Solve the Triangle A=35 , C=105 , a=21

$A = 35$ , $C = 105$ , $a = 21$

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 $c$ .

$\frac{\sin (105)}{c} = \frac{\sin (35)}{21}$

Solve the equation for $c$ .

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Factor each term.

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The exact value of $\sin (105)$ is $\frac{\sqrt{2} + \sqrt{6}}{4}$ .

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Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$\frac{\sin (75)}{c} = \frac{\sin (35)}{21}$

Split $75$ into two angles where the values of the six trigonometric functions are known.

$\frac{\sin (30 + 45)}{c} = \frac{\sin (35)}{21}$

Apply the sum of angles identity.

$\frac{\sin (30) \cos (45) + \cos (30) \sin (45)}{c} = \frac{\sin (35)}{21}$

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

$\frac{\frac{1}{2} \cos (45) + \cos (30) \sin (45)}{c} = \frac{\sin (35)}{21}$

The exact value of $\cos (45)$ is $\frac{\sqrt{2}}{2}$ .

$\frac{\frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \cos (30) \sin (45)}{c} = \frac{\sin (35)}{21}$

The exact value of $\cos (30)$ is $\frac{\sqrt{3}}{2}$ .

$\frac{\frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2} \sin (45)}{c} = \frac{\sin (35)}{21}$

The exact value of $\sin (45)$ is $\frac{\sqrt{2}}{2}$ .

$\frac{\frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}}{c} = \frac{\sin (35)}{21}$

Simplify $\frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}$ .

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Simplify each term.

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Multiply $\frac{1}{2} \cdot \frac{\sqrt{2}}{2}$ .

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

$\frac{\frac{\sqrt{2}}{2 \cdot 2} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}}{c} = \frac{\sin (35)}{21}$

Multiply $2$ by $2$ .

$\frac{\frac{\sqrt{2}}{4} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}}{c} = \frac{\sin (35)}{21}$

Multiply $\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}$ .

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Multiply $\frac{\sqrt{3}}{2}$ and $\frac{\sqrt{2}}{2}$ .

$\frac{\frac{\sqrt{2}}{4} + \frac{\sqrt{3} \sqrt{2}}{2 \cdot 2}}{c} = \frac{\sin (35)}{21}$

Combine using the product rule for radicals.

$\frac{\frac{\sqrt{2}}{4} + \frac{\sqrt{3 \cdot 2}}{2 \cdot 2}}{c} = \frac{\sin (35)}{21}$

Multiply $3$ by $2$ .

$\frac{\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{2 \cdot 2}}{c} = \frac{\sin (35)}{21}$

Multiply $2$ by $2$ .

$\frac{\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}}{c} = \frac{\sin (35)}{21}$

Combine the numerators over the common denominator.

$\frac{\frac{\sqrt{2} + \sqrt{6}}{4}}{c} = \frac{\sin (35)}{21}$

Multiply the numerator by the reciprocal of the denominator.

$\frac{\sqrt{2} + \sqrt{6}}{4} \cdot \frac{1}{c} = \frac{\sin (35)}{21}$

Multiply $\frac{\sqrt{2} + \sqrt{6}}{4}$ and $\frac{1}{c}$ .

$\frac{\sqrt{2} + \sqrt{6}}{4 c} = \frac{\sin (35)}{21}$

Evaluate $\sin (35)$ .

$\frac{\sqrt{2} + \sqrt{6}}{4 c} = \frac{0.57357643}{21}$

Divide $0.57357643$ by $21$ .

$\frac{\sqrt{2} + \sqrt{6}}{4 c} = 0.02731316$

Solve for $c$ .

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Multiply each term by $c$ and simplify.

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Multiply each term in $\frac{\sqrt{2} + \sqrt{6}}{4 c} = 0.02731316$ by $c$ .

$\frac{\sqrt{2} + \sqrt{6}}{4 c} \cdot c = 0.02731316 \cdot c$

Cancel the common factor of $c$ .

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Factor $c$ out of $4 c$ .

$\frac{\sqrt{2} + \sqrt{6}}{c \cdot 4} \cdot c = 0.02731316 \cdot c$

Cancel the common factor.

