Trigonometry Examples

$3 \cos (y) = 2 \sec (y)$

Step 1

Divide each term in $3 \cos (y) = 2 \sec (y)$ by $3 \cos (y)$ and simplify.

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Step 1.1

Divide each term in $3 \cos (y) = 2 \sec (y)$ by $3 \cos (y)$ .

$\frac{3 \cos (y)}{3 \cos (y)} = \frac{2 \sec (y)}{3 \cos (y)}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $3$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{3 \cos (y)}{3 \cos (y)} = \frac{2 \sec (y)}{3 \cos (y)}$

Step 1.2.1.2

Rewrite the expression.

$\frac{\cos (y)}{\cos (y)} = \frac{2 \sec (y)}{3 \cos (y)}$

Step 1.2.2

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

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Step 1.2.2.1

Cancel the common factor.

$\frac{\cos (y)}{\cos (y)} = \frac{2 \sec (y)}{3 \cos (y)}$

Step 1.2.2.2

Rewrite the expression.

$1 = \frac{2 \sec (y)}{3 \cos (y)}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Separate fractions.

$1 = \frac{2}{3} \cdot \frac{\sec (y)}{\cos (y)}$

Step 1.3.2

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

$1 = \frac{2}{3} \cdot \frac{\frac{1}{\cos (y)}}{\cos (y)}$

Step 1.3.3

Rewrite $\frac{\frac{1}{\cos (y)}}{\cos (y)}$ as a product.

$1 = \frac{2}{3} (\frac{1}{\cos (y)} \cdot \frac{1}{\cos (y)})$

Step 1.3.4

Multiply $\frac{1}{\cos (y)}$ by $\frac{1}{\cos (y)}$ .

$1 = \frac{2}{3} \cdot \frac{1}{\cos (y) \cos (y)}$

Step 1.3.5

Simplify the denominator.

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Step 1.3.5.1

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

$1 = \frac{2}{3} \cdot \frac{1}{\cos^{1} (y) \cos (y)}$

Step 1.3.5.2

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

$1 = \frac{2}{3} \cdot \frac{1}{\cos^{1} (y) \cos^{1} (y)}$

Step 1.3.5.3

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

$1 = \frac{2}{3} \cdot \frac{1}{{\cos (y)}^{1 + 1}}$

Step 1.3.5.4

Add $1$ and $1$ .

$1 = \frac{2}{3} \cdot \frac{1}{\cos^{2} (y)}$

Step 1.3.6

Combine fractions.

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Step 1.3.6.1

Combine.

$1 = \frac{2 \cdot 1}{3 \cos^{2} (y)}$

Step 1.3.6.2

Multiply $2$ by $1$ .

$1 = \frac{2}{3 \cos^{2} (y)}$

Step 1.3.7

Multiply by $1$ .

$1 = \frac{2}{3 (\cos^{2} (y) \cdot 1)}$

Step 1.3.8

Separate fractions.

$1 = \frac{2}{3 \cdot (1)} \cdot \frac{1}{\cos^{2} (y)}$

Step 1.3.9

Convert from $\frac{1}{\cos^{2} (y)}$ to $\sec^{2} (y)$ .

$1 = \frac{2}{3 \cdot (1)} \sec^{2} (y)$

Step 1.3.10

Multiply $3$ by $1$ .

$1 = \frac{2}{3} \sec^{2} (y)$

Step 1.3.11

Combine $\frac{2}{3}$ and $\sec^{2} (y)$ .

$1 = \frac{2 \sec^{2} (y)}{3}$

Step 2

Rewrite the equation as $\frac{2 \sec^{2} (y)}{3} = 1$ .

$\frac{2 \sec^{2} (y)}{3} = 1$

Step 3

Multiply both sides of the equation by $\frac{3}{2}$ .

$\frac{3}{2} \cdot \frac{2 \sec^{2} (y)}{3} = \frac{3}{2} \cdot 1$

Step 4

Simplify both sides of the equation.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Simplify $\frac{3}{2} \cdot \frac{2 \sec^{2} (y)}{3}$ .

