Pearson Further Pure Mathematics 10: Trigonometry

Trigonometry connects angle, ratio, circular motion, geometry and periodic functions. Pearson International GCSE Further Pure Mathematics (4PM1) requires radians, arc length and sector area; sine, cosine and tangent for angles of any magnitude; exact values and graphs; two- and three-dimensional applications; sine and cosine formulae; core identities and addition formulae; and complete solutions of simple trigonometric equations on stated intervals. The goal is not a single calculator angle but a complete, unit-consistent argument.

1. Degrees, radians and circles

One full turn is 360° = 2π radians, so

degrees × π/180 = radians and radians × 180/π = degrees.

Radians measure an angle as arc length divided by radius. This definition gives the circle formulas

s = rθ and A = (1/2)r²θ,

where θ must be in radians. Here s is arc length and A is sector area. If an angle is supplied in degrees, convert it before using these forms. For a major sector, ensure the selected angle matches the requested arc rather than automatically using the smaller central angle.

Radians carry no physical dimension, but stating “radians” clarifies angle mode. Calculator settings must match the question. A numerically plausible answer can still be wrong if degrees were entered in radian mode.

2. Ratios for angles of any magnitude

On the unit circle, a point at angle θ has coordinates (cos θ, sin θ). Therefore cosine is the horizontal coordinate and sine the vertical coordinate. Tangent is sin θ/cos θ where cosine is non-zero. Signs follow the quadrant:

• quadrant I: all three positive;
• quadrant II: sine positive;
• quadrant III: tangent positive;
• quadrant IV: cosine positive.

Reference angles connect related values. For example, sin 120° = sin 60° = √3/2, while cos 120° = -cos 60° = -1/2. Determine the reference angle, then apply the sign from the actual quadrant.

Exact values for 30°, 45° and 60°, and their radian equivalents, should be known. They can be derived from an equilateral triangle split in half and an isosceles right triangle. Preserve exact surds and fractions unless a decimal is requested.

Angles outside one turn are reduced using periodicity. Sine and cosine repeat every 360° or 2π. Tangent repeats every 180° or π. Negative angles travel clockwise and can also be related using