Trigonometry Examples

$5 \sin (t) + 5 \cos (t) = - 5$

Step 1

Square both sides of the equation.

${(5 \sin (t) + 5 \cos (t))}^{2} = {(- 5)}^{2}$

Step 2

Simplify ${(5 \sin (t) + 5 \cos (t))}^{2}$ .

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Step 2.1

Rewrite ${(5 \sin (t) + 5 \cos (t))}^{2}$ as $(5 \sin (t) + 5 \cos (t)) (5 \sin (t) + 5 \cos (t))$ .

$(5 \sin (t) + 5 \cos (t)) (5 \sin (t) + 5 \cos (t)) = {(- 5)}^{2}$

Step 2.2

Expand $(5 \sin (t) + 5 \cos (t)) (5 \sin (t) + 5 \cos (t))$ using the FOIL Method.

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Step 2.2.1

Apply the distributive property.

$5 \sin (t) (5 \sin (t) + 5 \cos (t)) + 5 \cos (t) (5 \sin (t) + 5 \cos (t)) = {(- 5)}^{2}$

Step 2.2.2

Apply the distributive property.

$5 \sin (t) (5 \sin (t)) + 5 \sin (t) (5 \cos (t)) + 5 \cos (t) (5 \sin (t) + 5 \cos (t)) = {(- 5)}^{2}$

Step 2.2.3

Apply the distributive property.

$5 \sin (t) (5 \sin (t)) + 5 \sin (t) (5 \cos (t)) + 5 \cos (t) (5 \sin (t)) + 5 \cos (t) (5 \cos (t)) = {(- 5)}^{2}$

Step 2.3

Simplify and combine like terms.

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Step 2.3.1

Simplify each term.

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Step 2.3.1.1

Multiply $5 \sin (t) (5 \sin (t))$ .

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Step 2.3.1.1.1

Multiply $5$ by $5$ .

$25 \sin (t) \sin (t) + 5 \sin (t) (5 \cos (t)) + 5 \cos (t) (5 \sin (t)) + 5 \cos (t) (5 \cos (t)) = {(- 5)}^{2}$

Step 2.3.1.1.2

Raise $\sin (t)$ to the power of $1$ .

$25 (\sin^{1} (t) \sin (t)) + 5 \sin (t) (5 \cos (t)) + 5 \cos (t) (5 \sin (t)) + 5 \cos (t) (5 \cos (t)) = {(- 5)}^{2}$

Step 2.3.1.1.3

Raise $\sin (t)$ to the power of $1$ .

$25 (\sin^{1} (t) \sin^{1} (t)) + 5 \sin (t) (5 \cos (t)) + 5 \cos (t) (5 \sin (t)) + 5 \cos (t) (5 \cos (t)) = {(- 5)}^{2}$

Step 2.3.1.1.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$25 {\sin (t)}^{1 + 1} + 5 \sin (t) (5 \cos (t)) + 5 \cos (t) (5 \sin (t)) + 5 \cos (t) (5 \cos (t)) = {(- 5)}^{2}$

Step 2.3.1.1.5

Add $1$ and $1$ .

$25 \sin^{2} (t) + 5 \sin (t) (5 \cos (t)) + 5 \cos (t) (5 \sin (t)) + 5 \cos (t) (5 \cos (t)) = {(- 5)}^{2}$

Step 2.3.1.2

Multiply $5$ by $5$ .

$25 \sin^{2} (t) + 25 \sin (t) \cos (t) + 5 \cos (t) (5 \sin (t)) + 5 \cos (t) (5 \cos (t)) = {(- 5)}^{2}$

Step 2.3.1.3

Multiply $5$ by $5$ .

$25 \sin^{2} (t) + 25 \sin (t) \cos (t) + 25 \cos (t) \sin (t) + 5 \cos (t) (5 \cos (t)) = {(- 5)}^{2}$

Step 2.3.1.4

Multiply $5 \cos (t) (5 \cos (t))$ .

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Step 2.3.1.4.1

Multiply $5$ by $5$ .

$25 \sin^{2} (t) + 25 \sin (t) \cos (t) + 25 \cos (t) \sin (t) + 25 \cos (t) \cos (t) = {(- 5)}^{2}$

Step 2.3.1.4.2

Raise $\cos (t)$ to the power of $1$ .

$25 \sin^{2} (t) + 25 \sin (t) \cos (t) + 25 \cos (t) \sin (t) + 25 (\cos^{1} (t) \cos (t)) = {(- 5)}^{2}$

Step 2.3.1.4.3

Raise $\cos (t)$ to the power of $1$ .

$25 \sin^{2} (t) + 25 \sin (t) \cos (t) + 25 \cos (t) \sin (t) + 25 (\cos^{1} (t) \cos^{1} (t)) = {(- 5)}^{2}$

Step 2.3.1.4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$25 \sin^{2} (t) + 25 \sin (t) \cos (t) + 25 \cos (t) \sin (t) + 25 {\cos (t)}^{1 + 1} = {(- 5)}^{2}$

Step 2.3.1.4.5

Add $1$ and $1$ .

