Precalculus Examples

$\sin (t) = \frac{\sqrt{2}}{7}$ , $\sin^{2} (t) + \cos^{2} (t) = 1$

Step 1

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

$t = \arcsin (\frac{\sqrt{2}}{7})$

$\sin^{2} (t) + \cos^{2} (t) = 1$

Step 2

Simplify the right side.

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Step 2.1

Evaluate $\arcsin (\frac{\sqrt{2}}{7})$ .

$t = 0.20343074$

$\sin^{2} (t) + \cos^{2} (t) = 1$

$t = 0.20343074$

$\sin^{2} (t) + \cos^{2} (t) = 1$

Step 3

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.

$t = (3.14159265) - 0.20343074$

$\sin^{2} (t) + \cos^{2} (t) = 1$

Step 4

Solve for $t$ .

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Step 4.1

Remove parentheses.

$t = 3.14159265 - 0.20343074$

$\sin^{2} (t) + \cos^{2} (t) = 1$

Step 4.2

Remove parentheses.

$t = (3.14159265) - 0.20343074$

$\sin^{2} (t) + \cos^{2} (t) = 1$

Step 4.3

Subtract $0.20343074$ from $3.14159265$ .

$t = 2.93816191$

$\sin^{2} (t) + \cos^{2} (t) = 1$

$t = 2.93816191$

$\sin^{2} (t) + \cos^{2} (t) = 1$

Step 5

Find the period of $\sin (t)$ .

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Step 5.1

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

$\frac{2 π}{| b |}$

Step 5.2

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

$\frac{2 π}{| 1 |}$

Step 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 5.4

Divide $2 π$ by $1$ .

$2 π$

Step 6

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

$t = 0.20343074 + 2 π n, 2.93816191 + 2 π n$

$\sin^{2} (t) + \cos^{2} (t) = 1$

Step 7

Replace the $\sin^{2} (t)$ with $1 - \cos^{2} (t)$ based on the $\sin^{2} (x) + \cos^{2} (x) = 1$ identity.

$(1 - \cos^{2} (t)) + \cos^{2} (t) = 1$

$t = 0.20343074 + 2 π n, 2.93816191 + 2 π n$

Step 8

Combine the opposite terms in $1 - \cos^{2} (t) + \cos^{2} (t)$ .

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Step 8.1

Add $- \cos^{2} (t)$ and $\cos^{2} (t)$ .

$1 + 0 = 1$

$t = 0.20343074 + 2 π n, 2.93816191 + 2 π n$

Step 8.2

Add $1$ and $0$ .

$1 = 1$

$t = 0.20343074 + 2 π n, 2.93816191 + 2 π n$

$1 = 1$

$t = 0.20343074 + 2 π n, 2.93816191 + 2 π n$

Step 9

The equation is always true.

All real numbers

$t = 0.20343074 + 2 π n, 2.93816191 + 2 π n$

Step 10

Since the roots of an equation are the points where the solution is $0$ , set each root as a factor of the equation that equals $0$ .

Step 11

Simplify.

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Step 11.1

Simplify by multiplying through.

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Step 11.1.1

Apply the distributive property.

$(t - 0.20343074 + 2 π n, 2.93816191 + 2 π n) A + (t - 0.20343074 + 2 π n, 2.93816191 + 2 π n) (- A l^{3} r^{2} a n u m b e^{2} s) = 0$

Step 11.1.2

Rewrite using the commutative property of multiplication.

$(t - 0.20343074 + 2 π n, 2.93816191 + 2 π n) A - e^{2} (t - 0.20343074 + 2 π n, 2.93816191 + 2 π n) (A l^{3} r^{2} a n u m b s) = 0$

Step 11.2

Simplify each term.

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Step 11.2.1

Multiply $- e^{2}$ by each element of the matrix.