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Step 4.1.1.1

Combine.

$\frac{3 (2 \sec^{2} (y))}{2 \cdot 3} = \frac{3}{2} \cdot 1$

Step 4.1.1.2

Cancel the common factor of $3$ .

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Step 4.1.1.2.1

Cancel the common factor.

$\frac{3 (2 \sec^{2} (y))}{2 \cdot 3} = \frac{3}{2} \cdot 1$

Step 4.1.1.2.2

Rewrite the expression.

$\frac{2 \sec^{2} (y)}{2} = \frac{3}{2} \cdot 1$

Step 4.1.1.3

Cancel the common factor of $2$ .

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Step 4.1.1.3.1

Cancel the common factor.

$\frac{2 \sec^{2} (y)}{2} = \frac{3}{2} \cdot 1$

Step 4.1.1.3.2

Divide $\sec^{2} (y)$ by $1$ .

$\sec^{2} (y) = \frac{3}{2} \cdot 1$

Step 4.2

Simplify the right side.

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Step 4.2.1

Multiply $\frac{3}{2}$ by $1$ .

$\sec^{2} (y) = \frac{3}{2}$

Step 5

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$\sec (y) = \pm \sqrt{\frac{3}{2}}$

Step 6

Simplify $\pm \sqrt{\frac{3}{2}}$ .

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Step 6.1

Rewrite $\sqrt{\frac{3}{2}}$ as $\frac{\sqrt{3}}{\sqrt{2}}$ .

$\sec (y) = \pm \frac{\sqrt{3}}{\sqrt{2}}$

Step 6.2

Multiply $\frac{\sqrt{3}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\sec (y) = \pm \frac{\sqrt{3}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 6.3

Combine and simplify the denominator.

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Step 6.3.1

Multiply $\frac{\sqrt{3}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\sec (y) = \pm \frac{\sqrt{3} \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 6.3.2

Raise $\sqrt{2}$ to the power of $1$ .

$\sec (y) = \pm \frac{\sqrt{3} \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 6.3.3

Raise $\sqrt{2}$ to the power of $1$ .

$\sec (y) = \pm \frac{\sqrt{3} \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 6.3.4

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

$\sec (y) = \pm \frac{\sqrt{3} \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 6.3.5

Add $1$ and $1$ .

$\sec (y) = \pm \frac{\sqrt{3} \sqrt{2}}{{\sqrt{2}}^{2}}$

Step 6.3.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 6.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$\sec (y) = \pm \frac{\sqrt{3} \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 6.3.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\sec (y) = \pm \frac{\sqrt{3} \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 6.3.6.3

Combine $\frac{1}{2}$ and $2$ .

$\sec (y) = \pm \frac{\sqrt{3} \sqrt{2}}{2^{\frac{2}{2}}}$

Step 6.3.6.4

Cancel the common factor of $2$ .

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Step 6.3.6.4.1

Cancel the common factor.

$\sec (y) = \pm \frac{\sqrt{3} \sqrt{2}}{2^{\frac{2}{2}}}$

Step 6.3.6.4.2

Rewrite the expression.

$\sec (y) = \pm \frac{\sqrt{3} \sqrt{2}}{2^{1}}$

Step 6.3.6.5

Evaluate the exponent.

$\sec (y) = \pm \frac{\sqrt{3} \sqrt{2}}{2}$

Step 6.4

Simplify the numerator.

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Step 6.4.1

Combine using the product rule for radicals.

$\sec (y) = \pm \frac{\sqrt{3 \cdot 2}}{2}$

Step 6.4.2

Multiply $3$ by $2$ .

$\sec (y) = \pm \frac{\sqrt{6}}{2}$

Step 7

The complete solution is the result of both the positive and negative portions of the solution.

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Step 7.1

First, use the positive value of the $\pm$ to find the first solution.

$\sec (y) = \frac{\sqrt{6}}{2}$

Step 7.2

Next, use the negative value of the $\pm$ to find the second solution.

