Question 1

How do I remember the exact trig values for 30°, 45°, and 60°?

Accepted Answer

Use the mnemonic 'Sin and Cos are the same but swapped' or the hand trick: for sin, count fingers below the angle (0°=0, 30°=1, 45°=2, 60°=3, 90°=4) and take square root over 2. For cos, count fingers above. Alternatively, remember the triangles: 30-60-90 has sides 1, √3, 2; 45-45-90 has sides 1, 1, √2. Practice writing them out until they become automatic.

Question 2

What is the cast diagram and how do I use it?

Accepted Answer

The cast diagram (or ASTC) shows which trig functions are positive in each quadrant: All in Q1, Sin in Q2, Tan in Q3, Cos in Q4. To solve an equation like sinθ = 0.5, first find the acute reference angle (30°). Then, since sin is positive in Q1 and Q2, the solutions in 0-360° are 30° and 180°-30°=150°. For negative values, use the quadrants where the function is negative. This method works for any angle range.

Question 3

When should I use the sine rule versus the cosine rule?

Accepted Answer

Use the sine rule when you know two angles and one side (AAS or ASA) or two sides and a non-included angle (SSA). Use the cosine rule when you know two sides and the included angle (SAS) or all three sides (SSS). For SSA, be cautious of the ambiguous case: if the given angle is acute and the opposite side is shorter than the adjacent side, there may be two possible triangles.

Question 4

How do I solve trigonometric equations that involve quadratics?

Accepted Answer

If the equation contains sin²θ or cos²θ, use the identity sin²θ + cos²θ = 1 to rewrite it in terms of one function. For example, 2cos²θ + sinθ = 1 becomes 2(1 - sin²θ) + sinθ = 1, which simplifies to 2sin²θ - sinθ - 1 = 0. Factor or use the quadratic formula to solve for sinθ, then find θ using inverse sine and the cast diagram. Always check for extraneous solutions if you squared both sides.

Question 5

What are compound angle formulas and when do I use them?

Accepted Answer

Compound angle formulas express sin(A±B) and cos(A±B) in terms of sinA, cosA, sinB, cosB. For example, sin(A+B) = sinA cosB + cosA sinB. Use them to simplify expressions like sin(θ+30°) or to prove identities. They are also essential for solving equations like sin2θ = cosθ by rewriting sin2θ as 2sinθ cosθ. Memorise the four formulas: sin(A±B), cos(A±B), and the double-angle versions (e.g., sin2A = 2sinA cosA).

Question 6

How do I find the general solution of a trigonometric equation?

Accepted Answer

For equations like sinθ = k, the general solution is θ = nπ + (-1)ⁿ α, where α is the principal value (in radians) and n is an integer. For cosθ = k, use θ = 2nπ ± α. For tanθ = k, use θ = nπ + α. Always convert degrees to radians if needed. Practice by finding all solutions for a simple equation like sinθ = 0.5, then generalise. This is common in A-Level questions asking for 'all solutions' or 'general solution'.

E: Trigonometry

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Related Topics in AQA vocational Mathematics

A: Proof

B: Algebra and functions

C: Coordinate geometry in the ( x , y ) plane

D: Sequences and series

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