Trigonometry Examples

$a = 15$ , $c = 7$ , $B = 64$

Step 1

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 2

Solve the equation.

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

Step 3

Substitute the known values into the equation.

$b = \sqrt{{(15)}^{2} + {(7)}^{2} - 2 \cdot 15 \cdot 7 \cos (64)}$

Step 4

Simplify the results.

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Step 4.1

Raise $15$ to the power of $2$ .

$b = \sqrt{225 + {(7)}^{2} - 2 \cdot 15 \cdot (7 \cos (64))}$

Step 4.2

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

$b = \sqrt{225 + 49 - 2 \cdot 15 \cdot (7 \cos (64))}$

Step 4.3

Multiply $- 2 \cdot 15 \cdot 7$ .

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Step 4.3.1

Multiply $- 2$ by $15$ .

$b = \sqrt{225 + 49 - 30 \cdot (7 \cos (64))}$

Step 4.3.2

Multiply $- 30$ by $7$ .

$b = \sqrt{225 + 49 - 210 \cos (64)}$

Step 4.4

Evaluate $\cos (64)$ .

$b = \sqrt{225 + 49 - 210 \cdot 0.43837114}$

Step 4.5

Multiply $- 210$ by $0.43837114$ .

$b = \sqrt{225 + 49 - 92.05794082}$

Step 4.6

Add $225$ and $49$ .

$b = \sqrt{274 - 92.05794082}$

Step 4.7

Subtract $92.05794082$ from $274$ .

$b = \sqrt{181.94205917}$

Step 4.8

Evaluate the root.

$b = 13.48858996$

Step 5

The law of sines is based on the proportionality of sides and angles in triangles. The law states that for the angles of a non-right triangle, each angle of the triangle has the same ratio of angle measure to sine value.

$\frac{\sin (A)}{a} = \frac{\sin (B)}{b} = \frac{\sin (C)}{c}$

Step 6

Substitute the known values into the law of sines to find $A$ .

$\frac{\sin (A)}{15} = \frac{\sin (64)}{13.48858996}$

Step 7

Solve the equation for $A$ .

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Step 7.1

Multiply both sides of the equation by $15$ .

$15 \frac{\sin (A)}{15} = 15 \frac{\sin (64)}{13.48858996}$

Step 7.2

Simplify both sides of the equation.

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Step 7.2.1

Simplify the left side.

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Step 7.2.1.1

Cancel the common factor of $15$ .

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Step 7.2.1.1.1

Cancel the common factor.

$15 \frac{\sin (A)}{15} = 15 \frac{\sin (64)}{13.48858996}$

Step 7.2.1.1.2

Rewrite the expression.

$\sin (A) = 15 \frac{\sin (64)}{13.48858996}$

Step 7.2.2

Simplify the right side.

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Step 7.2.2.1

Simplify $15 \frac{\sin (64)}{13.48858996}$ .

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Step 7.2.2.1.1

Evaluate $\sin (64)$ .

$\sin (A) = 15 (\frac{0.89879404}{13.48858996})$

Step 7.2.2.1.2

Divide $0.89879404$ by $13.48858996$ .

$\sin (A) = 15 \cdot 0.06663365$

Step 7.2.2.1.3

Multiply $15$ by $0.06663365$ .

$\sin (A) = 0.99950482$

Step 7.3

Take the inverse sine of both sides of the equation to extract $A$ from inside the sine.

$A = \arcsin (0.99950482)$

Step 7.4

Simplify the right side.

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Step 7.4.1

Evaluate $\arcsin (0.99950482)$ .

$A = 88.19682985$

Step 7.5

The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from $180$ to find the solution in the second quadrant.

$A = 180 - 88.19682985$

Step 7.6

Subtract $88.19682985$ from $180$ .

$A = 91.80317014$

Step 7.7

The solution to the equation $A = 88.19682985$ .

$A = 88.19682985, 91.80317014$

Step 8

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

$88.19682985 + C + 64 = 180$

Step 9

Solve the equation for $C$ .

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Step 9.1

Add $88.19682985$ and $64$ .

$C + 152.19682985 = 180$

Step 9.2

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

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Step 9.2.1

Subtract $152.19682985$ from both sides of the equation.

$C = 180 - 152.19682985$

Step 9.2.2

Subtract $152.19682985$ from $180$ .

$C = 27.80317014$

Step 10

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

$A = 88.19682985$

$B = 64$

$C = 27.80317014$

$a = 15$

$b = 13.48858996$

$c = 7$