Find All Complex Solutions cos(x)^2+2cos(x)-3=0

$\cos^{2} (x) + 2 \cos (x) - 3 = 0$

Factor $\cos^{2} (x) + 2 \cos (x) - 3$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 3$ and whose sum is $2$ .

$- 1, 3 = 0$

Write the factored form using these integers.

$(\cos (x) - 1) (\cos (x) + 3) = 0$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$\cos (x) - 1 = 0$

$\cos (x) + 3 = 0$

Set $\cos (x) - 1$ equal to $0$ and solve for $x$ .

Set $\cos (x) - 1$ equal to $0$ .

$\cos (x) - 1 = 0$

Solve $\cos (x) - 1 = 0$ for $x$ .

Add $1$ to both sides of the equation.

$\cos (x) = 1$

Take the inverse cosine of both sides of the equation to extract $x$ from inside the cosine.

$x = \arccos (1)$

Simplify the right side.

The exact value of $\arccos (1)$ is $0$ .

$x = 0$

The cosine function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the fourth quadrant.

$x = 2 π - 0$

Subtract $0$ from $2 π$ .

$x = 2 π$

Find the period of $\cos (x)$ .

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Divide $2 π$ by $1$ .

$2 π$

The period of the $\cos (x)$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

$x = 2 π n, 2 π + 2 π n$ , for any integer $n$

Set $\cos (x) + 3$ equal to $0$ and solve for $x$ .

Set $\cos (x) + 3$ equal to $0$ .

$\cos (x) + 3 = 0$

Solve $\cos (x) + 3 = 0$ for $x$ .

Subtract $3$ from both sides of the equation.

$\cos (x) = - 3$

The range of cosine is $- 1 \leq y \leq 1$ . Since $- 3$ does not fall in this range, there is no solution.

No solution

The final solution is all the values that make $(\cos (x) - 1) (\cos (x) + 3) = 0$ true.

$x = 2 π n, 2 π + 2 π n$ , for any integer $n$

Find All Complex Solutions cos(x)^2+2cos(x)-3=0

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