Trigonometry Examples

$A = 60$ ,

$c = 6$ ,

$b = 2 p$

Step 1

Use the law of cosines to find the unknown side of the triangle, given the other two sides and the included angle.

$a^{2} = b^{2} + c^{2} - 2 b c \cos (A)$

Step 2

Solve the equation.

$a = \sqrt{b^{2} + c^{2} - 2 b c \cos (A)}$

Step 3

Substitute the known values into the equation.

$a = \sqrt{{(2 p)}^{2} + {(6)}^{2} - 2 (2 p) \cdot 6 \cos (60)}$

Step 4

Simplify the results.

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Step 4.1

Apply the product rule to

$2 p$ .

$a = \sqrt{2^{2} p^{2} + {(6)}^{2} - 2 (2 p) \cdot (6 \cos (60))}$

Step 4.2

Raise

$2$ to the power of

$2$ .

$a = \sqrt{4 p^{2} + {(6)}^{2} - 2 (2 p) \cdot (6 \cos (60))}$

Step 4.3

Raise

$6$ to the power of

$2$ .

$a = \sqrt{4 p^{2} + 36 - 2 (2 p) \cdot (6 \cos (60))}$

Step 4.4

Multiply

$2$ by

$- 2$ .

$a = \sqrt{4 p^{2} + 36 - 4 p \cdot (6 \cos (60))}$

Step 4.5

Multiply

$6$ by

$- 4$ .

$a = \sqrt{4 p^{2} + 36 - 24 p \cos (60)}$

Step 4.6

The exact value of

$\cos (60)$ is

$\frac{1}{2}$ .

$a = \sqrt{4 p^{2} + 36 - 24 p \frac{1}{2}}$

Step 4.7

Cancel the common factor of

$2$ .

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Step 4.7.1

Factor

$2$ out of

$- 24 p$ .

$a = \sqrt{4 p^{2} + 36 + 2 (- 12 p) (\frac{1}{2})}$

Step 4.7.2

Cancel the common factor.

$a = \sqrt{4 p^{2} + 36 + 2 (- 12 p) (\frac{1}{2})}$

Step 4.7.3

Rewrite the expression.

$a = \sqrt{4 p^{2} + 36 - 12 p}$

Step 4.8

Reorder terms.

$a = \sqrt{4 p^{2} - 12 p + 36}$

Step 4.9

Factor

$4$ out of

$4 p^{2} - 12 p + 36$ .

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Step 4.9.1

Factor

$4$ out of

$4 p^{2}$ .

$a = \sqrt{4 (p^{2}) - 12 p + 36}$

Step 4.9.2

Factor

$4$ out of

$- 12 p$ .

$a = \sqrt{4 (p^{2}) + 4 (- 3 p) + 36}$

Step 4.9.3

Factor

$4$ out of

$36$ .

$a = \sqrt{4 p^{2} + 4 (- 3 p) + 4 \cdot 9}$

Step 4.9.4

Factor

$4$ out of

$4 p^{2} + 4 (- 3 p)$ .

$a = \sqrt{4 (p^{2} - 3 p) + 4 \cdot 9}$

Step 4.9.5

Factor

$4$ out of

$4 (p^{2} - 3 p) + 4 \cdot 9$ .

$a = \sqrt{4 (p^{2} - 3 p + 9)}$

Step 4.10

Rewrite

$4 (p^{2} - 3 p + 9)$ as

$2^{2} (p^{2} - 3 p + 3^{2})$ .

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Step 4.10.1

Rewrite

$4$ as

$2^{2}$ .

$a = \sqrt{2^{2} (p^{2} - 3 p + 9)}$

Step 4.10.2

Rewrite

$9$ as

$3^{2}$ .

$a = \sqrt{2^{2} (p^{2} - 3 p + 3^{2})}$

Step 4.11

Pull terms out from under the radical.

$a = 2 \sqrt{p^{2} - 3 p + 3^{2}}$

Step 4.12

Raise

$3$ to the power of

$2$ .

$a = 2 \sqrt{p^{2} - 3 p + 9}$

Step 5

There are not enough parameters given to solve the triangle.

Unknown triangle