Trigonometry Examples

$f (x) = 2 \cos (2 x - π) + 4$

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 $x$ .

$0 = 2 \cos (2 x - π) + 4$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $2 \cos (2 x - π) + 4 = 0$ .

$2 \cos (2 x - π) + 4 = 0$

Step 1.2.2

Subtract $4$ from both sides of the equation.

$2 \cos (2 x - π) = - 4$

Step 1.2.3

Divide each term in $2 \cos (2 x - π) = - 4$ by $2$ and simplify.

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Step 1.2.3.1

Divide each term in $2 \cos (2 x - π) = - 4$ by $2$ .

$\frac{2 \cos (2 x - π)}{2} = \frac{- 4}{2}$

Step 1.2.3.2

Simplify the left side.

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Step 1.2.3.2.1

Cancel the common factor of $2$ .

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Step 1.2.3.2.1.1

Cancel the common factor.

$\frac{2 \cos (2 x - π)}{2} = \frac{- 4}{2}$

Step 1.2.3.2.1.2

Divide $\cos (2 x - π)$ by $1$ .

$\cos (2 x - π) = \frac{- 4}{2}$

Step 1.2.3.3

Simplify the right side.

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Step 1.2.3.3.1

Divide $- 4$ by $2$ .

$\cos (2 x - π) = - 2$

Step 1.2.4

The range of cosine is $- 1 \leq y \leq 1$ . Since $- 2$ does not fall in this range, there is no solution.

No solution

Step 1.3

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

x-intercept(s): $None$

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 $y$ .

$y = 2 \cos (2 (0) - π) + 4$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = 2 \cos (2 (0) - π) + 4$

Step 2.2.2

Simplify $2 \cos (2 (0) - π) + 4$ .

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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 $0$ .

$y = 2 \cos (0 - π) + 4$

Step 2.2.2.1.2

Subtract $π$ from $0$ .

$y = 2 \cos (- π) + 4$

Step 2.2.2.1.3

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

$y = 2 (- \cos (0)) + 4$

Step 2.2.2.1.4

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

$y = 2 (- 1 \cdot 1) + 4$

Step 2.2.2.1.5

Multiply $2 (- 1 \cdot 1)$ .

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Step 2.2.2.1.5.1

Multiply $- 1$ by $1$ .

$y = 2 \cdot - 1 + 4$

Step 2.2.2.1.5.2

Multiply $2$ by $- 1$ .

$y = - 2 + 4$

Step 2.2.2.2

Add $- 2$ and $4$ .

$y = 2$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 2)$

Step 3

List the intersections.

x-intercept(s): $None$

y-intercept(s): $(0, 2)$

Step 4