Trigonometry Examples

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

Step 1

Divide each term in the equation by $\cos (x)$ .

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

Step 2

Replace $\cos (x)$ with an equivalent expression in the numerator.

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

Step 3

Apply the distributive property.

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

Step 4

Multiply $4$ by $1$ .

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

Step 5

Rewrite $\sec (x)$ in terms of sines and cosines.

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

Step 6

Apply the distributive property.

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

Step 7

Combine $4$ and $\frac{1}{\cos (x)}$ .

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

Step 8

Multiply $4 \sin (x) \frac{1}{\cos (x)}$ .

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Step 8.1

Combine $\frac{1}{\cos (x)}$ and $4$ .

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

Step 8.2

Combine $\frac{4}{\cos (x)}$ and $\sin (x)$ .

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

Step 9

Simplify each term.

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Step 9.1

Separate fractions.

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

Step 9.2

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

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

Step 9.3

Divide $4$ by $1$ .

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

Step 9.4

Separate fractions.

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

Step 9.5

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

$4 \sec (x) + \frac{4}{1} \cdot \tan (x) = \frac{\cos^{2} (x)}{\cos (x)}$

Step 9.6

Divide $4$ by $1$ .

$4 \sec (x) + 4 \tan (x) = \frac{\cos^{2} (x)}{\cos (x)}$

Step 10

Cancel the common factor of $\cos^{2} (x)$ and $\cos (x)$ .

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Step 10.1

Factor $\cos (x)$ out of $\cos^{2} (x)$ .

$4 \sec (x) + 4 \tan (x) = \frac{\cos (x) \cos (x)}{\cos (x)}$

Step 10.2

Cancel the common factors.

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Step 10.2.1

Multiply by $1$ .

$4 \sec (x) + 4 \tan (x) = \frac{\cos (x) \cos (x)}{\cos (x) \cdot 1}$

Step 10.2.2

Cancel the common factor.

$4 \sec (x) + 4 \tan (x) = \frac{\cos (x) \cos (x)}{\cos (x) \cdot 1}$

Step 10.2.3

Rewrite the expression.

$4 \sec (x) + 4 \tan (x) = \frac{\cos (x)}{1}$

Step 10.2.4

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

$4 \sec (x) + 4 \tan (x) = \cos (x)$

Step 11

Simplify the left side.

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Step 11.1

Simplify each term.

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Step 11.1.1

Rewrite $\sec (x)$ in terms of sines and cosines.

$4 \frac{1}{\cos (x)} + 4 \tan (x) = \cos (x)$

Step 11.1.2

Combine $4$ and $\frac{1}{\cos (x)}$ .

$\frac{4}{\cos (x)} + 4 \tan (x) = \cos (x)$

Step 11.1.3

Rewrite $\tan (x)$ in terms of sines and cosines.

$\frac{4}{\cos (x)} + 4 \frac{\sin (x)}{\cos (x)} = \cos (x)$

Step 11.1.4

Combine $4$ and $\frac{\sin (x)}{\cos (x)}$ .

$\frac{4}{\cos (x)} + \frac{4 \sin (x)}{\cos (x)} = \cos (x)$

Step 12

Multiply both sides of the equation by $\cos (x)$ .

$\cos (x) (\frac{4}{\cos (x)} + \frac{4 \sin (x)}{\cos (x)}) = \cos (x) \cos (x)$

Step 13

Apply the distributive property.

$\cos (x) \frac{4}{\cos (x)} + \cos (x) \frac{4 \sin (x)}{\cos (x)} = \cos (x) \cos (x)$

Step 14

Cancel the common factor of $\cos (x)$ .

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Step 14.1

Cancel the common factor.

$\cos (x) \frac{4}{\cos (x)} + \cos (x) \frac{4 \sin (x)}{\cos (x)} = \cos (x) \cos (x)$

Step 14.2

Rewrite the expression.

$4 + \cos (x) \frac{4 \sin (x)}{\cos (x)} = \cos (x) \cos (x)$

Step 15

Cancel the common factor of $\cos (x)$ .

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Step 15.1

Cancel the common factor.

$4 + \cos (x) \frac{4 \sin (x)}{\cos (x)} = \cos (x) \cos (x)$

Step 15.2

Rewrite the expression.

$4 + 4 \sin (x) = \cos (x) \cos (x)$

Step 16

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

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Step 16.1

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

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

Step 16.2

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

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

Step 16.3

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

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

Step 16.4

Add $1$ and $1$ .

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

Step 17

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

$4 + 4 \sin (x) - \cos^{2} (x) = 0$

Step 18

Replace $\cos^{2} (x)$ with $1 - \sin^{2} (x)$ .

$4 + 4 \sin (x) - (1 - \sin^{2} (x)) = 0$

Step 19

Solve for $x$ .

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Step 19.1

Substitute $u$ for $\sin (x)$ .

$4 + 4 (u) - (1 - {(u)}^{2}) = 0$

Step 19.2

Simplify $4 + 4 u - (1 - u^{2})$ .

