θ / cos θ = (4/5) ÷ (−3/5) = −4/3.

Answer: sin θ = 4/5 and tan θ = −4/3 (negative, as expected in Quadrant II).

Quick revision checklist

  • ✓Radian ties an angle to arc length: θ = l/r, and it has no unit. π rad = 180°.
  • ✓To convert: degrees × π/180 = radians; radians × 180/π = degrees.
  • ✓On the unit circle: cos θ = x-coordinate, sin θ = y-coordinate; both stay within [−1, 1].
  • ✓Signs by quadrant: All, Sine, Tangent, Cosine positive (Add Sugar To Coffee).
  • ✓The three identities all flow from x² + y² = 1: sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, 1 + cot²θ = cosec²θ.
  • ✓When square-rooting to find a ratio, always fix the sign using the quadrant.

⚡ Quick check

The point for angle θ on the unit circle lies in the third quadrant. Which one of these ratios is positive?

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