Trigonometry Examples

$y = - 2 \cos (3 x) + 4$

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 $x$ .

$0 = - 2 \cos (3 x) + 4$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $- 2 \cos (3 x) + 4 = 0$ .

$- 2 \cos (3 x) + 4 = 0$

Step 1.2.2

Subtract $4$ from both sides of the equation.

$- 2 \cos (3 x) = - 4$

Step 1.2.3

Divide each term in $- 2 \cos (3 x) = - 4$ by $- 2$ and simplify.

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Step 1.2.3.1

Divide each term in $- 2 \cos (3 x) = - 4$ by $- 2$ .

$\frac{- 2 \cos (3 x)}{- 2} = \frac{- 4}{- 2}$

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.

$\frac{- 2 \cos (3 x)}{- 2} = \frac{- 4}{- 2}$

Step 1.2.3.2.1.2

Divide $\cos (3 x)$ by $1$ .

$\cos (3 x) = \frac{- 4}{- 2}$

Step 1.2.3.3

Simplify the right side.

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Step 1.2.3.3.1

Divide $- 4$ by $- 2$ .

$\cos (3 x) = 2$

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 $y$ .

$y = - 2 \cos (3 (0)) + 4$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = - 2 \cos (3 (0)) + 4$

Step 2.2.2

Simplify $- 2 \cos (3 (0)) + 4$ .

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Step 2.2.2.1

Simplify each term.

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Step 2.2.2.1.1

Multiply $3$ by $0$ .

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

Step 2.2.2.1.2

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

$y = - 2 \cdot 1 + 4$

Step 2.2.2.1.3

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