$\frac{\sqrt{2} + \sqrt{6}}{c \cdot 4} \cdot c = 0.02731316 \cdot c$

Rewrite the expression.

$\frac{\sqrt{2} + \sqrt{6}}{4} = 0.02731316 \cdot c$

$\frac{\sqrt{2} + \sqrt{6}}{4} = 0.02731316 c$

Rewrite the equation as $0.02731316 c = \frac{\sqrt{2} + \sqrt{6}}{4}$ .

$0.02731316 c = \frac{\sqrt{2} + \sqrt{6}}{4}$

Divide each term by $0.02731316$ and simplify.

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Divide each term in $0.02731316 c = \frac{\sqrt{2} + \sqrt{6}}{4}$ by $0.02731316$ .

$\frac{0.02731316 c}{0.02731316} = \frac{\frac{\sqrt{2} + \sqrt{6}}{4}}{0.02731316}$

Cancel the common factor of $0.02731316$ .

$c = \frac{\frac{\sqrt{2} + \sqrt{6}}{4}}{0.02731316}$

Simplify $\frac{\frac{\sqrt{2} + \sqrt{6}}{4}}{0.02731316}$ .

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Multiply the numerator by the reciprocal of the denominator.

$c = \frac{\sqrt{2} + \sqrt{6}}{4} \cdot \frac{1}{0.02731316}$

Divide $1$ by $0.02731316$ .

$c = \frac{\sqrt{2} + \sqrt{6}}{4} \cdot 36.6123827$

Multiply $\frac{\sqrt{2} + \sqrt{6}}{4} \cdot 36.6123827$ .

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Combine $\frac{\sqrt{2} + \sqrt{6}}{4}$ and $36.6123827$ .

$c = \frac{(\sqrt{2} + \sqrt{6}) \cdot 36.6123827}{4}$

Multiply $\sqrt{2} + \sqrt{6}$ by $36.6123827$ .

$c = \frac{141.45938407}{4}$

Divide $141.45938407$ by $4$ .

$c = 35.36484601$

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

$35 + 105 + B = 180$

Solve the equation for $B$ .

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Add $35$ and $105$ .

$140 + B = 180$

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

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Subtract $140$ from both sides of the equation.

$B = 180 - 140$

Subtract $140$ from $180$ .

$B = 40$

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

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

$\frac{\sin (40)}{b} = \frac{\sin (35)}{21}$

Solve the equation for $b$ .

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Factor each term.

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Evaluate $\sin (40)$ .

$\frac{0.6427876}{b} = \frac{\sin (35)}{21}$

Evaluate $\sin (35)$ .

$\frac{0.6427876}{b} = \frac{0.57357643}{21}$

Divide $0.57357643$ by $21$ .

$\frac{0.6427876}{b} = 0.02731316$

Solve for $b$ .

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Multiply each term by $b$ and simplify.

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Multiply each term in $\frac{0.6427876}{b} = 0.02731316$ by $b$ .

$\frac{0.6427876}{b} \cdot b = 0.02731316 \cdot b$

Cancel the common factor of $b$ .

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Cancel the common factor.

$\frac{0.6427876}{b} \cdot b = 0.02731316 \cdot b$

Rewrite the expression.

$0.6427876 = 0.02731316 \cdot b$

$0.6427876 = 0.02731316 b$

Rewrite the equation as $0.02731316 b = 0.6427876$ .

$0.02731316 b = 0.6427876$

Divide each term by $0.02731316$ and simplify.

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Divide each term in $0.02731316 b = 0.6427876$ by $0.02731316$ .

$\frac{0.02731316 b}{0.02731316} = \frac{0.6427876}{0.02731316}$

Cancel the common factor of $0.02731316$ .

$b = \frac{0.6427876}{0.02731316}$

Divide $0.6427876$ by $0.02731316$ .

$b = 23.53398596$

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

$A = 35$

$B = 40$

$C = 105$

$a = 21$

$b = 23.53398596$

$c = 35.36484601$

Solve the Triangle A=35 , C=105 , a=21

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