$\sec (y) = - \frac{\sqrt{6}}{2}$

Step 7.3

The complete solution is the result of both the positive and negative portions of the solution.

$\sec (y) = \frac{\sqrt{6}}{2}, - \frac{\sqrt{6}}{2}$

Step 8

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

$\sec (y) = \frac{\sqrt{6}}{2}$

$\sec (y) = - \frac{\sqrt{6}}{2}$

Step 9

Solve for $y$ in $\sec (y) = \frac{\sqrt{6}}{2}$ .

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Step 9.1

Take the inverse secant of both sides of the equation to extract $y$ from inside the secant.

$y = arcsec (\frac{\sqrt{6}}{2})$

Step 9.2

Simplify the right side.

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Step 9.2.1

Evaluate $arcsec (\frac{\sqrt{6}}{2})$ .

$y = 0.6154797$

Step 9.3

The secant 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.

$y = 2 (3.14159265) - 0.6154797$

Step 9.4

Solve for $y$ .

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Step 9.4.1

Remove parentheses.

$y = 2 (3.14159265) - 0.6154797$

Step 9.4.2

Simplify $2 (3.14159265) - 0.6154797$ .

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Step 9.4.2.1

Multiply $2$ by $3.14159265$ .

$y = 6.2831853 - 0.6154797$

Step 9.4.2.2

Subtract $0.6154797$ from $6.2831853$ .

$y = 5.66770559$

Step 9.5

Find the period of $\sec (y)$ .

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Step 9.5.1

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

$\frac{2 π}{| b |}$

Step 9.5.2

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

$\frac{2 π}{| 1 |}$

Step 9.5.3

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

$\frac{2 π}{1}$

Step 9.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 9.6

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

$y = 0.6154797 + 2 π n, 5.66770559 + 2 π n$ , for any integer $n$

Step 10

Solve for $y$ in $\sec (y) = - \frac{\sqrt{6}}{2}$ .

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Step 10.1

Take the inverse secant of both sides of the equation to extract $y$ from inside the secant.

$y = arcsec (- \frac{\sqrt{6}}{2})$

Step 10.2

Simplify the right side.

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Step 10.2.1

Evaluate $arcsec (- \frac{\sqrt{6}}{2})$ .

$y = 2.52611294$

Step 10.3

The secant 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.

$y = 2 (3.14159265) - 2.52611294$

Step 10.4

Solve for $y$ .

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Step 10.4.1

Remove parentheses.

$y = 2 (3.14159265) - 2.52611294$

Step 10.4.2

Simplify $2 (3.14159265) - 2.52611294$ .

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Step 10.4.2.1

Multiply $2$ by $3.14159265$ .

$y = 6.2831853 - 2.52611294$

Step 10.4.2.2

Subtract $2.52611294$ from $6.2831853$ .

$y = 3.75707236$

Step 10.5

Find the period of $\sec (y)$ .

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Step 10.5.1

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

$\frac{2 π}{| b |}$

Step 10.5.2

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

$\frac{2 π}{| 1 |}$

Step 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 10.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 10.6

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

$y = 2.52611294 + 2 π n, 3.75707236 + 2 π n$ , for any integer $n$

Step 11

List all of the solutions.

$y = 0.6154797 + 2 π n, 5.66770559 + 2 π n, 2.52611294 + 2 π n, 3.75707236 + 2 π n$ , for any integer $n$

Step 12

Consolidate the solutions.

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Step 12.1

Consolidate $0.6154797 + 2 π n$ and $3.75707236 + 2 π n$ to $0.6154797 + π n$ .

$y = 0.6154797 + π n, 5.66770559 + 2 π n, 2.52611294 + 2 π n$ , for any integer $n$

Step 12.2

Consolidate $5.66770559 + 2 π n$ and $2.52611294 + 2 π n$ to $2.52611294 + π n$ .

$y = 0.6154797 + π n, 2.52611294 + π n$ , for any integer $n$