Trig - equations - strategy

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Trigonometry - Trigonometric Equations.
Summary of strategies.

To solve trigonometric equations, follow these strategies to identify what type of equation you have and what the relevant strategy is:

Equation type	Characteristic of this type of equation.	Strategy
1.	One trig term and a constant.	move constant to the opposite side of = sign; divide by the number in front of the trig term; solve using exact values or acalculator; check what quadrant the angle lies in (using ASTC)
2.	One trig term squared and a constant.	move constant to the opposite side of = sign; divide by the number in front of the trig term; square root both sides (REMEMBER ±); solve usi ng exact values or a calculator; check what quadrant the angle lies in (using ASTC) - Here all 4 quadrants have a solution.
3.	Two trig terms - one a cos and one a sin - and NO constant.	separate the terms to opposite sides of = sign; divide by the cos term to create sin/cos = tan; solve using exact values or a calculator; check what quadrant the angle lies in (using ASTC)
4.	A quadratic equation with a squared trig ratio, a usual trig ratio and a constant term.	make all trig ratios the same - especially using one of the Pythagorean identities to replace the squared term; EITHER factorise directly or substitute u for the trig ratio; solve the factorisation; check what quadrant the angle lies in (using ASTC); test the values to determine if they make sense.
5.	Other formats.	Reduce other formats to their basic components with factorisation - then use one of the above strategies as appropriate.
6.	Double angle (sometimes triple) - for example - 2θ or 3θ - applies to all of the above	When the angle is expressed as a multiple: write down possible solutions beyond 360°; then divide by the multiple.

Examples.

Solve the following equations for 0 ≤ θ ≤ 360°.

Equation type	Characteristic of this type of equation.	Strategy
1.	2 sin θ - 1 = 0 One trig term and a constant.	2 sin θ = 1 sin θ = ½ θ = 30°, 150°.
2.	4 cos²θ - 1 = 0 One trig term squared and a constant.	4 cos²θ = 1 cos²θ = ¼ cos θ = ± ½ θ = 60°, 120°, 240°, 300°.
3.	sin θ - cos θ = 0 Two trig terms - one a cos and one a sin - and NO constant.	sin θ = cos θ tan θ = 1 θ = 45°, 225°.
4:	3sin²θ - sin θ - 4 = 0 . A quadratic equation with a squared trig ratio, a usual trig ratio and a constant term.	(3sin θ - 4)(sin θ + 1) = 0 sin θ = 4/3 (not possible) or sin θ = -1 ∴ θ = 270°
5:	sin θ cos θ + sin θ = 0 Other formats.	sin θ (cos θ + 1) = 0 sin θ = 0 or cos θ = -1 θ = 0°, 180°.
6.	sin 2θ = ½ for 0 ≤ θ ≤ 360° The angle is a multiple of θ.	sin 2θ = ½ (need to go two revolutions) ∴ 2θ = 30°, 150°, 390°, 510°. θ = 15°, 75°, 195°, 255°.