Trigonometry Examples

$y = \cos (x - 3)$

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 $x$ .

$0 = \cos (x - 3)$

Step 1.2

Solve the equation.

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Step 1.2.1

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

$\cos (x - 3) = 0$

Step 1.2.2

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

$x - 3 = \arccos (0)$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

The exact value of $\arccos (0)$ is $\frac{π}{2}$ .

$x - 3 = \frac{π}{2}$

Step 1.2.4

Add $3$ to both sides of the equation.

$x = \frac{π}{2} + 3$

Step 1.2.5

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.

$x - 3 = 2 π - \frac{π}{2}$

Step 1.2.6

Solve for $x$ .

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Step 1.2.6.1

Simplify $2 π - \frac{π}{2}$ .

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Step 1.2.6.1.1

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

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

Step 1.2.6.1.2

Combine fractions.

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Step 1.2.6.1.2.1

Combine $2 π$ and $\frac{2}{2}$ .

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

Step 1.2.6.1.2.2

Combine the numerators over the common denominator.

$x - 3 = \frac{2 π \cdot 2 - π}{2}$

Step 1.2.6.1.3

Simplify the numerator.

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Step 1.2.6.1.3.1

Multiply $2$ by $2$ .

$x - 3 = \frac{4 π - π}{2}$

Step 1.2.6.1.3.2

Subtract $π$ from $4 π$ .

$x - 3 = \frac{3 π}{2}$

Step 1.2.6.2

Add $3$ to both sides of the equation.

$x = \frac{3 π}{2} + 3$

Step 1.2.7

Find the period of $\cos (x - 3)$ .

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Step 1.2.7.1

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

$\frac{2 π}{| b |}$

Step 1.2.7.2

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

$\frac{2 π}{| 1 |}$

Step 1.2.7.3

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

$\frac{2 π}{1}$

Step 1.2.7.4

Divide $2 π$ by $1$ .

$2 π$

Step 1.2.8

The period of the $\cos (x - 3)$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

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

Step 1.2.9

Consolidate the answers.

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

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(\frac{π}{2} + 3 + π n, 0)$ , 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 $y$ .

$y = \cos ((0) - 3)$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = \cos (0 - 3)$

Step 2.2.2

Remove parentheses.

$y = \cos ((0) - 3)$

Step 2.2.3

Simplify $\cos ((0) - 3)$ .

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Step 2.2.3.1

Subtract $3$ from $0$ .

$y = \cos (- 3)$

Step 2.2.3.2

Evaluate $\cos (- 3)$ .

$y = - 0.98999249$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - 0.98999249)$

Step 3

List the intersections.

x-intercept(s): $(\frac{π}{2} + 3 + π n, 0)$ , for any integer $n$

y-intercept(s): $(0, - 0.98999249)$

Step 4