Trigonometry Examples

$\frac{12 \sin^{2} (x) + 15 \sin (x) + 3}{3 \sin^{2} (x) + 9 \sin (x) + 6}$

Step 1

Cancel the common factor of $12 \sin^{2} (x) + 15 \sin (x) + 3$ and $3 \sin^{2} (x) + 9 \sin (x) + 6$ .

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Step 1.1

Factor $3$ out of $12 \sin^{2} (x)$ .

$\frac{3 (4 \sin^{2} (x)) + 15 \sin (x) + 3}{3 \sin^{2} (x) + 9 \sin (x) + 6}$

Step 1.2

Factor $3$ out of $15 \sin (x)$ .

$\frac{3 (4 \sin^{2} (x)) + 3 (5 \sin (x)) + 3}{3 \sin^{2} (x) + 9 \sin (x) + 6}$

Step 1.3

Factor $3$ out of $3 (4 \sin^{2} (x)) + 3 (5 \sin (x))$ .

$\frac{3 (4 \sin^{2} (x) + 5 \sin (x)) + 3}{3 \sin^{2} (x) + 9 \sin (x) + 6}$

Step 1.4

Factor $3$ out of $3$ .

$\frac{3 (4 \sin^{2} (x) + 5 \sin (x)) + 3 \cdot 1}{3 \sin^{2} (x) + 9 \sin (x) + 6}$

Step 1.5

Factor $3$ out of $3 (4 \sin^{2} (x) + 5 \sin (x)) + 3 (1)$ .

$\frac{3 (4 \sin^{2} (x) + 5 \sin (x) + 1)}{3 \sin^{2} (x) + 9 \sin (x) + 6}$

Step 1.6

Cancel the common factors.

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Step 1.6.1

Factor $3$ out of $3 \sin^{2} (x)$ .

$\frac{3 (4 \sin^{2} (x) + 5 \sin (x) + 1)}{3 (\sin^{2} (x)) + 9 \sin (x) + 6}$

Step 1.6.2

Factor $3$ out of $9 \sin (x)$ .

$\frac{3 (4 \sin^{2} (x) + 5 \sin (x) + 1)}{3 (\sin^{2} (x)) + 3 (3 \sin (x)) + 6}$

Step 1.6.3

Factor $3$ out of $3 (\sin^{2} (x)) + 3 (3 \sin (x))$ .

$\frac{3 (4 \sin^{2} (x) + 5 \sin (x) + 1)}{3 (\sin^{2} (x) + 3 \sin (x)) + 6}$

Step 1.6.4

Factor $3$ out of $6$ .

$\frac{3 (4 \sin^{2} (x) + 5 \sin (x) + 1)}{3 (\sin^{2} (x) + 3 \sin (x)) + 3 \cdot 2}$

Step 1.6.5

Factor $3$ out of $3 (\sin^{2} (x) + 3 \sin (x)) + 3 \cdot 2$ .

$\frac{3 (4 \sin^{2} (x) + 5 \sin (x) + 1)}{3 (\sin^{2} (x) + 3 \sin (x) + 2)}$

Step 1.6.6

Cancel the common factor.

$\frac{3 (4 \sin^{2} (x) + 5 \sin (x) + 1)}{3 (\sin^{2} (x) + 3 \sin (x) + 2)}$

Step 1.6.7

Rewrite the expression.

$\frac{4 \sin^{2} (x) + 5 \sin (x) + 1}{\sin^{2} (x) + 3 \sin (x) + 2}$

Step 2

Simplify the numerator.

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Step 2.1

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

$\frac{4 u^{2} + 5 u + 1}{\sin^{2} (x) + 3 \sin (x) + 2}$

Step 2.2

Factor by grouping.

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Step 2.2.1

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 $a \cdot c = 4 \cdot 1 = 4$ and whose sum is $b = 5$ .

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Step 2.2.1.1

Factor $5$ out of $5 u$ .

$\frac{4 u^{2} + 5 (u) + 1}{\sin^{2} (x) + 3 \sin (x) + 2}$

Step 2.2.1.2

Rewrite $5$ as $1$ plus $4$

$\frac{4 u^{2} + (1 + 4) u + 1}{\sin^{2} (x) + 3 \sin (x) + 2}$

Step 2.2.1.3

Apply the distributive property.

$\frac{4 u^{2} + 1 u + 4 u + 1}{\sin^{2} (x) + 3 \sin (x) + 2}$

Step 2.2.1.4

Multiply $u$ by $1$ .

$\frac{4 u^{2} + u + 4 u + 1}{\sin^{2} (x) + 3 \sin (x) + 2}$

Step 2.2.2

Factor out the greatest common factor from each group.

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Step 2.2.2.1

Group the first two terms and the last two terms.

$\frac{(4 u^{2} + u) + 4 u + 1}{\sin^{2} (x) + 3 \sin (x) + 2}$

Step 2.2.2.2

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

$\frac{u (4 u + 1) + 1 (4 u + 1)}{\sin^{2} (x) + 3 \sin (x) + 2}$

Step 2.2.3

Factor the polynomial by factoring out the greatest common factor, $4 u + 1$ .

$\frac{(4 u + 1) (u + 1)}{\sin^{2} (x) + 3 \sin (x) + 2}$

Step 2.3

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

$\frac{(4 \sin (x) + 1) (\sin (x) + 1)}{\sin^{2} (x) + 3 \sin (x) + 2}$

Step 3

Simplify the denominator.

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Step 3.1

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

$\frac{(4 \sin (x) + 1) (\sin (x) + 1)}{u^{2} + 3 u + 2}$

Step 3.2

Factor $u^{2} + 3 u + 2$ using the AC method.

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Step 3.2.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $2$ and whose sum is $3$ .

$1, 2$

Step 3.2.2

Write the factored form using these integers.

$\frac{(4 \sin (x) + 1) (\sin (x) + 1)}{(u + 1) (u + 2)}$

Step 3.3

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

$\frac{(4 \sin (x) + 1) (\sin (x) + 1)}{(\sin (x) + 1) (\sin (x) + 2)}$

Step 4

Cancel the common factor of $\sin (x) + 1$ .

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Step 4.1

Cancel the common factor.

$\frac{(4 \sin (x) + 1) (\sin (x) + 1)}{(\sin (x) + 1) (\sin (x) + 2)}$

Step 4.2

Rewrite the expression.

$\frac{4 \sin (x) + 1}{\sin (x) + 2}$