Trigonometry Examples

$\sin (2 t) + \sin (t) + 4 \cos (t) = - 2$

Step 1

Add $2$ to both sides of the equation.

$\sin (2 t) + \sin (t) + 4 \cos (t) + 2 = 0$

Step 2

Apply the sine double-angle identity.

$2 \sin (t) \cos (t) + \sin (t) + 4 \cos (t) + 2 = 0$

Step 3

Factor $2 \sin (t) \cos (t) + \sin (t) + 4 \cos (t) + 2$ .

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Step 3.1

Factor out the greatest common factor from each group.

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Step 3.1.1

Group the first two terms and the last two terms.

$2 \sin (t) \cos (t) + \sin (t) + 4 \cos (t) + 2 = 0$

Step 3.1.2

Factor out the greatest common factor (GCF) from each group.

$\sin (t) (2 \cos (t) + 1) + 2 (2 \cos (t) + 1) = 0$

Step 3.2

Factor the polynomial by factoring out the greatest common factor, $2 \cos (t) + 1$ .

$(2 \cos (t) + 1) (\sin (t) + 2) = 0$

Step 4

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

$2 \cos (t) + 1 = 0$

$\sin (t) + 2 = 0$

Step 5

Set $2 \cos (t) + 1$ equal to $0$ and solve for $t$ .

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Step 5.1

Set $2 \cos (t) + 1$ equal to $0$ .

$2 \cos (t) + 1 = 0$

Step 5.2

Solve $2 \cos (t) + 1 = 0$ for $t$ .

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Step 5.2.1

Subtract $1$ from both sides of the equation.

$2 \cos (t) = - 1$

Step 5.2.2

Divide each term in $2 \cos (t) = - 1$ by $2$ and simplify.

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Step 5.2.2.1

Divide each term in $2 \cos (t) = - 1$ by $2$ .

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

Step 5.2.2.2

Simplify the left side.

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Step 5.2.2.2.1

Cancel the common factor of $2$ .

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Step 5.2.2.2.1.1

Cancel the common factor.

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

Step 5.2.2.2.1.2

Divide $\cos (t)$ by $1$ .

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

Step 5.2.2.3

Simplify the right side.

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Step 5.2.2.3.1

Move the negative in front of the fraction.

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

Step 5.2.3

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

$t = \arccos (- \frac{1}{2})$

Step 5.2.4

Simplify the right side.

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Step 5.2.4.1

The exact value of $\arccos (- \frac{1}{2})$ is $\frac{2 π}{3}$ .

$t = \frac{2 π}{3}$

Step 5.2.5

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.

$t = 2 π - \frac{2 π}{3}$

Step 5.2.6

Simplify $2 π - \frac{2 π}{3}$ .

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Step 5.2.6.1

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$t = 2 π \cdot \frac{3}{3} - \frac{2 π}{3}$

Step 5.2.6.2

Combine fractions.

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Step 5.2.6.2.1

Combine $2 π$ and $\frac{3}{3}$ .

$t = \frac{2 π \cdot 3}{3} - \frac{2 π}{3}$

Step 5.2.6.2.2

Combine the numerators over the common denominator.

$t = \frac{2 π \cdot 3 - 2 π}{3}$

Step 5.2.6.3

Simplify the numerator.

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Step 5.2.6.3.1

Multiply $3$ by $2$ .

$t = \frac{6 π - 2 π}{3}$

Step 5.2.6.3.2

Subtract $2 π$ from $6 π$ .

$t = \frac{4 π}{3}$

Step 5.2.7

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

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Step 5.2.7.1

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

$\frac{2 π}{| b |}$

Step 5.2.7.2

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

$\frac{2 π}{| 1 |}$

Step 5.2.7.3

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

$\frac{2 π}{1}$

Step 5.2.7.4

Divide $2 π$ by $1$ .

$2 π$

Step 5.2.8

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

$t = \frac{2 π}{3} + 2 π n, \frac{4 π}{3} + 2 π n$ , for any integer $n$

Step 6

Set $\sin (t) + 2$ equal to $0$ and solve for $t$ .

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Step 6.1

Set $\sin (t) + 2$ equal to $0$ .

$\sin (t) + 2 = 0$

Step 6.2

Solve $\sin (t) + 2 = 0$ for $t$ .

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Step 6.2.1

Subtract $2$ from both sides of the equation.

$\sin (t) = - 2$

Step 6.2.2

The range of sine is $- 1 \leq y \leq 1$ . Since $- 2$ does not fall in this range, there is no solution.

No solution

Step 7

The final solution is all the values that make $(2 \cos (t) + 1) (\sin (t) + 2) = 0$ true.

$t = \frac{2 π}{3} + 2 π n, \frac{4 π}{3} + 2 π n$ , for any integer $n$