Trigonometry Examples

$5 \cos (π t) = - 3$

Step 1

Divide each term in $5 \cos (π t) = - 3$ by $5$ and simplify.

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Step 1.1

Divide each term in $5 \cos (π t) = - 3$ by $5$ .

$\frac{5 \cos (π t)}{5} = \frac{- 3}{5}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $5$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{5 \cos (π t)}{5} = \frac{- 3}{5}$

Step 1.2.1.2

Divide $\cos (π t)$ by $1$ .

$\cos (π t) = \frac{- 3}{5}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Move the negative in front of the fraction.

$\cos (π t) = - \frac{3}{5}$

Step 2

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

$π t = \arccos (- \frac{3}{5})$

Step 3

Simplify the right side.

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Step 3.1

Evaluate $\arccos (- \frac{3}{5})$ .

$π t = 2.21429743$

Step 4

Divide each term in $π t = 2.21429743$ by $π$ and simplify.

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Step 4.1

Divide each term in $π t = 2.21429743$ by $π$ .

$\frac{π t}{π} = \frac{2.21429743}{π}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $π$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{π t}{π} = \frac{2.21429743}{π}$

Step 4.2.1.2

Divide $t$ by $1$ .

$t = \frac{2.21429743}{π}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Replace $π$ with an approximation.

$t = \frac{2.21429743}{3.14159265}$

Step 4.3.2

Divide $2.21429743$ by $3.14159265$ .

$t = 0.70483276$

Step 5

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.

$π t = 2 (3.14159265) - 2.21429743$

Step 6

Solve for $t$ .

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Step 6.1

Simplify.

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Step 6.1.1

Multiply $2$ by $3.14159265$ .

$π t = 6.2831853 - 2.21429743$

Step 6.1.2

Subtract $2.21429743$ from $6.2831853$ .

$π t = 4.06888787$

Step 6.2

Divide each term in $π t = 4.06888787$ by $π$ and simplify.

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Step 6.2.1

Divide each term in $π t = 4.06888787$ by $π$ .

$\frac{π t}{π} = \frac{4.06888787}{π}$

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Cancel the common factor of $π$ .

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Step 6.2.2.1.1

Cancel the common factor.

$\frac{π t}{π} = \frac{4.06888787}{π}$

Step 6.2.2.1.2

Divide $t$ by $1$ .

$t = \frac{4.06888787}{π}$

Step 6.2.3

Simplify the right side.

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Step 6.2.3.1

Replace $π$ with an approximation.

$t = \frac{4.06888787}{3.14159265}$

Step 6.2.3.2

Divide $4.06888787$ by $3.14159265$ .

$t = 1.29516723$

Step 7

Find the period of $\cos (π t)$ .

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Step 7.1

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

$\frac{2 π}{| b |}$

Step 7.2

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

$\frac{2 π}{| π |}$

Step 7.3

$π$ is approximately $3.14159265$ which is positive so remove the absolute value

$\frac{2 π}{π}$

Step 7.4

Cancel the common factor of $π$ .

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Step 7.4.1

Cancel the common factor.

$\frac{2 π}{π}$

Step 7.4.2

Divide $2$ by $1$ .

$2$

Step 8

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

$t = 0.70483276 + 2 n, 1.29516723 + 2 n$ , for any integer $n$