Trigonometry Examples

$b = 10$ , $A = 63$ , $C = 35$

Step 1

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

$63 + 35 + B = 180$

Step 2

Solve the equation for $B$ .

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Step 2.1

Add $63$ and $35$ .

$98 + B = 180$

Step 2.2

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

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Step 2.2.1

Subtract $98$ from both sides of the equation.

$B = 180 - 98$

Step 2.2.2

Subtract $98$ from $180$ .

$B = 82$

Step 3

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 4

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

$\frac{\sin (63)}{a} = \frac{\sin (82)}{10}$

Step 5

Solve the equation for $a$ .

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Step 5.1

Factor each term.

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Step 5.1.1

Evaluate $\sin (63)$ .

$\frac{0.89100652}{a} = \frac{\sin (82)}{10}$

Step 5.1.2

Evaluate $\sin (82)$ .

$\frac{0.89100652}{a} = \frac{0.99026806}{10}$

Step 5.1.3

Divide $0.99026806$ by $10$ .

$\frac{0.89100652}{a} = 0.0990268$

Step 5.2

Find the LCD of the terms in the equation.

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Step 5.2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$a, 1$

Step 5.2.2

The LCM of one and any expression is the expression.

$a$

Step 5.3

Multiply each term in $\frac{0.89100652}{a} = 0.0990268$ by $a$ to eliminate the fractions.

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Step 5.3.1

Multiply each term in $\frac{0.89100652}{a} = 0.0990268$ by $a$ .

$\frac{0.89100652}{a} a = 0.0990268 a$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $a$ .

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Step 5.3.2.1.1

Cancel the common factor.

$\frac{0.89100652}{a} a = 0.0990268 a$

Step 5.3.2.1.2

Rewrite the expression.

$0.89100652 = 0.0990268 a$

Step 5.4

Solve the equation.

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Step 5.4.1

Rewrite the equation as $0.0990268 a = 0.89100652$ .

$0.0990268 a = 0.89100652$

Step 5.4.2

Divide each term in $0.0990268 a = 0.89100652$ by $0.0990268$ and simplify.

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Step 5.4.2.1

Divide each term in $0.0990268 a = 0.89100652$ by $0.0990268$ .

$\frac{0.0990268 a}{0.0990268} = \frac{0.89100652}{0.0990268}$

Step 5.4.2.2

Simplify the left side.

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Step 5.4.2.2.1

Cancel the common factor of $0.0990268$ .

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Step 5.4.2.2.1.1

Cancel the common factor.

$\frac{0.0990268 a}{0.0990268} = \frac{0.89100652}{0.0990268}$

Step 5.4.2.2.1.2

Divide $a$ by $1$ .

$a = \frac{0.89100652}{0.0990268}$

Step 5.4.2.3

Simplify the right side.

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Step 5.4.2.3.1

Divide $0.89100652$ by $0.0990268$ .

$a = 8.99762955$

Step 6

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

Step 7

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

$\frac{\sin (35)}{c} = \frac{\sin (63)}{8.99762955}$

Step 8

Solve the equation for $c$ .

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Step 8.1

Factor each term.

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Step 8.1.1

Evaluate $\sin (35)$ .

$\frac{0.57357643}{c} = \frac{\sin (63)}{8.99762955}$

Step 8.1.2

Evaluate $\sin (63)$ .

$\frac{0.57357643}{c} = \frac{0.89100652}{8.99762955}$

Step 8.1.3

Divide $0.89100652$ by $8.99762955$ .

$\frac{0.57357643}{c} = 0.0990268$

Step 8.2

Find the LCD of the terms in the equation.

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Step 8.2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$c, 1$

Step 8.2.2

The LCM of one and any expression is the expression.

$c$

Step 8.3

Multiply each term in $\frac{0.57357643}{c} = 0.0990268$ by $c$ to eliminate the fractions.

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Step 8.3.1

Multiply each term in $\frac{0.57357643}{c} = 0.0990268$ by $c$ .

$\frac{0.57357643}{c} c = 0.0990268 c$

Step 8.3.2

Simplify the left side.

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Step 8.3.2.1

Cancel the common factor of $c$ .

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Step 8.3.2.1.1

Cancel the common factor.

$\frac{0.57357643}{c} c = 0.0990268 c$

Step 8.3.2.1.2

Rewrite the expression.

$0.57357643 = 0.0990268 c$

Step 8.4

Solve the equation.

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Step 8.4.1

Rewrite the equation as $0.0990268 c = 0.57357643$ .

$0.0990268 c = 0.57357643$

Step 8.4.2

Divide each term in $0.0990268 c = 0.57357643$ by $0.0990268$ and simplify.

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Step 8.4.2.1

Divide each term in $0.0990268 c = 0.57357643$ by $0.0990268$ .

$\frac{0.0990268 c}{0.0990268} = \frac{0.57357643}{0.0990268}$

Step 8.4.2.2

Simplify the left side.

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Step 8.4.2.2.1

Cancel the common factor of $0.0990268$ .

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Step 8.4.2.2.1.1

Cancel the common factor.

$\frac{0.0990268 c}{0.0990268} = \frac{0.57357643}{0.0990268}$

Step 8.4.2.2.1.2

Divide $c$ by $1$ .

$c = \frac{0.57357643}{0.0990268}$

Step 8.4.2.3

Simplify the right side.

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Step 8.4.2.3.1

Divide $0.57357643$ by $0.0990268$ .

$c = 5.792133$

Step 9

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

$A = 63$

$B = 82$

$C = 35$

$a = 8.99762955$

$b = 10$

$c = 5.792133$