Trigonometry Examples

$\cos (θ) - 4 = - 3$ , $[0, 2 π)$

Step 1

Move all terms not containing $\cos (θ)$ to the right side of the equation.

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Step 1.1

Add $4$ to both sides of the equation.

$\cos (θ) = - 3 + 4$

Step 1.2

Add $- 3$ and $4$ .

$\cos (θ) = 1$

Step 2

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

$θ = \arccos (1)$

Step 3

Simplify the right side.

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Step 3.1

The exact value of $\arccos (1)$ is $0$ .

$θ = 0$

Step 4

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.

$θ = 2 π - 0$

Step 5

Subtract $0$ from $2 π$ .

$θ = 2 π$

Step 6

Find the period of $\cos (θ)$ .

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Step 6.1

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

$\frac{2 π}{| b |}$

Step 6.2

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

$\frac{2 π}{| 1 |}$

Step 6.3

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

$\frac{2 π}{1}$

Step 6.4

Divide $2 π$ by $1$ .

$2 π$

Step 7

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

$θ = 2 π n, 2 π + 2 π n$ , for any integer $n$

Step 8

Consolidate the answers.

$θ = 2 π n$ , for any integer $n$

Step 9

Plug in $0$ for $n$ and simplify to see if the solution is contained in $[0, 2 π)$ .

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Step 9.1

Plug in $0$ for $n$ .

$2 π (0)$

Step 9.2

Multiply $2 π (0)$ .

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Step 9.2.1

Multiply $0$ by $2$ .

$0 π$

Step 9.2.2

Multiply $0$ by $π$ .

$0$

Step 9.3

The interval $[0, 2 π)$ contains $0$ .

$θ = 0$