Solving Trigonometric Equations
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The various trigonometric formulae and identities can be used to help solve trigonometric equations. Here is a summary of the most important trigonometric formulae you should know:
sin˛x + cos˛x = 1
1 + cot˛x = cosec˛x
tan˛x + 1 = sec˛x
cos2x = cos˛x - sin˛x = 2cos˛x - 1 = 1 - 2sin˛x
sin2x = 2sinx cosx
tanx = sinx
         cosx

Example:
Solve 2cos˛x + 3sinx = 3, giving your answer in radians for 0< x <p.
\ 2cos˛x + 3sinx - 3 = 0
We need to get everything in terms of sinx or everything in terms of cosx. Since we know that cos˛x = 1 - sin˛x:
\ 2(1 - sin˛x) + 3sinx - 3 = 0
\ 2 - 2sin˛x + 3sinx - 3 = 0
\ -2sin˛x + 3sinx - 1 = 0
\ 2sin˛x - 3sinx + 1 = 0
\ (2sinx - 1)(sinx - 1) = 0
\ sin x = ˝ or sin x = 1
x = p/6, 5p/6, p/2

Remember, if sinx = 1,  x = p/2, 5p/2, 9p/2, ... and the same is true for arcsin(˝). In the question, you are asked for values of x between 0 and p. You must write down all of the appropriate solutions.

© Matthew Pinkey

Other Notes in this Category

1. Double angle formulae
2. Pythagorean Identities
3. Radians
4. Sec, cosec, cot
5. Sin, Cos, Tan
6. Sine and Cosine Formulae
7. Solving Trigonometric Equations

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