Trigonometry Examples

$2 \cos^{2} (t) \tan (t) - \tan (t) = 0$

Step 1

Simplify $2 \cos^{2} (t) \tan (t) - \tan (t)$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Rewrite $\tan (t)$ in terms of sines and cosines.

$2 \cos^{2} (t) \frac{\sin (t)}{\cos (t)} - \tan (t) = 0$

Step 1.1.2

Cancel the common factor of $\cos (t)$ .

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Step 1.1.2.1

Factor $\cos (t)$ out of $2 \cos^{2} (t)$ .

$\cos (t) (2 \cos (t)) \frac{\sin (t)}{\cos (t)} - \tan (t) = 0$

Step 1.1.2.2

Cancel the common factor.

$\cos (t) (2 \cos (t)) \frac{\sin (t)}{\cos (t)} - \tan (t) = 0$

Step 1.1.2.3

Rewrite the expression.

$2 \cos (t) \sin (t) - \tan (t) = 0$

Step 1.1.3

Reorder $2 \cos (t)$ and $\sin (t)$ .

$\sin (t) (2 \cos (t)) - \tan (t) = 0$

Step 1.1.4

Reorder $\sin (t)$ and $2$ .

$2 \cdot \sin (t) \cos (t) - \tan (t) = 0$

Step 1.1.5

Apply the sine double-angle identity.

$\sin (2 t) - \tan (t) = 0$

Step 1.1.6

Rewrite $\tan (t)$ in terms of sines and cosines.

$\sin (2 t) - \frac{\sin (t)}{\cos (t)} = 0$

Step 1.2

Convert from $\frac{\sin (t)}{\cos (t)}$ to $\tan (t)$ .

$\sin (2 t) - \tan (t) = 0$

Step 2

Graph each side of the equation. The solution is the x-value of the point of intersection.

$t = π n, \frac{π}{4} + π n, \frac{3 π}{4} + π n$ , for any integer $n$

Step 3