$25 \sin^{2} (t) + 25 \sin (t) \cos (t) + 25 \cos (t) \sin (t) + 25 \cos^{2} (t) = {(- 5)}^{2}$

Step 2.3.2

Reorder the factors of $25 \sin (t) \cos (t)$ .

$25 \sin^{2} (t) + 25 \cos (t) \sin (t) + 25 \cos (t) \sin (t) + 25 \cos^{2} (t) = {(- 5)}^{2}$

Step 2.3.3

Add $25 \cos (t) \sin (t)$ and $25 \cos (t) \sin (t)$ .

$25 \sin^{2} (t) + 50 \cos (t) \sin (t) + 25 \cos^{2} (t) = {(- 5)}^{2}$

Step 2.4

Simplify with factoring out.

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Step 2.4.1

Move $25 \cos^{2} (t)$ .

$25 \sin^{2} (t) + 25 \cos^{2} (t) + 50 \cos (t) \sin (t) = {(- 5)}^{2}$

Step 2.4.2

Factor $25$ out of $25 \sin^{2} (t)$ .

$25 (\sin^{2} (t)) + 25 \cos^{2} (t) + 50 \cos (t) \sin (t) = {(- 5)}^{2}$

Step 2.4.3

Factor $25$ out of $25 \cos^{2} (t)$ .

$25 (\sin^{2} (t)) + 25 (\cos^{2} (t)) + 50 \cos (t) \sin (t) = {(- 5)}^{2}$

Step 2.4.4

Factor $25$ out of $25 (\sin^{2} (t)) + 25 (\cos^{2} (t))$ .

$25 (\sin^{2} (t) + \cos^{2} (t)) + 50 \cos (t) \sin (t) = {(- 5)}^{2}$

Step 2.5

Apply pythagorean identity.

$25 \cdot 1 + 50 \cos (t) \sin (t) = {(- 5)}^{2}$

Step 2.6

Multiply $25$ by $1$ .

$25 + 50 \cos (t) \sin (t) = {(- 5)}^{2}$

Step 3

Raise $- 5$ to the power of $2$ .

$25 + 50 \cos (t) \sin (t) = 25$

Step 4

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

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Step 4.1

Subtract $25$ from both sides of the equation.

$50 \cos (t) \sin (t) = 25 - 25$

Step 4.2

Subtract $25$ from $25$ .

$50 \cos (t) \sin (t) = 0$

Step 5

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

$\cos (t) = 0$

$\sin (t) = 0$

Step 6

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

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Step 6.1

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

$\cos (t) = 0$

Step 6.2

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

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Step 6.2.1

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

$t = \arccos (0)$

Step 6.2.2

Simplify the right side.

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Step 6.2.2.1

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

$t = \frac{π}{2}$

Step 6.2.3

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.

$t = 2 π - \frac{π}{2}$

Step 6.2.4

Simplify $2 π - \frac{π}{2}$ .

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Step 6.2.4.1

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

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

Step 6.2.4.2

Combine fractions.

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Step 6.2.4.2.1

Combine $2 π$ and $\frac{2}{2}$ .

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

Step 6.2.4.2.2

Combine the numerators over the common denominator.

$t = \frac{2 π \cdot 2 - π}{2}$

Step 6.2.4.3

Simplify the numerator.

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Step 6.2.4.3.1

Multiply $2$ by $2$ .

$t = \frac{4 π - π}{2}$

Step 6.2.4.3.2

Subtract $π$ from $4 π$ .

$t = \frac{3 π}{2}$

Step 6.2.5

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

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Step 6.2.5.1

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

$\frac{2 π}{| b |}$

Step 6.2.5.2

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

$\frac{2 π}{| 1 |}$

Step 6.2.5.3

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

$\frac{2 π}{1}$

Step 6.2.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 6.2.6

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

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

Step 7

Set $\sin (t)$ equal to $0$ and solve for $t$ .

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Step 7.1

Set $\sin (t)$ equal to $0$ .

$\sin (t) = 0$

Step 7.2

Solve $\sin (t) = 0$ for $t$ .

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Step 7.2.1

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

$t = \arcsin (0)$

Step 7.2.2

Simplify the right side.

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Step 7.2.2.1

The exact value of $\arcsin (0)$ is $0$ .

$t = 0$

Step 7.2.3

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.

$t = π - 0$

Step 7.2.4

Subtract $0$ from $π$ .

$t = π$

Step 7.2.5

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

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Step 7.2.5.1

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

$\frac{2 π}{| b |}$

Step 7.2.5.2

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

$\frac{2 π}{| 1 |}$

Step 7.2.5.3

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

$\frac{2 π}{1}$

Step 7.2.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 7.2.6

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

$t = 2 π n, π + 2 π n$ , for any integer $n$

Step 8

The final solution is all the values that make $50 \cos (t) \sin (t) = 0$ true.

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

Step 9

Consolidate the answers.

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

Step 10

Verify each of the solutions by substituting them into $5 \sin (t) + 5 \cos (t) = - 5$ and solving.

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