Trigonometry Examples

$8 \cos^{2} (2 x) + 7 \cos (2 x) - 1 = 0$

Step 1

Factor the left side of the equation.

Tap for more steps...

Step 1.1

Let $u = \cos (2 x)$ . Substitute $u$ for all occurrences of $\cos (2 x)$ .

$8 u^{2} + 7 u - 1 = 0$

Step 1.2

Factor by grouping.

Tap for more steps...

Step 1.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 8 \cdot - 1 = - 8$ and whose sum is $b = 7$ .

Tap for more steps...

Step 1.2.1.1

Factor $7$ out of $7 u$ .

$8 u^{2} + 7 (u) - 1 = 0$

Step 1.2.1.2

Rewrite $7$ as $- 1$ plus $8$

$8 u^{2} + (- 1 + 8) u - 1 = 0$

Step 1.2.1.3

Apply the distributive property.

$8 u^{2} - 1 u + 8 u - 1 = 0$

Step 1.2.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 1.2.2.1

Group the first two terms and the last two terms.

$(8 u^{2} - 1 u) + 8 u - 1 = 0$

Step 1.2.2.2

Factor out the greatest common factor (GCF) from each group.

$u (8 u - 1) + 1 (8 u - 1) = 0$

Step 1.2.3

Factor the polynomial by factoring out the greatest common factor, $8 u - 1$ .

$(8 u - 1) (u + 1) = 0$

Step 1.3

Replace all occurrences of $u$ with $\cos (2 x)$ .

$(8 \cos (2 x) - 1) (\cos (2 x) + 1) = 0$

Step 2

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

$8 \cos (2 x) - 1 = 0$

$\cos (2 x) + 1 = 0$

Step 3

Set $8 \cos (2 x) - 1$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 3.1

Set $8 \cos (2 x) - 1$ equal to $0$ .

$8 \cos (2 x) - 1 = 0$

Step 3.2

Solve $8 \cos (2 x) - 1 = 0$ for $x$ .

Tap for more steps...

Step 3.2.1

Add $1$ to both sides of the equation.

$8 \cos (2 x) = 1$

Step 3.2.2

Divide each term in $8 \cos (2 x) = 1$ by $8$ and simplify.

Tap for more steps...

Step 3.2.2.1

Divide each term in $8 \cos (2 x) = 1$ by $8$ .

$\frac{8 \cos (2 x)}{8} = \frac{1}{8}$

Step 3.2.2.2

Simplify the left side.

Tap for more steps...

Step 3.2.2.2.1

Cancel the common factor of $8$ .

Tap for more steps...

Step 3.2.2.2.1.1

Cancel the common factor.

$\frac{8 \cos (2 x)}{8} = \frac{1}{8}$

Step 3.2.2.2.1.2

Divide $\cos (2 x)$ by $1$ .

$\cos (2 x) = \frac{1}{8}$

Step 3.2.3

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

$2 x = \arccos (\frac{1}{8})$

Step 3.2.4

Simplify the right side.

Tap for more steps...

Step 3.2.4.1

Evaluate $\arccos (\frac{1}{8})$ .

$2 x = 1.44546849$

Step 3.2.5

Divide each term in $2 x = 1.44546849$ by $2$ and simplify.

Tap for more steps...

Step 3.2.5.1

Divide each term in $2 x = 1.44546849$ by $2$ .

$\frac{2 x}{2} = \frac{1.44546849}{2}$

Step 3.2.5.2

Simplify the left side.

Tap for more steps...

Step 3.2.5.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.2.5.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{1.44546849}{2}$

Step 3.2.5.2.1.2

Divide $x$ by $1$ .

$x = \frac{1.44546849}{2}$

Step 3.2.5.3

Simplify the right side.

Tap for more steps...

Step 3.2.5.3.1

Divide $1.44546849$ by $2$ .

$x = 0.72273424$

Step 3.2.6

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.

$2 x = 2 (3.14159265) - 1.44546849$

Step 3.2.7

Solve for $x$ .

Tap for more steps...

Step 3.2.7.1

Simplify.

Tap for more steps...

