Calculus Examples

$2.1 + 1.5 \sin (1.8 t + 0.3) = 3$

Step 1

Move all terms not containing $\sin (1.8 t + 0.3)$ to the right side of the equation.

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Step 1.1

Subtract $2.1$ from both sides of the equation.

$1.5 \sin (1.8 t + 0.3) = 3 - 2.1$

Step 1.2

Subtract $2.1$ from $3$ .

$1.5 \sin (1.8 t + 0.3) = 0.9$

Step 2

Divide each term in $1.5 \sin (1.8 t + 0.3) = 0.9$ by $1.5$ and simplify.

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Step 2.1

Divide each term in $1.5 \sin (1.8 t + 0.3) = 0.9$ by $1.5$ .

$\frac{1.5 \sin (1.8 t + 0.3)}{1.5} = \frac{0.9}{1.5}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $1.5$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{1.5 \sin (1.8 t + 0.3)}{1.5} = \frac{0.9}{1.5}$

Step 2.2.1.2

Divide $\sin (1.8 t + 0.3)$ by $1$ .

$\sin (1.8 t + 0.3) = \frac{0.9}{1.5}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide $0.9$ by $1.5$ .

$\sin (1.8 t + 0.3) = 0.6$

Step 3

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

$1.8 t + 0.3 = \arcsin (0.6)$

Step 4

Simplify the right side.

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Step 4.1

Evaluate $\arcsin (0.6)$ .

$1.8 t + 0.3 = 0.6435011$

Step 5

Move all terms not containing $t$ to the right side of the equation.

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Step 5.1

Subtract $0.3$ from both sides of the equation.

$1.8 t = 0.6435011 - 0.3$

Step 5.2

Subtract $0.3$ from $0.6435011$ .

$1.8 t = 0.3435011$

Step 6

Divide each term in $1.8 t = 0.3435011$ by $1.8$ and simplify.

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Step 6.1

Divide each term in $1.8 t = 0.3435011$ by $1.8$ .

$\frac{1.8 t}{1.8} = \frac{0.3435011}{1.8}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $1.8$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{1.8 t}{1.8} = \frac{0.3435011}{1.8}$

Step 6.2.1.2

Divide $t$ by $1$ .

$t = \frac{0.3435011}{1.8}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Divide $0.3435011$ by $1.8$ .

$t = 0.19083394$

Step 7

The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from $π$ to find the solution in the second quadrant.

$1.8 t + 0.3 = (3.14159265) - 0.6435011$

Step 8

Solve for $t$ .

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Step 8.1

Subtract $0.6435011$ from $3.14159265$ .

$1.8 t + 0.3 = 2.49809154$

Step 8.2

Move all terms not containing $t$ to the right side of the equation.

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Step 8.2.1

Subtract $0.3$ from both sides of the equation.

$1.8 t = 2.49809154 - 0.3$

Step 8.2.2

Subtract $0.3$ from $2.49809154$ .

$1.8 t = 2.19809154$

Step 8.3

Divide each term in $1.8 t = 2.19809154$ by $1.8$ and simplify.

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Step 8.3.1

Divide each term in $1.8 t = 2.19809154$ by $1.8$ .

$\frac{1.8 t}{1.8} = \frac{2.19809154}{1.8}$

Step 8.3.2

Simplify the left side.

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Step 8.3.2.1

Cancel the common factor of $1.8$ .

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Step 8.3.2.1.1

Cancel the common factor.

$\frac{1.8 t}{1.8} = \frac{2.19809154}{1.8}$

Step 8.3.2.1.2

Divide $t$ by $1$ .

$t = \frac{2.19809154}{1.8}$

Step 8.3.3

Simplify the right side.

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Step 8.3.3.1

Divide $2.19809154$ by $1.8$ .

$t = 1.22116196$

Step 9

Find the period of $\sin (1.8 t + 0.3)$ .

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Step 9.1

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

$\frac{2 π}{| b |}$

Step 9.2

Replace $b$ with $1.8$ in the formula for period.

$\frac{2 π}{| 1.8 |}$

Step 9.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1.8$ is $1.8$ .

$\frac{2 π}{1.8}$

Step 9.4

Replace $π$ with an approximation.

$\frac{2 \cdot 3.14159265}{1.8}$

Step 9.5

Multiply $2$ by $3.14159265$ .

$\frac{6.2831853}{1.8}$

Step 9.6

Divide $6.2831853$ by $1.8$ .

$3.4906585$

Step 10

The period of the $\sin (1.8 t + 0.3)$ function is $3.4906585$ so values will repeat every $3.4906585$ radians in both directions.

$t = 0.19083394 + 3.4906585 n, 1.22116196 + 3.4906585 n$ , for any integer $n$