Trigonometry
What this domain covers
Trigonometry in 8227 combines identities and equation solving with radian measure: arc length, sector and segment areas, graph sketching and transformations of sine, cosine and tangent, and exact values.
Core rules
- Radians: π radians = 180°; arc length s = rθ; sector area = ½r²θ; segment area = sector − triangle = ½r²θ − ½r²sin θ.
- Identities: tan θ = sin θ / cos θ; sin²θ + cos²θ = 1; sin 2θ = 2 sin θ cos θ; cos 2θ = cos²θ − sin²θ = 1 − 2sin²θ.
- Equation solving: take the principal value from the inverse function, then use graph symmetry (or a CAST diagram) to list every solution in the given interval. If the argument is 2x or x + 30°, solve for the whole argument first, then divide.
- Graphs: sine and cosine have period 2π and amplitude 1; tan has period π with asymptotes at θ = π/2 + kπ.
Worked example
Solve 2sin²x + sin x − 1 = 0 for 0 ≤ x < 2π. Factorise as a quadratic in sin x: (2 sin x − 1)(sin x + 1) = 0. Hence sin x = ½, giving x = π/6 and x = π − π/6 = 5π/6; or sin x = −1, giving x = 3π/2. The solution set is {π/6, 5π/6, 3π/2}.
Traps
- Leaving the calculator in degree mode while the interval is given in radians.
- Dividing through by cos x and losing the solutions where cos x = 0.
- Reporting only the principal value after an inverse sine or cosine instead of all solutions in the interval.
- Using s = rθ with θ in degrees; the formula requires radians, otherwise use s = rθπ/180.