Trigonometry Examples

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

Step 1

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

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Step 1.1

Simplify each term.

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Step 1.1.1

Expand $(\cos (x) - 2) (\cos (x) - 2)$ using the FOIL Method.

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Step 1.1.1.1

Apply the distributive property.

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

Step 1.1.1.2

Apply the distributive property.

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

Step 1.1.1.3

Apply the distributive property.

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

Step 1.1.2

Simplify and combine like terms.

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Step 1.1.2.1

Simplify each term.

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Step 1.1.2.1.1

Multiply $\cos (x) \cos (x)$ .

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Step 1.1.2.1.1.1

Raise $\cos (x)$ to the power of $1$ .

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

Step 1.1.2.1.1.2

Raise $\cos (x)$ to the power of $1$ .

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

Step 1.1.2.1.1.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 1.1.2.1.1.4

Add $1$ and $1$ .

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

Step 1.1.2.1.2

Move $- 2$ to the left of $\cos (x)$ .

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

Step 1.1.2.1.3

Multiply $- 2$ by $- 2$ .

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

Step 1.1.2.2

Subtract $2 \cos (x)$ from $- 2 \cos (x)$ .

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

Step 1.2

Move $\sin^{2} (x)$ .

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

Step 1.3

Rearrange terms.

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

Step 1.4

Apply pythagorean identity.

$1 - 4 \cos (x) + 4 = 4$

Step 1.5

Add $1$ and $4$ .

$5 - 4 \cos (x) = 4$

Step 2

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

$x \approx 1.31811607 + 2 π n, 4.96506923 + 2 π n$ , for any integer $n$

Step 3