Trigonometry Examples

$f (x) = 3 \cos (2 x) + 1$

Step 1

Find the x-intercepts.

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Step 1.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = 3 \cos (2 x) + 1$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $3 \cos (2 x) + 1 = 0$ .

$3 \cos (2 x) + 1 = 0$

Step 1.2.2

Subtract $1$ from both sides of the equation.

$3 \cos (2 x) = - 1$

Step 1.2.3

Divide each term in $3 \cos (2 x) = - 1$ by $3$ and simplify.

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Step 1.2.3.1

Divide each term in $3 \cos (2 x) = - 1$ by $3$ .

$\frac{3 \cos (2 x)}{3} = \frac{- 1}{3}$

Step 1.2.3.2

Simplify the left side.

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Step 1.2.3.2.1

Cancel the common factor of $3$ .

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Step 1.2.3.2.1.1

Cancel the common factor.

$\frac{3 \cos (2 x)}{3} = \frac{- 1}{3}$

Step 1.2.3.2.1.2

Divide $\cos (2 x)$ by $1$ .

$\cos (2 x) = \frac{- 1}{3}$

Step 1.2.3.3

Simplify the right side.

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Step 1.2.3.3.1

Move the negative in front of the fraction.

$\cos (2 x) = - \frac{1}{3}$

Step 1.2.4

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

$2 x = \arccos (- \frac{1}{3})$

Step 1.2.5

Simplify the right side.

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Step 1.2.5.1

Evaluate $\arccos (- \frac{1}{3})$ .

$2 x = 1.91063323$

Step 1.2.6

Divide each term in $2 x = 1.91063323$ by $2$ and simplify.

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Step 1.2.6.1

Divide each term in $2 x = 1.91063323$ by $2$ .

$\frac{2 x}{2} = \frac{1.91063323}{2}$

Step 1.2.6.2

Simplify the left side.

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Step 1.2.6.2.1

Cancel the common factor of $2$ .

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Step 1.2.6.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{1.91063323}{2}$

Step 1.2.6.2.1.2

Divide $x$ by $1$ .

$x = \frac{1.91063323}{2}$

Step 1.2.6.3

Simplify the right side.

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Step 1.2.6.3.1

Divide $1.91063323$ by $2$ .

$x = 0.95531661$

Step 1.2.7

The cosine 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.

$2 x = 2 (3.14159265) - 1.91063323$

Step 1.2.8

Solve for $x$ .

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Step 1.2.8.1

Simplify.

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Step 1.2.8.1.1

Multiply $2$ by $3.14159265$ .

$2 x = 6.2831853 - 1.91063323$

Step 1.2.8.1.2

Subtract $1.91063323$ from $6.2831853$ .

$2 x = 4.37255207$

Step 1.2.8.2

Divide each term in $2 x = 4.37255207$ by $2$ and simplify.

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Step 1.2.8.2.1

Divide each term in $2 x = 4.37255207$ by $2$ .

$\frac{2 x}{2} = \frac{4.37255207}{2}$

Step 1.2.8.2.2

Simplify the left side.

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Step 1.2.8.2.2.1

Cancel the common factor of $2$ .

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Step 1.2.8.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{4.37255207}{2}$

Step 1.2.8.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{4.37255207}{2}$

Step 1.2.8.2.3

Simplify the right side.

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Step 1.2.8.2.3.1

Divide $4.37255207$ by $2$ .

$x = 2.18627603$

Step 1.2.9

Find the period of $\cos (2 x)$ .

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Step 1.2.9.1

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

$\frac{2 π}{| b |}$

Step 1.2.9.2

Replace $b$ with $2$ in the formula for period.

$\frac{2 π}{| 2 |}$

Step 1.2.9.3

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$\frac{2 π}{2}$

Step 1.2.9.4

Cancel the common factor of $2$ .

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Step 1.2.9.4.1

Cancel the common factor.

$\frac{2 π}{2}$

Step 1.2.9.4.2

Divide $π$ by $1$ .

$π$

Step 1.2.10

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

$x = 0.95531661 + π n, 2.18627603 + π n$ , for any integer $n$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(0.95531661 + π n, 0), (2.18627603 + π n, 0)$ , for any integer $n$

Step 2

Find the y-intercepts.

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Step 2.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = 3 \cos (2 (0)) + 1$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = 3 \cos (2 (0)) + 1$

Step 2.2.2

Simplify $3 \cos (2 (0)) + 1$ .

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Step 2.2.2.1

Simplify each term.

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Step 2.2.2.1.1

Multiply $2$ by $0$ .

$y = 3 \cos (0) + 1$

Step 2.2.2.1.2

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

$y = 3 \cdot 1 + 1$

Step 2.2.2.1.3

Multiply $3$ by $1$ .

$y = 3 + 1$

Step 2.2.2.2

Add $3$ and $1$ .

$y = 4$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 4)$

Step 3

List the intersections.

x-intercept(s): $(0.95531661 + π n, 0), (2.18627603 + π n, 0)$ , for any integer $n$

y-intercept(s): $(0, 4)$

Step 4