Trigonometry Examples

$0.4 = \cos (2 π t)$

Step 1

Rewrite the equation as $\cos (2 π t) = 0.4$ .

$\cos (2 π t) = 0.4$

Step 2

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

$2 π t = \arccos (0.4)$

Step 3

Simplify the right side.

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Step 3.1

Evaluate $\arccos (0.4)$ .

$2 π t = 1.15927948$

Step 4

Divide each term in $2 π t = 1.15927948$ by $2 π$ and simplify.

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Step 4.1

Divide each term in $2 π t = 1.15927948$ by $2 π$ .

$\frac{2 π t}{2 π} = \frac{1.15927948}{2 π}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $2$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{2 π t}{2 π} = \frac{1.15927948}{2 π}$

Step 4.2.1.2

Rewrite the expression.

$\frac{π t}{π} = \frac{1.15927948}{2 π}$

Step 4.2.2

Cancel the common factor of $π$ .

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Step 4.2.2.1

Cancel the common factor.

$\frac{π t}{π} = \frac{1.15927948}{2 π}$

Step 4.2.2.2

Divide $t$ by $1$ .

$t = \frac{1.15927948}{2 π}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Replace $π$ with an approximation.

$t = \frac{1.15927948}{2 \cdot 3.14159265}$

Step 4.3.2

Multiply $2$ by $3.14159265$ .

$t = \frac{1.15927948}{6.2831853}$

Step 4.3.3

Divide $1.15927948$ by $6.2831853$ .

$t = 0.18450505$

Step 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.

$2 π t = 2 (3.14159265) - 1.15927948$

Step 6

Solve for $t$ .

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Step 6.1

Simplify.

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Step 6.1.1

Multiply $2$ by $3.14159265$ .

$2 π t = 6.2831853 - 1.15927948$

Step 6.1.2

Subtract $1.15927948$ from $6.2831853$ .

$2 π t = 5.12390582$

Step 6.2

Divide each term in $2 π t = 5.12390582$ by $2 π$ and simplify.

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Step 6.2.1

Divide each term in $2 π t = 5.12390582$ by $2 π$ .

$\frac{2 π t}{2 π} = \frac{5.12390582}{2 π}$

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Cancel the common factor of $2$ .

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Step 6.2.2.1.1

Cancel the common factor.

$\frac{2 π t}{2 π} = \frac{5.12390582}{2 π}$

Step 6.2.2.1.2

Rewrite the expression.

$\frac{π t}{π} = \frac{5.12390582}{2 π}$

Step 6.2.2.2

Cancel the common factor of $π$ .

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Step 6.2.2.2.1

Cancel the common factor.

$\frac{π t}{π} = \frac{5.12390582}{2 π}$

Step 6.2.2.2.2

Divide $t$ by $1$ .

$t = \frac{5.12390582}{2 π}$

Step 6.2.3

Simplify the right side.

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Step 6.2.3.1

Replace $π$ with an approximation.

$t = \frac{5.12390582}{2 \cdot 3.14159265}$

Step 6.2.3.2

Multiply $2$ by $3.14159265$ .

$t = \frac{5.12390582}{6.2831853}$

Step 6.2.3.3

Divide $5.12390582$ by $6.2831853$ .

$t = 0.81549494$

Step 7

Find the period of $\cos (2 π t)$ .

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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 $2 π$ in the formula for period.

$\frac{2 π}{| 2 π |}$

Step 7.3

$2 π$ is approximately $6.2831853$ which is positive so remove the absolute value

$\frac{2 π}{2 π}$

Step 7.4

Cancel the common factor of $2$ .

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Step 7.4.1

Cancel the common factor.

$\frac{2 π}{2 π}$

Step 7.4.2

Rewrite the expression.

$\frac{π}{π}$

Step 7.5

Cancel the common factor of $π$ .

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Step 7.5.1

Cancel the common factor.

$\frac{π}{π}$

Step 7.5.2

Rewrite the expression.

$1$

Step 8

The period of the $\cos (2 π t)$ function is $1$ so values will repeat every $1$ radians in both directions.

$t = 0.18450505 + n, 0.81549494 + n$ , for any integer $n$