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. |
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2. | One trig term squared and a constant. |
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3. | Two trig terms - one a cos and one a sin - and NO constant. |
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4. | A quadratic equation with a squared trig ratio, a usual trig ratio and a constant term. |
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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:
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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. |
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2. | 4 cos2θ - 1 = 0 One trig term squared and a constant. |
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3. | sin θ - cos θ = 0 Two trig terms - one a cos and one a sin - and NO constant. |
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4: | 3sin2θ - sin θ - 4 = 0 . A quadratic equation with a squared trig ratio, a usual trig ratio and a constant term. |
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5: | sin θ cos θ + sin θ = 0 Other formats. |
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6. | sin 2θ = ½ The angle is a multiple of θ.
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sin 2θ = ½ (need to go two revolutions) ∴ 2θ = 30°, 150°, 390°, 510°. θ = 15°, 75°, 195°, 255°. |