Trigonometry Examples

$7 \sin^{2} (x) + 5 \cos^{2} (x) = 5 + 2 \sin^{2} (x)$

Step 1

Move all the expressions to the left side of the equation.

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Step 1.1

Subtract $5$ from both sides of the equation.

$7 \sin^{2} (x) + 5 \cos^{2} (x) - 5 = 2 \sin^{2} (x)$

Step 1.2

Subtract $2 \sin^{2} (x)$ from both sides of the equation.

$7 \sin^{2} (x) + 5 \cos^{2} (x) - 5 - 2 \sin^{2} (x) = 0$

Step 2

Simplify $7 \sin^{2} (x) + 5 \cos^{2} (x) - 5 - 2 \sin^{2} (x)$ .

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Step 2.1

Reorder $5 \cos^{2} (x)$ and $- 5$ .

$7 \sin^{2} (x) - 5 + 5 \cos^{2} (x) - 2 \sin^{2} (x) = 0$

Step 2.2

Factor $- 5$ out of $- 5$ .

$7 \sin^{2} (x) - 5 (1) + 5 \cos^{2} (x) - 2 \sin^{2} (x) = 0$

Step 2.3

Factor $- 5$ out of $5 \cos^{2} (x)$ .

$7 \sin^{2} (x) - 5 (1) - 5 (- \cos^{2} (x)) - 2 \sin^{2} (x) = 0$

Step 2.4

Factor $- 5$ out of $- 5 (1) - 5 (- \cos^{2} (x))$ .

$7 \sin^{2} (x) - 5 (1 - \cos^{2} (x)) - 2 \sin^{2} (x) = 0$

Step 2.5

Apply pythagorean identity.

$7 \sin^{2} (x) - 5 \sin^{2} (x) - 2 \sin^{2} (x) = 0$

Step 2.6

Subtract $5 \sin^{2} (x)$ from $7 \sin^{2} (x)$ .

$2 \sin^{2} (x) - 2 \sin^{2} (x) = 0$

Step 2.7

Subtract $2 \sin^{2} (x)$ from $2 \sin^{2} (x)$ .

$0 = 0$

Step 3

Since $0 = 0$ , the equation will always be true for any value of $x$ .

All real numbers

Step 4

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$