Trigonometry Examples

$a = 25$ , $b = 7$ , $c = 24$

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.

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

Step 3

Substitute the known values into the equation.

$A = \arccos (\frac{{(7)}^{2} + {(24)}^{2} - {(25)}^{2}}{2 (7) (24)})$

Step 4

Simplify the results.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $7$ to the power of $2$ .

$A = \arccos (\frac{49 + 24^{2} - 25^{2}}{2 (7) \cdot 24})$

Step 4.1.2

Raise $24$ to the power of $2$ .

$A = \arccos (\frac{49 + 576 - 25^{2}}{2 (7) \cdot 24})$

Step 4.1.3

Raise $25$ to the power of $2$ .

$A = \arccos (\frac{49 + 576 - 1 \cdot 625}{2 (7) \cdot 24})$

Step 4.1.4

Multiply $- 1$ by $625$ .

$A = \arccos (\frac{49 + 576 - 625}{2 (7) \cdot 24})$

Step 4.1.5

Add $49$ and $576$ .

$A = \arccos (\frac{625 - 625}{2 (7) \cdot 24})$

Step 4.1.6

Subtract $625$ from $625$ .

$A = \arccos (\frac{0}{2 (7) \cdot 24})$

Step 4.2

Simplify the denominator.

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Step 4.2.1

Multiply $2$ by $7$ .

$A = \arccos (\frac{0}{14 \cdot 24})$

Step 4.2.2

Multiply $14$ by $24$ .

$A = \arccos (\frac{0}{336})$

Step 4.3

Divide $0$ by $336$ .

$A = \arccos (0)$

Step 4.4

The exact value of $\arccos (0)$ is $90$ .

$A = 90$

Step 5

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

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

Step 6

Solve the equation.

$B = \arccos (\frac{a^{2} + c^{2} - b^{2}}{2 a c})$

Step 7

Substitute the known values into the equation.

$B = \arccos (\frac{{(25)}^{2} + {(24)}^{2} - {(7)}^{2}}{2 (25) (24)})$

Step 8

Simplify the results.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Raise $25$ to the power of $2$ .

$B = \arccos (\frac{625 + 24^{2} - 7^{2}}{2 (25) \cdot 24})$

Step 8.1.2

Raise $24$ to the power of $2$ .

$B = \arccos (\frac{625 + 576 - 7^{2}}{2 (25) \cdot 24})$

Step 8.1.3

Raise $7$ to the power of $2$ .

$B = \arccos (\frac{625 + 576 - 1 \cdot 49}{2 (25) \cdot 24})$

Step 8.1.4

Multiply $- 1$ by $49$ .

$B = \arccos (\frac{625 + 576 - 49}{2 (25) \cdot 24})$

Step 8.1.5

Add $625$ and $576$ .

$B = \arccos (\frac{1201 - 49}{2 (25) \cdot 24})$

Step 8.1.6

Subtract $49$ from $1201$ .

$B = \arccos (\frac{1152}{2 (25) \cdot 24})$

Step 8.2

Simplify the denominator.

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Step 8.2.1

Multiply $2$ by $25$ .

$B = \arccos (\frac{1152}{50 \cdot 24})$

Step 8.2.2

Multiply $50$ by $24$ .

$B = \arccos (\frac{1152}{1200})$

Step 8.3

Cancel the common factor of $1152$ and $1200$ .

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Step 8.3.1

Factor $48$ out of $1152$ .

$B = \arccos (\frac{48 (24)}{1200})$

Step 8.3.2

Cancel the common factors.

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Step 8.3.2.1

Factor $48$ out of $1200$ .

$B = \arccos (\frac{48 \cdot 24}{48 \cdot 25})$

Step 8.3.2.2

Cancel the common factor.

$B = \arccos (\frac{48 \cdot 24}{48 \cdot 25})$

Step 8.3.2.3

Rewrite the expression.

$B = \arccos (\frac{24}{25})$

Step 8.4

Evaluate $\arccos (\frac{24}{25})$ .

$B = 16.2602047$

Step 9

The sum of all the angles in a triangle is $180$ degrees.

$90 + C + 16.2602047 = 180$

Step 10

Solve the equation for $C$ .

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Step 10.1

Add $90$ and $16.2602047$ .

$C + 106.2602047 = 180$

Step 10.2

Move all terms not containing $C$ to the right side of the equation.

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Step 10.2.1

Subtract $106.2602047$ from both sides of the equation.

$C = 180 - 106.2602047$

Step 10.2.2

Subtract $106.2602047$ from $180$ .

$C = 73.73979529$

Step 11

These are the results for all angles and sides for the given triangle.

$A = 90$

$B = 16.2602047$

$C = 73.73979529$

$a = 25$

$b = 7$

$c = 24$