Trigonometry Examples

$\tan (x) + \sec (x) = 2 \cos (x)$

Step 1

Simplify the left side.

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Step 1.1

Simplify each term.

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Step 1.1.1

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

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

Step 1.1.2

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

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

Step 2

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

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

Step 3

Apply the distributive property.

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

Step 4

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

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Step 4.1

Cancel the common factor.

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

Step 4.2

Rewrite the expression.

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

Step 5

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

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Step 5.1

Cancel the common factor.

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

Step 5.2

Rewrite the expression.

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

Step 6

Rewrite using the commutative property of multiplication.

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

Step 7

Multiply $2 \cos (x) \cos (x)$ .

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Step 7.1

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

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

Step 7.2

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

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

Step 7.3

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

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

Step 7.4

Add $1$ and $1$ .

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

Step 8

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

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

Step 9

Replace the $- 2 \cos^{2} (x)$ with $- 2 (1 - \sin^{2} (x))$ based on the $\sin^{2} (x) + \cos^{2} (x) = 1$ identity.

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

Step 10

Simplify each term.

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Step 10.1

Apply the distributive property.

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

Step 10.2

Multiply $- 2$ by $1$ .

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

Step 10.3

Multiply $- 1$ by $- 2$ .

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

Step 11

Subtract $2$ from $1$ .

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

Step 12

Reorder the polynomial.

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

Step 13

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

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

Step 14

Factor by grouping.

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Step 14.1

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 $a \cdot c = 2 \cdot - 1 = - 2$ and whose sum is $b = 1$ .

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Step 14.1.1

Multiply by $1$ .

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

Step 14.1.2

Rewrite $1$ as $- 1$ plus $2$

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

Step 14.1.3

Apply the distributive property.

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

Step 14.2

Factor out the greatest common factor from each group.

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Step 14.2.1

Group the first two terms and the last two terms.

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

Step 14.2.2

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

$u (2 u - 1) + 1 (2 u - 1) = 0$

Step 14.3

Factor the polynomial by factoring out the greatest common factor, $2 u - 1$ .

$(2 u - 1) (u + 1) = 0$

Step 15

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

$2 u - 1 = 0$

$u + 1 = 0$

Step 16

Set $2 u - 1$ equal to $0$ and solve for $u$ .

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Step 16.1

Set $2 u - 1$ equal to $0$ .

$2 u - 1 = 0$

Step 16.2

Solve $2 u - 1 = 0$ for $u$ .

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Step 16.2.1

Add $1$ to both sides of the equation.

$2 u = 1$

Step 16.2.2

Divide each term in $2 u = 1$ by $2$ and simplify.

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Step 16.2.2.1

Divide each term in $2 u = 1$ by $2$ .

$\frac{2 u}{2} = \frac{1}{2}$

Step 16.2.2.2

Simplify the left side.

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Step 16.2.2.2.1

Cancel the common factor of $2$ .

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Step 16.2.2.2.1.1

Cancel the common factor.

$\frac{2 u}{2} = \frac{1}{2}$

Step 16.2.2.2.1.2

Divide $u$ by $1$ .

$u = \frac{1}{2}$

Step 17

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

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Step 17.1

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

$u + 1 = 0$

Step 17.2

Subtract $1$ from both sides of the equation.

$u = - 1$

Step 18

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

$u = \frac{1}{2}, - 1$

Step 19

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

$\sin (x) = \frac{1}{2}, - 1$

Step 20

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

$\sin (x) = \frac{1}{2}$

$\sin (x) = - 1$

Step 21

Solve for $x$ in $\sin (x) = \frac{1}{2}$ .

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Step 21.1

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

$x = \arcsin (\frac{1}{2})$

Step 21.2

Simplify the right side.

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Step 21.2.1

The exact value of $\arcsin (\frac{1}{2})$ is $\frac{π}{6}$ .

$x = \frac{π}{6}$

Step 21.3

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

$x = π - \frac{π}{6}$

Step 21.4

Simplify $π - \frac{π}{6}$ .

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Step 21.4.1

To write $π$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$x = π \cdot \frac{6}{6} - \frac{π}{6}$

Step 21.4.2

Combine fractions.

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Step 21.4.2.1

Combine $π$ and $\frac{6}{6}$ .

$x = \frac{π \cdot 6}{6} - \frac{π}{6}$

Step 21.4.2.2

Combine the numerators over the common denominator.

$x = \frac{π \cdot 6 - π}{6}$

Step 21.4.3

Simplify the numerator.

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Step 21.4.3.1

Move $6$ to the left of $π$ .

$x = \frac{6 \cdot π - π}{6}$

Step 21.4.3.2

Subtract $π$ from $6 π$ .

$x = \frac{5 π}{6}$

Step 21.5

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

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Step 21.5.1

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

$\frac{2 π}{| b |}$

Step 21.5.2

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

$\frac{2 π}{| 1 |}$

Step 21.5.3

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

$\frac{2 π}{1}$

Step 21.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 21.6

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

$x = \frac{π}{6} + 2 π n, \frac{5 π}{6} + 2 π n$ , for any integer $n$

Step 22

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

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Step 22.1

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

$x = \arcsin (- 1)$

Step 22.2

Simplify the right side.

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Step 22.2.1

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

$x = - \frac{π}{2}$

Step 22.3

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 22.4

Simplify the expression to find the second solution.

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Step 22.4.1

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

$x = 2 π + \frac{π}{2} + π - 2 π$

Step 22.4.2

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 22.5

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

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Step 22.5.1

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

$\frac{2 π}{| b |}$

Step 22.5.2

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

$\frac{2 π}{| 1 |}$

Step 22.5.3

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

$\frac{2 π}{1}$

Step 22.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 22.6

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

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Step 22.6.1

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

$- \frac{π}{2} + 2 π$

Step 22.6.2

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 22.6.3

Combine fractions.

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Step 22.6.3.1

Combine $2 π$ and $\frac{2}{2}$ .

$\frac{2 π \cdot 2}{2} - \frac{π}{2}$

Step 22.6.3.2

Combine the numerators over the common denominator.

$\frac{2 π \cdot 2 - π}{2}$

Step 22.6.4

Simplify the numerator.

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Step 22.6.4.1

Multiply $2$ by $2$ .

$\frac{4 π - π}{2}$

Step 22.6.4.2

Subtract $π$ from $4 π$ .

$\frac{3 π}{2}$

Step 22.6.5

List the new angles.

$x = \frac{3 π}{2}$

Step 22.7

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 23

List all of the solutions.

$x = \frac{π}{6} + 2 π n, \frac{5 π}{6} + 2 π n, \frac{3 π}{2} + 2 π n, \frac{3 π}{2} + 2 π n$ , for any integer $n$

Step 24

Consolidate the answers.

$x = \frac{π}{6} + \frac{2 π n}{3}$ , for any integer $n$