Precalculus Examples

\cos^{2} (x) - 2.4 \cos (x) - 0.81 = 0

$\cos^{2} (x) - 2.4 \cos (x) - 0.81 = 0$

Step 1

Substitute

u

$u$ for

\cos (x)

$\cos (x)$ .

{(u)}^{2} - 2.4 u - 0.81 = 0

${(u)}^{2} - 2.4 u - 0.81 = 0$

Step 2

Use the quadratic formula to find the solutions.

\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 3

Substitute the values

a = 1

$a = 1$ ,

b = - 2.4

$b = - 2.4$ , and

c = - 0.81

$c = - 0.81$ into the quadratic formula and solve for

u

$u$ .

\frac{2.4 \pm \sqrt{{(- 2.4)}^{2} - 4 \cdot (1 \cdot - 0.81)}}{2 \cdot 1}

$\frac{2.4 \pm \sqrt{{(- 2.4)}^{2} - 4 \cdot (1 \cdot - 0.81)}}{2 \cdot 1}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise

- 2.4

$- 2.4$ to the power of

2

$2$ .

u = \frac{2.4 \pm \sqrt{5.76 - 4 \cdot 1 \cdot - 0.81}}{2 \cdot 1}

$u = \frac{2.4 \pm \sqrt{5.76 - 4 \cdot 1 \cdot - 0.81}}{2 \cdot 1}$

Step 4.1.2

Multiply

- 4 \cdot 1 \cdot - 0.81

$- 4 \cdot 1 \cdot - 0.81$ .

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Step 4.1.2.1

Multiply

- 4

$- 4$ by

1

$1$ .

u = \frac{2.4 \pm \sqrt{5.76 - 4 \cdot - 0.81}}{2 \cdot 1}

$u = \frac{2.4 \pm \sqrt{5.76 - 4 \cdot - 0.81}}{2 \cdot 1}$

Step 4.1.2.2

Multiply

- 4

$- 4$ by

- 0.81

$- 0.81$ .

u = \frac{2.4 \pm \sqrt{5.76 + 3.24}}{2 \cdot 1}

$u = \frac{2.4 \pm \sqrt{5.76 + 3.24}}{2 \cdot 1}$

u = \frac{2.4 \pm \sqrt{5.76 + 3.24}}{2 \cdot 1}

$u = \frac{2.4 \pm \sqrt{5.76 + 3.24}}{2 \cdot 1}$

Step 4.1.3

Add

5.76

$5.76$ and

3.24

$3.24$ .

u = \frac{2.4 \pm \sqrt{9}}{2 \cdot 1}

$u = \frac{2.4 \pm \sqrt{9}}{2 \cdot 1}$

Step 4.1.4

Rewrite

9

$9$ as

3^{2}

$3^{2}$ .

u = \frac{2.4 \pm \sqrt{3^{2}}}{2 \cdot 1}

$u = \frac{2.4 \pm \sqrt{3^{2}}}{2 \cdot 1}$

Step 4.1.5

Pull terms out from under the radical, assuming positive real numbers.

u = \frac{2.4 \pm 3}{2 \cdot 1}

$u = \frac{2.4 \pm 3}{2 \cdot 1}$

u = \frac{2.4 \pm 3}{2 \cdot 1}

$u = \frac{2.4 \pm 3}{2 \cdot 1}$

Step 4.2

Multiply

2

$2$ by

1

$1$ .

u = \frac{2.4 \pm 3}{2}

$u = \frac{2.4 \pm 3}{2}$

u = \frac{2.4 \pm 3}{2}

$u = \frac{2.4 \pm 3}{2}$

Step 5

The final answer is the combination of both solutions.

u = 2.7, - 0.3

$u = 2.7, - 0.3$

Step 6

Substitute

\cos (x)

$\cos (x)$ for

u

$u$ .

\cos (x) = 2.7, - 0.3

$\cos (x) = 2.7, - 0.3$

Step 7

Set up each of the solutions to solve for

x

$x$ .

\cos (x) = 2.7

$\cos (x) = 2.7$

\cos (x) = - 0.3

$\cos (x) = - 0.3$

Step 8

Solve for

x

$x$ in

\cos (x) = 2.7

$\cos (x) = 2.7$ .

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Step 8.1

The range of cosine is

- 1 \leq y \leq 1

$- 1 \leq y \leq 1$ . Since

2.7

$2.7$ does not fall in this range, there is no solution.

No solution

Step 9

Solve for

x

$x$ in

\cos (x) = - 0.3

$\cos (x) = - 0.3$ .

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Step 9.1

Take the inverse cosine of both sides of the equation to extract

x

$x$ from inside the cosine.

x = \arccos (- 0.3)

$x = \arccos (- 0.3)$

Step 9.2

Simplify the right side.

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Step 9.2.1

Evaluate

\arccos (- 0.3)

$\arccos (- 0.3)$ .

x = 1.87548898

$x = 1.87548898$

x = 1.87548898

$x = 1.87548898$

Step 9.3

The cosine function is negative in the second and third quadrants. To find the second solution, subtract the reference angle from

2 π

$2 π$ to find the solution in the third quadrant.

x = 2 (3.14159265) - 1.87548898

$x = 2 (3.14159265) - 1.87548898$

Step 9.4

Solve for

x

$x$ .

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Step 9.4.1

Remove parentheses.

x = 2 (3.14159265) - 1.87548898

$x = 2 (3.14159265) - 1.87548898$

Step 9.4.2

Simplify

2 (3.14159265) - 1.87548898

$2 (3.14159265) - 1.87548898$ .

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Step 9.4.2.1

Multiply

2

$2$ by

3.14159265

$3.14159265$ .

x = 6.2831853 - 1.87548898

$x = 6.2831853 - 1.87548898$

Step 9.4.2.2

Subtract

1.87548898

$1.87548898$ from

6.2831853

$6.2831853$ .

x = 4.40769632

$x = 4.40769632$

x = 4.40769632

$x = 4.40769632$

x = 4.40769632

$x = 4.40769632$

Step 9.5

Find the period of

\cos (x)

$\cos (x)$ .

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Step 9.5.1

The period of the function can be calculated using

\frac{2 π}{| b |}

$\frac{2 π}{| b |}$ .

\frac{2 π}{| b |}

$\frac{2 π}{| b |}$

Step 9.5.2

Replace

b

$b$ with

1

$1$ in the formula for period.

\frac{2 π}{| 1 |}

$\frac{2 π}{| 1 |}$

Step 9.5.3

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

0

$0$ and

1

$1$ is

1

$1$ .

\frac{2 π}{1}

$\frac{2 π}{1}$

Step 9.5.4

Divide

2 π

$2 π$ by

1

$1$ .

2 π

$2 π$

2 π

$2 π$

Step 9.6

The period of the

\cos (x)

$\cos (x)$ function is

2 π

$2 π$ so values will repeat every

2 π

$2 π$ radians in both directions.

x = 1.87548898 + 2 π n, 4.40769632 + 2 π n

$x = 1.87548898 + 2 π n, 4.40769632 + 2 π n$ , for any integer

n

$n$

x = 1.87548898 + 2 π n, 4.40769632 + 2 π n

$x = 1.87548898 + 2 π n, 4.40769632 + 2 π n$ , for any integer

n

$n$

Step 10

List all of the solutions.

x = 1.87548898 + 2 π n, 4.40769632 + 2 π n

$x = 1.87548898 + 2 π n, 4.40769632 + 2 π n$ , for any integer

n

$n$