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Solving trigonometric equations
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A trig equation usually has more than one solution in a given interval. Find the first with your calculator, then use symmetry to find the rest.
The method
1. Find the principal value with sin−1, cos−1 or tan−1.
2. Find the other solutions:
sin: 180° − θ
cos: 360° − θ
tan: θ + 180°
3. Add or subtract 360° to stay in the interval.
2. Find the other solutions:
sin: 180° − θ
cos: 360° − θ
tan: θ + 180°
3. Add or subtract 360° to stay in the interval.
Example
sin x = 0.5, 0° ≤ x ≤ 360°
Principal value 30°. Second: 180° − 30° = 150°.
x = 30° or 150°
Principal value 30°. Second: 180° − 30° = 150°.
x = 30° or 150°
Harder forms
sin 2x = 0.5, 0° ≤ x ≤ 180°: solve for 2x in 0° to 360° first: 2x = 30°, 150°, so x = 15°, 75°.
sin x = 3 cos x: divide by cos x to get tan x = 3.
Quadratics in sin x: factorise. 2 sin2 x − sin x − 1 = 0 gives (2 sin x + 1)(sin x − 1) = 0.
In radians the method is the same with π in place of 180°.
sin x = 3 cos x: divide by cos x to get tan x = 3.
Quadratics in sin x: factorise. 2 sin2 x − sin x − 1 = 0 gives (2 sin x + 1)(sin x − 1) = 0.
In radians the method is the same with π in place of 180°.
Solve 2 sin2 x − sin x − 1 = 0 for 0° ≤ x ≤ 360°.
- (2 sin x + 1)(sin x − 1) = 0
- sin x = 1 gives x = 90°
- sin x = −12 gives x = 210° or 330°
Answer: x = 90°, 210°, 330°
Principal value, then symmetry: sin uses 180° − θ, cos uses 360° − θ, tan adds 180°. For sin kx, widen the interval by k first.
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