Trigonometry Examples

$10 \cos^{2} (x) + 13 \cos (x) + 4$

Step 1

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

$10 u^{2} + 13 u + 4$

Step 2

Factor by grouping.

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Step 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 $a \cdot c = 10 \cdot 4 = 40$ and whose sum is $b = 13$ .

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Step 2.1.1

Factor $13$ out of $13 u$ .

$10 u^{2} + 13 (u) + 4$

Step 2.1.2

Rewrite $13$ as $5$ plus $8$

$10 u^{2} + (5 + 8) u + 4$

Step 2.1.3

Apply the distributive property.

$10 u^{2} + 5 u + 8 u + 4$

Step 2.2

Factor out the greatest common factor from each group.

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Step 2.2.1

Group the first two terms and the last two terms.

$(10 u^{2} + 5 u) + 8 u + 4$

Step 2.2.2

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

$5 u (2 u + 1) + 4 (2 u + 1)$

Step 2.3

Factor the polynomial by factoring out the greatest common factor, $2 u + 1$ .

$(2 u + 1) (5 u + 4)$

Step 3

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

$(2 \cos (x) + 1) (5 \cos (x) + 4)$