Trigonometric Equations

Formula 1: 

`\sin\theta=0\Rightarrow\theta=n\pi`.

Example:

(1) Solve `\sin3x=0`.

Given that,

`\sin3x=0`

`\Rightarrow3x=n\pi`

`\Rightarrowx=\frac(n\pi)(3)`.

(2) Solve `2\sin3x\cos3x=0`

Solution: Try yourself.

(3) Solve `\sin2x\cos2x=0`.

Solution: Try yourself.

(4) Solve `\sin^2(\frac(x)(2))=0`.

Solution: Try yourself.

(5) Solve `\sin(\frac(x)(4))\cos(\frac(x)(4))=0`.

Solution: Try yourself.

Formula 2: `\cos\theta=0\Rightarrow\theta=(2n+1)\frac(\pi)(2)`.

Example:

(1) Solve `\cos\5theta=0`.

Solution:

Given that,

`\cos\5theta=0`

`\Rightarrow5\theta=(2n+1)\frac(\pi)(2)`

`\Rightarrow\theta=(2n+1)\frac(\pi)(10)`

(2) Solve `\cos^2\5theta=0`.

Solution: Try yourself.

Formula 3:

`\tan\theta=0\Rightarrow\theta=n\pi`.

Formula 4:

`\sin\theta=\sin\alpha\Rightarrow\theta=n\pi+(-1)^n\alpha`.

Formula 5:

`\cos\theta=\cos\alpha\Rightarrow\theta=(2n\pi\pm\alpha)`.

Formula 6:

`\tan\theta=\tan\alpha\Rightarrow\theta=(n\pi+\alpha)`.

Formula 7:

`\sin\theta=1\Rightarrow\theta=(4n+1)\frac(\pi)(2)`.

Formula 8:

`\cos\theta=1\Rightarrow\theta=2n\pi`.

Formula 9:

`\sin\theta=-1\Rightarrow\theta=(4n-1)\frac(\pi)(2)`.

Formula 10:

`\cos\theta=-1\Rightarrow\theta=(2n\pm1)\pi`.

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