Dr. J's Maths.com
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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°.