Trigonometry Examples

$2 \sin^{2} (x) + \sin (x) - 6 = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

Let $u = \sin (x)$ . Substitute $u$ for all occurrences of $\sin (x)$ .

$2 u^{2} + u - 6 = 0$

Step 1.2

Factor by grouping.

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Step 1.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot - 6 = - 12$ and whose sum is $b = 1$ .

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Step 1.2.1.1

Multiply by $1$ .

$2 u^{2} + 1 u - 6 = 0$

Step 1.2.1.2

Rewrite $1$ as $- 3$ plus $4$

$2 u^{2} + (- 3 + 4) u - 6 = 0$

Step 1.2.1.3

Apply the distributive property.

$2 u^{2} - 3 u + 4 u - 6 = 0$

Step 1.2.2

Factor out the greatest common factor from each group.

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Step 1.2.2.1

Group the first two terms and the last two terms.

$(2 u^{2} - 3 u) + 4 u - 6 = 0$

Step 1.2.2.2

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

$u (2 u - 3) + 2 (2 u - 3) = 0$

Step 1.2.3

Factor the polynomial by factoring out the greatest common factor, $2 u - 3$ .

$(2 u - 3) (u + 2) = 0$

Step 1.3

Replace all occurrences of $u$ with $\sin (x)$ .

$(2 \sin (x) - 3) (\sin (x) + 2) = 0$

Step 2

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

$2 \sin (x) - 3 = 0$

$\sin (x) + 2 = 0$

Step 3

Set $2 \sin (x) - 3$ equal to $0$ and solve for $x$ .

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Step 3.1

Set $2 \sin (x) - 3$ equal to $0$ .

$2 \sin (x) - 3 = 0$

Step 3.2

Solve $2 \sin (x) - 3 = 0$ for $x$ .

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Step 3.2.1

Add $3$ to both sides of the equation.

$2 \sin (x) = 3$

Step 3.2.2

Divide each term in $2 \sin (x) = 3$ by $2$ and simplify.

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Step 3.2.2.1

Divide each term in $2 \sin (x) = 3$ by $2$ .

$\frac{2 \sin (x)}{2} = \frac{3}{2}$

Step 3.2.2.2

Simplify the left side.

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Step 3.2.2.2.1

Cancel the common factor of $2$ .

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Step 3.2.2.2.1.1

Cancel the common factor.

$\frac{2 \sin (x)}{2} = \frac{3}{2}$

Step 3.2.2.2.1.2

Divide $\sin (x)$ by $1$ .

$\sin (x) = \frac{3}{2}$

Step 3.2.3

The range of sine is $- 1 \leq y \leq 1$ . Since $\frac{3}{2}$ does not fall in this range, there is no solution.

No solution

Step 4

The final solution is all the values that make $(2 \sin (x) - 3) (\sin (x) + 2) = 0$ true.

No solution