Trigonometry Examples

$\cos (θ) + \sin (θ) \cdot \tan (θ) = 2$

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

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

Step 1.1.2

Multiply $\sin (θ) \frac{\sin (θ)}{\cos (θ)}$ .

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Step 1.1.2.1

Raise $\sin (θ)$ to the power of $1$ .

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

Step 1.1.2.2

Raise $\sin (θ)$ to the power of $1$ .

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

Step 1.1.2.3

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

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

Step 1.1.2.4

Add $1$ and $1$ .

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

Step 2

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

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

Step 3

Apply the distributive property.

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

Step 4

Multiply $\cos (θ) \cos (θ)$ .

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

Add $1$ and $1$ .

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

Step 5

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

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Step 5.1

Cancel the common factor.

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

Step 5.2

Rewrite the expression.

$\cos^{2} (θ) + \sin^{2} (θ) = \cos (θ) \cdot 2$

Step 6

Rearrange terms.

$\sin^{2} (θ) + \cos^{2} (θ) = \cos (θ) \cdot 2$

Step 7

Apply pythagorean identity.

$1 = \cos (θ) \cdot 2$

Step 8

Move $2$ to the left of $\cos (θ)$ .

$1 = 2 \cos (θ)$

Step 9

Rewrite the equation as $2 \cos (θ) = 1$ .

$2 \cos (θ) = 1$

Step 10

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

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Step 10.1

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

$\frac{2 \cos (θ)}{2} = \frac{1}{2}$

Step 10.2

Simplify the left side.

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Step 10.2.1

Cancel the common factor of $2$ .

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Step 10.2.1.1

Cancel the common factor.

$\frac{2 \cos (θ)}{2} = \frac{1}{2}$

Step 10.2.1.2

Divide $\cos (θ)$ by $1$ .

$\cos (θ) = \frac{1}{2}$

Step 11

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

$θ = \arccos (\frac{1}{2})$

Step 12

Simplify the right side.

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Step 12.1

The exact value of $\arccos (\frac{1}{2})$ is $\frac{π}{3}$ .

$θ = \frac{π}{3}$

Step 13

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.

$θ = 2 π - \frac{π}{3}$

Step 14

Simplify $2 π - \frac{π}{3}$ .

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Step 14.1

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$θ = 2 π \cdot \frac{3}{3} - \frac{π}{3}$

Step 14.2

Combine fractions.

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Step 14.2.1

Combine $2 π$ and $\frac{3}{3}$ .

$θ = \frac{2 π \cdot 3}{3} - \frac{π}{3}$

Step 14.2.2

Combine the numerators over the common denominator.

$θ = \frac{2 π \cdot 3 - π}{3}$

Step 14.3

Simplify the numerator.

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Step 14.3.1

Multiply $3$ by $2$ .

$θ = \frac{6 π - π}{3}$

Step 14.3.2

Subtract $π$ from $6 π$ .

$θ = \frac{5 π}{3}$

Step 15

Find the period of $\cos (θ)$ .

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Step 15.1

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

$\frac{2 π}{| b |}$

Step 15.2

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

$\frac{2 π}{| 1 |}$

Step 15.3

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

$\frac{2 π}{1}$

Step 15.4

Divide $2 π$ by $1$ .

$2 π$

Step 16

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

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