Step 3.2.7.1.1

Multiply $2$ by $3.14159265$ .

$2 x = 6.2831853 - 1.44546849$

Step 3.2.7.1.2

Subtract $1.44546849$ from $6.2831853$ .

$2 x = 4.83771681$

Step 3.2.7.2

Divide each term in $2 x = 4.83771681$ by $2$ and simplify.

Tap for more steps...

Step 3.2.7.2.1

Divide each term in $2 x = 4.83771681$ by $2$ .

$\frac{2 x}{2} = \frac{4.83771681}{2}$

Step 3.2.7.2.2

Simplify the left side.

Tap for more steps...

Step 3.2.7.2.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.2.7.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{4.83771681}{2}$

Step 3.2.7.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{4.83771681}{2}$

Step 3.2.7.2.3

Simplify the right side.

Tap for more steps...

Step 3.2.7.2.3.1

Divide $4.83771681$ by $2$ .

$x = 2.4188584$

Step 3.2.8

Find the period of $\cos (2 x)$ .

Tap for more steps...

Step 3.2.8.1

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

$\frac{2 π}{| b |}$

Step 3.2.8.2

Replace $b$ with $2$ in the formula for period.

$\frac{2 π}{| 2 |}$

Step 3.2.8.3

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$\frac{2 π}{2}$

Step 3.2.8.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.2.8.4.1

Cancel the common factor.

$\frac{2 π}{2}$

Step 3.2.8.4.2

Divide $π$ by $1$ .

$π$

Step 3.2.9

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

$x = 0.72273424 + π n, 2.4188584 + π n$ , for any integer $n$

Step 4

Set $\cos (2 x) + 1$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 4.1

Set $\cos (2 x) + 1$ equal to $0$ .

$\cos (2 x) + 1 = 0$

Step 4.2

Solve $\cos (2 x) + 1 = 0$ for $x$ .

Tap for more steps...

Step 4.2.1

Subtract $1$ from both sides of the equation.

$\cos (2 x) = - 1$

Step 4.2.2

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

$2 x = \arccos (- 1)$

Step 4.2.3

Simplify the right side.

Tap for more steps...

Step 4.2.3.1

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

$2 x = π$

Step 4.2.4

Divide each term in $2 x = π$ by $2$ and simplify.

Tap for more steps...

Step 4.2.4.1

Divide each term in $2 x = π$ by $2$ .

$\frac{2 x}{2} = \frac{π}{2}$

Step 4.2.4.2

Simplify the left side.

Tap for more steps...

Step 4.2.4.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.2.4.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{π}{2}$

Step 4.2.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{π}{2}$

Step 4.2.5

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

$2 x = 2 π - π$

Step 4.2.6

Solve for $x$ .

Tap for more steps...

Step 4.2.6.1

Subtract $π$ from $2 π$ .

$2 x = π$

Step 4.2.6.2

Divide each term in $2 x = π$ by $2$ and simplify.

Tap for more steps...

Step 4.2.6.2.1

Divide each term in $2 x = π$ by $2$ .

$\frac{2 x}{2} = \frac{π}{2}$

Step 4.2.6.2.2

Simplify the left side.

Tap for more steps...

Step 4.2.6.2.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.2.6.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{π}{2}$

Step 4.2.6.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{π}{2}$

Step 4.2.7

Find the period of $\cos (2 x)$ .

Tap for more steps...

Step 4.2.7.1

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

$\frac{2 π}{| b |}$

Step 4.2.7.2

Replace $b$ with $2$ in the formula for period.

$\frac{2 π}{| 2 |}$

Step 4.2.7.3

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$\frac{2 π}{2}$

Step 4.2.7.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.2.7.4.1

Cancel the common factor.

$\frac{2 π}{2}$

Step 4.2.7.4.2

Divide $π$ by $1$ .

$π$

Step 4.2.8

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

$x = \frac{π}{2} + π n$ , for any integer $n$

Step 5

The final solution is all the values that make $(8 \cos (2 x) - 1) (\cos (2 x) + 1) = 0$ true.

$x = 0.72273424 + π n, 2.4188584 + π n, \frac{π}{2} + π n$ , for any integer $n$