What this chapter is about
This chapter extends your understanding of trigonometry from earlier classes, where you worked mainly with acute angles in right triangles, to a broader framework using any real angle measured in radians. You learn to treat trigonometric ratios as functions defined on the real number line, study their domains, ranges, graphs and periodic behaviour, and solve equations involving them.
The chapter matters now because trigonometric functions appear throughout higher mathematics, physics and engineering. Oscillations, waves, circular motion, alternating current and many natural cycles are modelled with sine and cosine. Mastering these functions here prepares you for calculus, where you will differentiate and integrate them, and for coordinate geometry of curves.
After working through the chapter you should be able to convert between degrees and radians, evaluate trigonometric functions for any standard angle, sketch their graphs, use fundamental identities to simplify expressions, and find all solutions of basic trigonometric equations within a given interval.
Key ideas
- Radian measure: One radian is the angle subtended at the centre of a circle by an arc equal in length to the radius. A full circle is 2π radians, so π radians = 180°.
- Sign convention for angles: Angles measured anticlockwise from the positive x-axis are positive; clockwise angles are negative. This lets every real number correspond to an angle.
- Trigonometric functions as ratios on the unit circle: For a point P(x, y) on the unit circle at angle θ from the positive x-axis, cos θ = x and sin θ = y. The other four functions follow: tan θ = sin θ / cos θ, cot θ = cos θ / sin θ, sec θ = 1 / cos θ, cosec θ = 1 / sin θ.
- Domain and range: sin and cos are defined for all real θ with range [−1, 1]. tan and sec are undefined where cos θ = 0. cot and cosec are undefined where sin θ = 0.
- Periodicity: sin and cos repeat every 2π; tan and cot repeat every π.
- Fundamental identity: sin²θ + cos²θ = 1 for all θ. Dividing through gives 1 + tan²θ = sec²θ and 1 + cot²θ = cosec²θ.
- Addition and subtraction formulas: These let you find exact values for sums or differences of angles, for example sin(A + B) = sin A cos B + cos A sin B.
- General solutions of trigonometric equations: Because the functions are periodic, equations have infinitely many solutions expressible in terms of an integer n.
Formulas and facts to remember
- Formula: π rad = 180° · Meaning: Conversion between radians and degrees.
- Formula: Arc length s = rθ (θ in radians) · Meaning: The arc of a circle equals radius times angle in radians.
- Formula: sin²θ + cos²θ = 1 · Meaning: Pythagorean identity; basis for the other two.
- Formula: 1 + tan²θ = sec²θ · Meaning: Derived by dividing the Pythagorean identity by cos²θ.
- Formula: 1 + cot²θ = cosec²θ · Meaning: Derived by dividing the Pythagorean identity by sin²θ.
- Formula: sin(A + B) = sin A cos B + cos A sin B · Meaning: Sine of a sum.
- Formula: cos(A + B) = cos A cos B − sin A sin B · Meaning: Cosine of a sum.
- Formula: tan(A + B) = (tan A + tan B) / (1 − tan A tan B) · Meaning: Tangent of a sum (provided the denominator ≠ 0).
- Formula: sin 2A = 2 sin A cos A · Meaning: Double-angle formula for sine.
- Formula: cos 2A = cos²A − sin²A = 2 cos²A − 1 = 1 − 2 sin²A · Meaning: Three equivalent forms of the double-angle cosine.
- Formula: General solution of sin θ = sin α: θ = nπ + (−1)ⁿ α, n ∈ Z · Meaning: Captures all angles with the same sine value.
- Formula: General solution of cos θ = cos α: θ = 2nπ ± α, n ∈ Z · Meaning: Captures all angles with the same cosine value.
- Formula: General solution of tan θ = tan α: θ = nπ + α, n ∈ Z · Meaning: Captures all angles with the same tangent value.
Worked examples
Example 1: Converting and finding arc length
A circular running track has radius 50 metres. A runner moves through an angle of 72° measured at the centre. Find the arc length covered.
Step 1 Convert 72° to radians. θ = 72 × (π / 180) = 72π / 180 = 2π / 5 radians.
Step 2 Use s = rθ. s = 50 × (2π / 5) = 100π / 5 = 20π metres ≈ 62.8 metres.
The runner covers about 62.8 metres along the track.
Example 2: Simplifying with identities
Simplify (1 − cos²θ) / sin θ for sin θ ≠ 0.
Step 1 Replace 1 − cos²θ using the Pythagorean identity: 1 − cos²θ = sin²θ.
Step 2 Substitute. (sin²θ) / sin θ = sin θ.
Hence the expression simplifies to sin θ.
Example 3: Solving a trigonometric equation
Find all solutions of 2 sin θ − 1 = 0 in the interval [0, 2π].
Step 1 Rearrange: sin θ = 1/2.
Step 2 Recall sin(π/6) = 1/2. So the principal value α = π/6.
Step 3 In [0, 2π], sine is positive in the first and second quadrants. First quadrant: θ = π/6. Second quadrant: θ = π − π/6 = 5π/6.
Solutions: θ = π/6 and θ = 5π/6.
Common mistakes
- Forgetting to convert degrees to radians before using the arc-length formula s = rθ → always check that θ is in radians when the formula requires it.
- Writing sin(A + B) as sin A + sin B → remember the addition formula involves products sin A cos B and cos A sin B.
- Giving only the principal solution of a trigonometric equation and ignoring the general solution → state the family of solutions using n ∈ Z.
- Using sec θ or tan θ at angles where cos θ = 0 → first check the domain; these functions are undefined at θ = π/2 + nπ.
- Mixing up signs in the quadrant rule (ASTC) → draw a quick unit-circle sketch to confirm which function is positive or negative in each quadrant.
Quick revision
- Radian measure: π rad = 180°; arc length s = rθ with θ in radians.
- Pythagorean identity sin²θ + cos²θ = 1 is the foundation of all trigonometric identities.
- sin and cos have period 2π and range [−1, 1]; tan has period π and range all real numbers.
- Use the ASTC rule (All, Sin, Tan, Cos positive in quadrants I–IV) to fix signs.
- General solutions involve an integer n; memorise the standard forms for sin θ = sin α, cos θ = cos α, tan θ = tan α.