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Step 19.2.1

Simplify each term.

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Step 19.2.1.1

Apply the distributive property.

$4 + 4 u - 1 \cdot 1 - - u^{2} = 0$

Step 19.2.1.2

Multiply $- 1$ by $1$ .

$4 + 4 u - 1 - - u^{2} = 0$

Step 19.2.1.3

Multiply $- - u^{2}$ .

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Step 19.2.1.3.1

Multiply $- 1$ by $- 1$ .

$4 + 4 u - 1 + 1 u^{2} = 0$

Step 19.2.1.3.2

Multiply $u^{2}$ by $1$ .

$4 + 4 u - 1 + u^{2} = 0$

Step 19.2.2

Subtract $1$ from $4$ .

$4 u + 3 + u^{2} = 0$

Step 19.3

Factor $4 u + 3 + u^{2}$ using the AC method.

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Step 19.3.1

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 $4$ .

$1, 3$

Step 19.3.2

Write the factored form using these integers.

$(u + 1) (u + 3) = 0$

Step 19.4

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

$u + 1 = 0$

$u + 3 = 0$

Step 19.5

Set $u + 1$ equal to $0$ and solve for $u$ .

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Step 19.5.1

Set $u + 1$ equal to $0$ .

$u + 1 = 0$

Step 19.5.2

Subtract $1$ from both sides of the equation.

$u = - 1$

Step 19.6

Set $u + 3$ equal to $0$ and solve for $u$ .

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Step 19.6.1

Set $u + 3$ equal to $0$ .

$u + 3 = 0$

Step 19.6.2

Subtract $3$ from both sides of the equation.

$u = - 3$

Step 19.7

The final solution is all the values that make $(u + 1) (u + 3) = 0$ true.

$u = - 1, - 3$

Step 19.8

Substitute $\sin (x)$ for $u$ .

$\sin (x) = - 1, - 3$

Step 19.9

Set up each of the solutions to solve for $x$ .

$\sin (x) = - 1$

$\sin (x) = - 3$

Step 19.10

Solve for $x$ in $\sin (x) = - 1$ .

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Step 19.10.1

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

$x = \arcsin (- 1)$

Step 19.10.2

Simplify the right side.

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Step 19.10.2.1

The exact value of $\arcsin (- 1)$ is $- \frac{π}{2}$ .

$x = - \frac{π}{2}$

Step 19.10.3

The sine function is negative in the third and fourth quadrants. To find the second solution, subtract the solution from $2 π$ , to find a reference angle. Next, add this reference angle to $π$ to find the solution in the third quadrant.

$x = 2 π + \frac{π}{2} + π$

Step 19.10.4

Simplify the expression to find the second solution.

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Step 19.10.4.1

Subtract $2 π$ from $2 π + \frac{π}{2} + π$ .

$x = 2 π + \frac{π}{2} + π - 2 π$

Step 19.10.4.2

The resulting angle of $\frac{3 π}{2}$ is positive, less than $2 π$ , and coterminal with $2 π + \frac{π}{2} + π$ .

$x = \frac{3 π}{2}$

Step 19.10.5

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

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Step 19.10.5.1

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

$\frac{2 π}{| b |}$

Step 19.10.5.2

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

$\frac{2 π}{| 1 |}$

Step 19.10.5.3

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

$\frac{2 π}{1}$

Step 19.10.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 19.10.6

Add $2 π$ to every negative angle to get positive angles.

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Step 19.10.6.1

Add $2 π$ to $- \frac{π}{2}$ to find the positive angle.

$- \frac{π}{2} + 2 π$

Step 19.10.6.2

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$2 π \cdot \frac{2}{2} - \frac{π}{2}$

Step 19.10.6.3

Combine fractions.

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Step 19.10.6.3.1

Combine $2 π$ and $\frac{2}{2}$ .

$\frac{2 π \cdot 2}{2} - \frac{π}{2}$

Step 19.10.6.3.2

Combine the numerators over the common denominator.

$\frac{2 π \cdot 2 - π}{2}$

Step 19.10.6.4

Simplify the numerator.

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Step 19.10.6.4.1

Multiply $2$ by $2$ .

$\frac{4 π - π}{2}$

Step 19.10.6.4.2

Subtract $π$ from $4 π$ .

$\frac{3 π}{2}$

Step 19.10.6.5

List the new angles.

$x = \frac{3 π}{2}$

Step 19.10.7

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

$x = \frac{3 π}{2} + 2 π n, \frac{3 π}{2} + 2 π n$ , for any integer $n$

Step 19.11

Solve for $x$ in $\sin (x) = - 3$ .

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Step 19.11.1

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

No solution

Step 19.12

List all of the solutions.

$x = \frac{3 π}{2} + 2 π n, \frac{3 π}{2} + 2 π n$ , for any integer $n$

Step 19.13

Consolidate the answers.

$x = \frac{3 π}{2} + 2 π n$ , for any integer $n$