Trigonometry Examples

$C = 41$ , $A = 35$ , $a = 26$

Step 1

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 2

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

$\frac{\sin (41)}{c} = \frac{\sin (35)}{26}$

Step 3

Solve the equation for $c$ .

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Step 3.1

Factor each term.

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Step 3.1.1

Evaluate $\sin (41)$ .

$\frac{0.65605902}{c} = \frac{\sin (35)}{26}$

Step 3.1.2

Evaluate $\sin (35)$ .

$\frac{0.65605902}{c} = \frac{0.57357643}{26}$

Step 3.1.3

Divide $0.57357643$ by $26$ .

$\frac{0.65605902}{c} = 0.02206063$

Step 3.2

Find the LCD of the terms in the equation.

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Step 3.2.1

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

$c, 1$

Step 3.2.2

The LCM of one and any expression is the expression.

$c$

Step 3.3

Multiply each term in $\frac{0.65605902}{c} = 0.02206063$ by $c$ to eliminate the fractions.

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Step 3.3.1

Multiply each term in $\frac{0.65605902}{c} = 0.02206063$ by $c$ .

$\frac{0.65605902}{c} c = 0.02206063 c$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $c$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{0.65605902}{c} c = 0.02206063 c$

Step 3.3.2.1.2

Rewrite the expression.

$0.65605902 = 0.02206063 c$

Step 3.4

Solve the equation.

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Step 3.4.1

Rewrite the equation as $0.02206063 c = 0.65605902$ .

$0.02206063 c = 0.65605902$

Step 3.4.2

Divide each term in $0.02206063 c = 0.65605902$ by $0.02206063$ and simplify.

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Step 3.4.2.1

Divide each term in $0.02206063 c = 0.65605902$ by $0.02206063$ .

$\frac{0.02206063 c}{0.02206063} = \frac{0.65605902}{0.02206063}$

Step 3.4.2.2

Simplify the left side.

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Step 3.4.2.2.1

Cancel the common factor of $0.02206063$ .

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Step 3.4.2.2.1.1

Cancel the common factor.

$\frac{0.02206063 c}{0.02206063} = \frac{0.65605902}{0.02206063}$

Step 3.4.2.2.1.2

Divide $c$ by $1$ .

$c = \frac{0.65605902}{0.02206063}$

Step 3.4.2.3

Simplify the right side.

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Step 3.4.2.3.1

Divide $0.65605902$ by $0.02206063$ .

$c = 29.7389043$

Step 4

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

$35 + 41 + B = 180$

Step 5

Solve the equation for $B$ .

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Step 5.1

Add $35$ and $41$ .

$76 + B = 180$

Step 5.2

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

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Step 5.2.1

Subtract $76$ from both sides of the equation.

$B = 180 - 76$

Step 5.2.2

Subtract $76$ from $180$ .

$B = 104$

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 $b$ .

$\frac{\sin (104)}{b} = \frac{\sin (35)}{26}$

Step 8

Solve the equation for $b$ .

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Step 8.1

Factor each term.

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Step 8.1.1

Evaluate $\sin (104)$ .

$\frac{0.97029572}{b} = \frac{\sin (35)}{26}$

Step 8.1.2

Evaluate $\sin (35)$ .

$\frac{0.97029572}{b} = \frac{0.57357643}{26}$

Step 8.1.3

Divide $0.57357643$ by $26$ .

$\frac{0.97029572}{b} = 0.02206063$

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.

$b, 1$

Step 8.2.2

The LCM of one and any expression is the expression.

$b$

Step 8.3

Multiply each term in $\frac{0.97029572}{b} = 0.02206063$ by $b$ to eliminate the fractions.

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Step 8.3.1

Multiply each term in $\frac{0.97029572}{b} = 0.02206063$ by $b$ .

$\frac{0.97029572}{b} b = 0.02206063 b$

Step 8.3.2

Simplify the left side.

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Step 8.3.2.1

Cancel the common factor of $b$ .

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Step 8.3.2.1.1

Cancel the common factor.

$\frac{0.97029572}{b} b = 0.02206063 b$

Step 8.3.2.1.2

Rewrite the expression.

$0.97029572 = 0.02206063 b$

Step 8.4

Solve the equation.

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Step 8.4.1

Rewrite the equation as $0.02206063 b = 0.97029572$ .

$0.02206063 b = 0.97029572$

Step 8.4.2

Divide each term in $0.02206063 b = 0.97029572$ by $0.02206063$ and simplify.

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Step 8.4.2.1

Divide each term in $0.02206063 b = 0.97029572$ by $0.02206063$ .

$\frac{0.02206063 b}{0.02206063} = \frac{0.97029572}{0.02206063}$

Step 8.4.2.2

Simplify the left side.

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Step 8.4.2.2.1

Cancel the common factor of $0.02206063$ .

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Step 8.4.2.2.1.1

Cancel the common factor.

$\frac{0.02206063 b}{0.02206063} = \frac{0.97029572}{0.02206063}$

Step 8.4.2.2.1.2

Divide $b$ by $1$ .

$b = \frac{0.97029572}{0.02206063}$

Step 8.4.2.3

Simplify the right side.

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Step 8.4.2.3.1

Divide $0.97029572$ by $0.02206063$ .

$b = 43.98313334$

Step 9

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

$A = 35$

$B = 104$

$C = 41$

$a = 26$

$b = 43.98313334$

$c = 29.7389043$