Solve the Triangle A=42 , B=76 , c=12

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Trigonometry

$A = 42$ , $B = 76$ , $c = 12$

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

$42 + C + 76 = 180$

Solve the equation for $C$ .

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Add $42$ and $76$ .

$C + 118 = 180$

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

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Subtract $118$ from both sides of the equation.

$C = 180 - 118$

Subtract $118$ from $180$ .

$C = 62$

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 $a$ .

$\frac{\sin (42)}{a} = \frac{\sin (62)}{12}$

Solve the equation for $a$ .

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Factor each term.

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Evaluate $\sin (42)$ .

$\frac{0.6691306}{a} = \frac{\sin (62)}{12}$

Evaluate $\sin (62)$ .

$\frac{0.6691306}{a} = \frac{0.88294759}{12}$

Divide $0.88294759$ by $12$ .

$\frac{0.6691306}{a} = 0.07357896$

Solve for $a$ .

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Multiply each term by $a$ and simplify.

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Multiply each term in $\frac{0.6691306}{a} = 0.07357896$ by $a$ .

$\frac{0.6691306}{a} \cdot a = 0.07357896 \cdot a$

Cancel the common factor of $a$ .

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Cancel the common factor.

$\frac{0.6691306}{a} \cdot a = 0.07357896 \cdot a$

Rewrite the expression.

$0.6691306 = 0.07357896 \cdot a$

$0.6691306 = 0.07357896 a$

Rewrite the equation as $0.07357896 a = 0.6691306$ .

$0.07357896 a = 0.6691306$

Divide each term by $0.07357896$ and simplify.

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Divide each term in $0.07357896 a = 0.6691306$ by $0.07357896$ .

$\frac{0.07357896 a}{0.07357896} = \frac{0.6691306}{0.07357896}$

Cancel the common factor of $0.07357896$ .

$a = \frac{0.6691306}{0.07357896}$

Divide $0.6691306$ by $0.07357896$ .

$a = 9.09404741$

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

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

$\frac{\sin (76)}{b} = \frac{\sin (42)}{9.09404741}$

Solve the equation for $b$ .

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Factor each term.

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Evaluate $\sin (76)$ .

$\frac{0.97029572}{b} = \frac{\sin (42)}{9.09404741}$

Evaluate $\sin (42)$ .

$\frac{0.97029572}{b} = \frac{0.6691306}{9.09404741}$

Divide $0.6691306$ by $9.09404741$ .

$\frac{0.97029572}{b} = 0.07357896$

Solve for $b$ .

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Multiply each term by $b$ and simplify.

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Multiply each term in $\frac{0.97029572}{b} = 0.07357896$ by $b$ .

$\frac{0.97029572}{b} \cdot b = 0.07357896 \cdot b$

Cancel the common factor of $b$ .

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Cancel the common factor.

$\frac{0.97029572}{b} \cdot b = 0.07357896 \cdot b$

Rewrite the expression.

$0.97029572 = 0.07357896 \cdot b$

$0.97029572 = 0.07357896 b$

Rewrite the equation as $0.07357896 b = 0.97029572$ .

$0.07357896 b = 0.97029572$

Divide each term by $0.07357896$ and simplify.

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Divide each term in $0.07357896 b = 0.97029572$ by $0.07357896$ .

$\frac{0.07357896 b}{0.07357896} = \frac{0.97029572}{0.07357896}$

Cancel the common factor of $0.07357896$ .

$b = \frac{0.97029572}{0.07357896}$

Divide $0.97029572$ by $0.07357896$ .

$b = 13.18713455$

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

$A = 42$

$B = 76$

$C = 62$

$a = 9.09404741$

$b = 13.18713455$

$c = 12$

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