Trigonometry

Sine, cosine and tangent are the main trigonometric functions and can be defined for all angles; understanding sin, cos and tan values is essential for solving A-Level trigonometry problems.

The sine rule relates the sides and opposite angles of a triangle using a/$\sin A = b/\sin B = c/\sin C$, and it is useful when sufficient side-angle information is known.

The cos rule, also called the cosine rule, can be written as c² = a² + b² - 2ab cos C, while the formula cosine rule may be rearranged to find an unknown angle.

The area of a triangle can be calculated using $Area = 1/2 ab \sin C$ when two sides and the included angle are known.

Radian measure expresses angles using the radius of a circle, with π radians equal to 180°, and the area of a sector in radians is $A = 1/2 r²θ$.

For an angle θ measured in radians, the arc length formula is $s = rθ$, while the area of a sector of a circle radians formula is $A = 1/2 r²θ$.

Small angle approximations are sin θ ≈ θ, tan θ ≈ θ and cos θ ≈ 1 - θ²/2 when θ is close to zero and measured in radians.

Trig values for standard angles should be known exactly, including sin cos tan values for 0, π/6, π/4, π/3, π/2 and related angles.

Trig graphs show the behaviour of sine, cosine and tangent functions, including their periodicity, symmetry, intercepts and maximum or minimum values.

When studying trigonometric graphs, sin cos tan graphs have different periods and asymptotes; sine and cosine have period 2π while tangent has period π.

Secant, cosecant and cotangent are reciprocal trigonometric functions, while arcsin, arccos and arctan are inverse trigonometric functions with specified domains and ranges.

Trig identities A Level include tan θ = sin θ/cos θ, sin²θ + cos²θ ≡ 1, sec²θ ≡ 1 + tan²θ and cosec²θ ≡ 1 + cot²θ.

The double angle formula includes sin 2θ = 2sin θ cos θ, cos 2θ = cos²θ - sin²θ and tan 2θ = 2tan θ/(1 - tan²θ); these double angle identities are widely used when simplifying expressions and solving equations.

The compound angle formula and addition formula give expressions for sin(A ± B), cos(A ± B) and tan(A ± B), while expressions such as a cos θ + b sin θ can be rewritten in an equivalent R-form.

Trigonometric equations, including trig equations that are quadratic in sin, cos or tan or involve multiple angles, are solved over a stated interval using exact trig values, identities and graphs; trigonometric identities can also be used to construct proofs and solve contextual problems involving vectors, kinematics and forces.