Trigonometry Examples

$3 \sin^{2} (x) + 4 \cos^{2} (x) = 3 + \cos^{2} (x)$

Step 1

Replace the $3 \sin^{2} (x)$ with $3 (1 - \cos^{2} (x))$ based on the $\sin^{2} (x) + \cos^{2} (x) = 1$ identity.

$3 (1 - \cos^{2} (x)) + 4 \cos^{2} (x) = 3 + \cos^{2} (x)$

Step 2

Simplify each term.

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Step 2.1

Apply the distributive property.

$3 \cdot 1 + 3 (- \cos^{2} (x)) + 4 \cos^{2} (x) = 3 + \cos^{2} (x)$

Step 2.2

Multiply $3$ by $1$ .

$3 + 3 (- \cos^{2} (x)) + 4 \cos^{2} (x) = 3 + \cos^{2} (x)$

Step 2.3

Multiply $- 1$ by $3$ .

$3 - 3 \cos^{2} (x) + 4 \cos^{2} (x) = 3 + \cos^{2} (x)$

Step 3

Add $- 3 \cos^{2} (x)$ and $4 \cos^{2} (x)$ .

$3 + \cos^{2} (x) = 3 + \cos^{2} (x)$

Step 4

Reorder the polynomial.

$\cos^{2} (x) + 3 = 3 + \cos^{2} (x)$

Step 5

Move all terms containing $\cos (x)$ to the left side of the equation.

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Step 5.1

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

$\cos^{2} (x) + 3 - \cos^{2} (x) = 3$

Step 5.2

Combine the opposite terms in $\cos^{2} (x) + 3 - \cos^{2} (x)$ .

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Step 5.2.1

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

$0 + 3 = 3$

Step 5.2.2

Add $0$ and $3$ .

$3 = 3$

Step 6

Since $3 = 3$ , the equation will always be true for any value of $x$ .

All real numbers

Step 7

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$