Trigonometry Examples

$\cos (x) \sec (x)$

Step 1

Write $\cos (x) \sec (x)$ as an equation.

$y = \cos (x) \sec (x)$

Step 2

Find the x-intercepts.

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Step 2.1

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

$0 = \cos (x) \sec (x)$

Step 2.2

Solve the equation.

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Step 2.2.1

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

$\cos (x) \sec (x) = 0$

Step 2.2.2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$\cos (x) = 0$

$\sec (x) = 0$

Step 2.2.3

Set $\cos (x)$ equal to $0$ and solve for $x$ .

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Step 2.2.3.1

Set $\cos (x)$ equal to $0$ .

$\cos (x) = 0$

Step 2.2.3.2

Solve $\cos (x) = 0$ for $x$ .

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Step 2.2.3.2.1

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

$x = \arccos (0)$

Step 2.2.3.2.2

Simplify the right side.

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Step 2.2.3.2.2.1

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

$x = \frac{π}{2}$

Step 2.2.3.2.3

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.

$x = 2 π - \frac{π}{2}$

Step 2.2.3.2.4

Simplify $2 π - \frac{π}{2}$ .

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Step 2.2.3.2.4.1

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

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

Step 2.2.3.2.4.2

Combine fractions.

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Step 2.2.3.2.4.2.1

Combine $2 π$ and $\frac{2}{2}$ .

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

Step 2.2.3.2.4.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 2 - π}{2}$

Step 2.2.3.2.4.3

Simplify the numerator.

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Step 2.2.3.2.4.3.1

Multiply $2$ by $2$ .

$x = \frac{4 π - π}{2}$

Step 2.2.3.2.4.3.2

Subtract $π$ from $4 π$ .

$x = \frac{3 π}{2}$

Step 2.2.3.2.5

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

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Step 2.2.3.2.5.1

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

$\frac{2 π}{| b |}$

Step 2.2.3.2.5.2

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

$\frac{2 π}{| 1 |}$

Step 2.2.3.2.5.3

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

$\frac{2 π}{1}$

Step 2.2.3.2.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 2.2.3.2.6

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

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

Step 2.2.4

Set $\sec (x)$ equal to $0$ and solve for $x$ .

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Step 2.2.4.1

Set $\sec (x)$ equal to $0$ .

$\sec (x) = 0$

Step 2.2.4.2

The range of secant is $y \leq - 1$ and $y \geq 1$ . Since $0$ does not fall in this range, there is no solution.

No solution

Step 2.2.5

The final solution is all the values that make $\cos (x) \sec (x) = 0$ true.

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

Step 2.2.6

Consolidate the answers.

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

Step 2.2.7

Exclude the solutions that do not make $0 = \cos (x) \sec (x)$ true.

$No solution$ , for any integer $n$

Step 2.3

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

x-intercept(s): $None$

Step 3

Find the y-intercepts.

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Step 3.1

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

$y = \cos (0) \sec (0)$

Step 3.2

Solve the equation.

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Step 3.2.1

Remove parentheses.

$y = \cos (0) \sec (0)$

Step 3.2.2

Rewrite in terms of sines and cosines, then cancel the common factors.

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Step 3.2.2.1

Reorder $\cos (0)$ and $\sec (0)$ .

$y = \sec (0) \cos (0)$

Step 3.2.2.2

Rewrite $\cos (0) \sec (0)$ in terms of sines and cosines.

$y = \frac{1}{\cos (0)} \cos (0)$

Step 3.2.2.3

Cancel the common factors.

$y = 1$

Step 3.3

y-intercept(s) in point form.

y-intercept(s): $(0, 1)$

Step 4

List the intersections.

x-intercept(s): $None$

y-intercept(s): $(0, 1)$

Step 5