E: Trigonometry: Understand and use the sine, cosine and tangent functions; their graphs, symmetries and periodicity. Know and use exact values of sin and cos for 0, π/6, π/4, π/3, π/2, π and multiples thereof, and exact values of tan for 0, π/6, π/4, π/3, π and multiples thereof.

The idea

The sine, cosine and tangent functions are defined for angles in radians and have characteristic graphs. Sin and cos oscillate between −1 and 1 with period 2π and amplitude 1; tan has period π, no amplitude, and vertical asymptotes where cos θ = 0, for example at θ = π/2. Symmetries include sin(π − θ) = sin θ, cos(−θ) = cos θ and tan(π + θ) = tan θ. Exact values follow from the special triangles: sin π/6 = 1/2, sin π/4 = √2/2, sin π/3 = √3/2, with cos values reversed, and tan π/6 = 1/√3, tan π/4 = 1, tan π/3 = √3. Multiples are handled using periodicity and symmetry, for example sin 5π/6 = sin π/6 = 1/2 and cos 4π/3 = −cos π/3 = −1/2. Note that tan π/2 is undefined, so it is excluded from the exact values required for tan.

What earns the marks

• Sketches or describes the graphs of sin, cos and tan, including period and asymptotes; note that sin and cos have amplitude 1, while tan has no amplitude.
• States the periodicity: sin and cos have period 2π, tan has period π.
• Uses symmetries such as sin(π − θ) = sin θ, cos(−θ) = cos θ and tan(π + θ) = tan θ to find values at related angles.
• Recalls and applies exact values of sin and cos for 0, π/6, π/4, π/3, π/2 and π, and exact values of tan for 0, π/6, π/4, π/3 and π, and their multiples, without a calculator; tan π/2 is undefined and is not required.
• Solves simple trigonometric equations over a given interval using exact values, periodicity and symmetry.
• Converts between degrees and radians where needed, for example recognising π/6 as 30°.

Common mistakes and how to put them right

• Mixing degrees and radians, for example treating the input π/6 as if it were 30 radians rather than 30°.

  Put it right: Convert consistently and work in radians when the question uses π.

• Forgetting the sign changes in different quadrants, for example giving cos 4π/3 as +1/2.

  Put it right: Use the symmetry cos(π + θ) = −cos θ to obtain −1/2.

• Assuming tan has period 2π like sin and cos.

  Put it right: Tan repeats every π, so tan(θ + π) = tan θ, and this affects the number of solutions in an interval.

• Attempting to quote a value for tan π/2.

  Put it right: It is undefined because cos π/2 = 0, so the graph has a vertical asymptote there.

Exam tips

• Sketch the relevant graph or unit circle to locate the quadrant and sign before quoting an exact value.
• List all solutions in the required interval by adding or subtracting the period, then check each lies within the range.
• Learn the special triangles (45-45-90 and 30-60-90) so exact values can be reconstructed quickly if memory fails.
• Leave exact values in surd form unless the question asks for a decimal approximation.