$(t - 0.20343074 + 2 π n, 2.93816191 + 2 π n) A + (- e^{2} (t - 0.20343074 + 2 π n), - e^{2} (2.93816191 + 2 π n)) (A l^{3} r^{2} a n u m b s) = 0$

Step 11.2.2

Simplify each element in the matrix.

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Step 11.2.2.1

Apply the distributive property.

$(t - 0.20343074 + 2 π n, 2.93816191 + 2 π n) A + (- e^{2} t - e^{2} \cdot - 0.20343074 - e^{2} (2 π n), - e^{2} (2.93816191 + 2 π n)) (A l^{3} r^{2} a n u m b s) = 0$

Step 11.2.2.2

Simplify.

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Step 11.2.2.2.1

Multiply $- e^{2} \cdot - 0.20343074$ .

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Step 11.2.2.2.1.1

Multiply $- 0.20343074$ by $- 1$ .

$(t - 0.20343074 + 2 π n, 2.93816191 + 2 π n) A + (- e^{2} t + 0.20343074 e^{2} - e^{2} (2 π n), - e^{2} (2.93816191 + 2 π n)) (A l^{3} r^{2} a n u m b s) = 0$

Step 11.2.2.2.1.2

Multiply $0.20343074$ by $e^{2}$ .

$(t - 0.20343074 + 2 π n, 2.93816191 + 2 π n) A + (- e^{2} t + 1.50316115 - e^{2} (2 π n), - e^{2} (2.93816191 + 2 π n)) (A l^{3} r^{2} a n u m b s) = 0$

Step 11.2.2.2.2

Multiply $2$ by $- 1$ .

$(t - 0.20343074 + 2 π n, 2.93816191 + 2 π n) A + (- e^{2} t + 1.50316115 - 2 e^{2} (π n), - e^{2} (2.93816191 + 2 π n)) (A l^{3} r^{2} a n u m b s) = 0$

Step 11.2.2.3

Apply the distributive property.

$(t - 0.20343074 + 2 π n, 2.93816191 + 2 π n) A + (- e^{2} t + 1.50316115 - 2 e^{2} π n, - e^{2} \cdot 2.93816191 - e^{2} (2 π n)) (A l^{3} r^{2} a n u m b s) = 0$

Step 11.2.2.4

Multiply $- e^{2} \cdot 2.93816191$ .

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Step 11.2.2.4.1

Multiply $2.93816191$ by $- 1$ .

$(t - 0.20343074 + 2 π n, 2.93816191 + 2 π n) A + (- e^{2} t + 1.50316115 - 2 e^{2} π n, - 2.93816191 e^{2} - e^{2} (2 π n)) (A l^{3} r^{2} a n u m b s) = 0$

Step 11.2.2.4.2

Multiply $- 2.93816191$ by $e^{2}$ .

$(t - 0.20343074 + 2 π n, 2.93816191 + 2 π n) A + (- e^{2} t + 1.50316115 - 2 e^{2} π n, - 21.7102432 - e^{2} (2 π n)) (A l^{3} r^{2} a n u m b s) = 0$

Step 11.2.2.5

Multiply $2$ by $- 1$ .

$(t - 0.20343074 + 2 π n, 2.93816191 + 2 π n) A + (- e^{2} t + 1.50316115 - 2 e^{2} π n, - 21.7102432 - 2 e^{2} π n) (A l^{3} r^{2} a n u m b s) = 0$

Step 11.3

Reorder factors in $(t - 0.20343074 + 2 π n, 2.93816191 + 2 π n) A + (- e^{2} t + 1.50316115 - 2 e^{2} π n, - 21.7102432 - 2 e^{2} π n) (A l^{3} r^{2} a n u m b s)$ .

$A (t - 0.20343074 + 2 π n, 2.93816191 + 2 π n) + (A l^{3} r^{2} a n u m b) s (- e^{2} t + 1.50316115 - 2 e^{2} π n, - 21.7102432 - 2 e^{2} π n) = 0$