Grade 12 Trigonometry: Identities, Equations and General Solutions

The identities to know by heart

• tan θ = sin θ / cos θ and sin²θ + cos²θ = 1.
• sin(A ± B) = sinA cosB ± cosA sinB.
• cos(A ± B) = cosA cosB ∓ sinA sinB.
• sin 2A = 2 sinA cosA.
• cos 2A = cos²A − sin²A = 1 − 2sin²A = 2cos²A − 1.
• Reduction: sin(180° − θ) = sin θ, cos(180° − θ) = −cos θ, sin(360° − θ) = −sin θ, cos(−θ) = cos θ.

Worked example: proving an identity

Prove that (1 − cos 2A)/sin 2A = tan A. Left side = (1 − (1 − 2sin²A))/(2 sinA cosA) = 2sin²A/(2 sinA cosA) = sinA/cosA = tan A.

Work down one side only, state restrictions such as cos A ≠ 0, and never move terms across the equals sign in a proof.

Worked example: general solution

Solve 2 sin x = 1. Then sin x = 0.5, so the reference angle is 30°. General solution: x = 30° + k·360° or x = 150° + k·360°, k ∈ ℤ.

For cosine, use x = ±ref + k·360°. For tangent, use x = ref + k·180°. If the question gives an interval such as x ∈ [0°; 360°], substitute k = −1, 0, 1 and list only the values inside the interval.

Graphs, the sine rule and the cosine rule

• Use the sine rule when you have a side and its opposite angle.
• Use the cosine rule when you have three sides, or two sides and the included angle.
• Area of a triangle = ½ ab sin C.
• For graphs, state amplitude, period and any shift before plotting; label intercepts and turning points.
• In 3D questions, identify the triangle you are working in first and redraw it separately.