Solve the Triangle A=115 , C=35 , c=200

$A = 115$ , $C = 35$ , $c = 200$

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.

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

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

$\frac{\sin (115)}{a} = \frac{\sin (35)}{200}$

Solve the equation for $a$ .

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Factor each term.

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Evaluate $\sin (115)$ .

$\frac{0.90630778}{a} = \frac{\sin (35)}{200}$

Evaluate $\sin (35)$ .

$\frac{0.90630778}{a} = \frac{0.57357643}{200}$

Divide $0.57357643$ by $200$ .

$\frac{0.90630778}{a} = 0.00286788$

Solve for $a$ .

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Multiply each term by $a$ and simplify.

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Multiply each term in $\frac{0.90630778}{a} = 0.00286788$ by $a$ .

$\frac{0.90630778}{a} \cdot a = 0.00286788 \cdot a$

Cancel the common factor of $a$ .

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Cancel the common factor.

$\frac{0.90630778}{a} \cdot a = 0.00286788 \cdot a$

Rewrite the expression.

$0.90630778 = 0.00286788 \cdot a$

$0.90630778 = 0.00286788 a$

Rewrite the equation as $0.00286788 a = 0.90630778$ .

$0.00286788 a = 0.90630778$

Divide each term by $0.00286788$ and simplify.

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Divide each term in $0.00286788 a = 0.90630778$ by $0.00286788$ .

$\frac{0.00286788 a}{0.00286788} = \frac{0.90630778}{0.00286788}$

Cancel the common factor of $0.00286788$ .

$a = \frac{0.90630778}{0.00286788}$

Divide $0.90630778$ by $0.00286788$ .

$a = 316.01988143$

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

$115 + 35 + B = 180$

Solve the equation for $B$ .

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Add $115$ and $35$ .

$150 + B = 180$

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

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Subtract $150$ from both sides of the equation.

$B = 180 - 150$

Subtract $150$ from $180$ .

$B = 30$

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

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

$\frac{\sin (30)}{b} = \frac{\sin (115)}{316.01988143}$

Solve the equation for $b$ .

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Factor each term.

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The exact value of $\sin (30)$ is $\frac{1}{2}$ .

$\frac{\frac{1}{2}}{b} = \frac{\sin (115)}{316.01988143}$

Multiply the numerator by the reciprocal of the denominator.

$\frac{1}{2} \cdot \frac{1}{b} = \frac{\sin (115)}{316.01988143}$

Multiply $\frac{1}{2}$ and $\frac{1}{b}$ .

$\frac{1}{2 b} = \frac{\sin (115)}{316.01988143}$

Evaluate $\sin (115)$ .

$\frac{1}{2 b} = \frac{0.90630778}{316.01988143}$

Divide $0.90630778$ by $316.01988143$ .

$\frac{1}{2 b} = 0.00286788$

Solve for $b$ .

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Multiply each term by $b$ and simplify.

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Multiply each term in $\frac{1}{2 b} = 0.00286788$ by $b$ .

$\frac{1}{2 b} \cdot b = 0.00286788 \cdot b$

Cancel the common factor of $b$ .

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Factor $b$ out of $2 b$ .

$\frac{1}{b \cdot 2} \cdot b = 0.00286788 \cdot b$

Cancel the common factor.

$\frac{1}{b \cdot 2} \cdot b = 0.00286788 \cdot b$

Rewrite the expression.

$\frac{1}{2} = 0.00286788 \cdot b$

$\frac{1}{2} = 0.00286788 b$

Rewrite the equation as $0.00286788 b = \frac{1}{2}$ .

$0.00286788 b = \frac{1}{2}$

Divide each term by $0.00286788$ and simplify.

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Divide each term in $0.00286788 b = \frac{1}{2}$ by $0.00286788$ .

$\frac{0.00286788 b}{0.00286788} = \frac{1}{2} \cdot \frac{1}{0.00286788}$

Cancel the common factor of $0.00286788$ .

$b = \frac{1}{2} \cdot \frac{1}{0.00286788}$

Simplify $\frac{1}{2} \cdot \frac{1}{0.00286788}$ .

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Divide $1$ by $0.00286788$ .

$b = \frac{1}{2} \cdot 348.68935912$

Combine $\frac{1}{2}$ and $348.68935912$ .

$b = \frac{348.68935912}{2}$

Divide $348.68935912$ by $2$ .

$b = 174.34467956$

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

$A = 115$

$B = 30$

$C = 35$

$a = 316.01988143$

$b = 174.34467956$

$c = 200$

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