Trigonometry Examples

$A = 76$ , $B = 34$ , $c = 9$

Step 1

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

$76 + C + 34 = 180$

Step 2

Solve the equation for $C$ .

Tap for more steps...

Step 2.1

Add $76$ and $34$ .

$C + 110 = 180$

Step 2.2

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

Tap for more steps...

Step 2.2.1

Subtract $110$ from both sides of the equation.

$C = 180 - 110$

Step 2.2.2

Subtract $110$ from $180$ .

$C = 70$

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 $a$ .

$\frac{\sin (76)}{a} = \frac{\sin (70)}{9}$

Step 5

Solve the equation for $a$ .

Tap for more steps...

Step 5.1

Factor each term.

Tap for more steps...

Step 5.1.1

Evaluate $\sin (76)$ .

$\frac{0.97029572}{a} = \frac{\sin (70)}{9}$

Step 5.1.2

Evaluate $\sin (70)$ .

$\frac{0.97029572}{a} = \frac{0.93969262}{9}$

Step 5.1.3

Divide $0.93969262$ by $9$ .

$\frac{0.97029572}{a} = 0.10441029$

Step 5.2

Find the LCD of the terms in the equation.

Tap for more steps...

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 $\frac{0.97029572}{a} = 0.10441029$ by $a$ to eliminate the fractions.

Tap for more steps...

Step 5.3.1

Multiply each term in $\frac{0.97029572}{a} = 0.10441029$ by $a$ .

$\frac{0.97029572}{a} a = 0.10441029 a$

Step 5.3.2

Simplify the left side.

Tap for more steps...

Step 5.3.2.1

Cancel the common factor of $a$ .

Tap for more steps...

Step 5.3.2.1.1

Cancel the common factor.

$\frac{0.97029572}{a} a = 0.10441029 a$

Step 5.3.2.1.2

Rewrite the expression.

$0.97029572 = 0.10441029 a$

Step 5.4

Solve the equation.

Tap for more steps...

Step 5.4.1

Rewrite the equation as $0.10441029 a = 0.97029572$ .

$0.10441029 a = 0.97029572$

Step 5.4.2

Divide each term in $0.10441029 a = 0.97029572$ by $0.10441029$ and simplify.

Tap for more steps...

Step 5.4.2.1

Divide each term in $0.10441029 a = 0.97029572$ by $0.10441029$ .

$\frac{0.10441029 a}{0.10441029} = \frac{0.97029572}{0.10441029}$

Step 5.4.2.2

Simplify the left side.

Tap for more steps...

Step 5.4.2.2.1

Cancel the common factor of $0.10441029$ .

Tap for more steps...

Step 5.4.2.2.1.1

Cancel the common factor.

$\frac{0.10441029 a}{0.10441029} = \frac{0.97029572}{0.10441029}$

Step 5.4.2.2.1.2

Divide $a$ by $1$ .

$a = \frac{0.97029572}{0.10441029}$

Step 5.4.2.3

Simplify the right side.

Tap for more steps...

Step 5.4.2.3.1

Divide $0.97029572$ by $0.10441029$ .

$a = 9.2931043$

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 $b$ .

$\frac{\sin (34)}{b} = \frac{\sin (76)}{9.2931043}$

Step 8

Solve the equation for $b$ .

Tap for more steps...

Step 8.1

Factor each term.

Tap for more steps...

Step 8.1.1

Evaluate $\sin (34)$ .

$\frac{0.5591929}{b} = \frac{\sin (76)}{9.2931043}$

Step 8.1.2

Evaluate $\sin (76)$ .

$\frac{0.5591929}{b} = \frac{0.97029572}{9.2931043}$

Step 8.1.3

Divide $0.97029572$ by $9.2931043$ .

$\frac{0.5591929}{b} = 0.10441029$

Step 8.2

Find the LCD of the terms in the equation.

Tap for more steps...

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 $\frac{0.5591929}{b} = 0.10441029$ by $b$ to eliminate the fractions.

Tap for more steps...

Step 8.3.1

Multiply each term in $\frac{0.5591929}{b} = 0.10441029$ by $b$ .

$\frac{0.5591929}{b} b = 0.10441029 b$

Step 8.3.2

Simplify the left side.

Tap for more steps...

Step 8.3.2.1

Cancel the common factor of $b$ .

Tap for more steps...

Step 8.3.2.1.1

Cancel the common factor.

$\frac{0.5591929}{b} b = 0.10441029 b$

Step 8.3.2.1.2

Rewrite the expression.

$0.5591929 = 0.10441029 b$

Step 8.4

Solve the equation.

Tap for more steps...

Step 8.4.1

Rewrite the equation as $0.10441029 b = 0.5591929$ .

$0.10441029 b = 0.5591929$

Step 8.4.2

Divide each term in $0.10441029 b = 0.5591929$ by $0.10441029$ and simplify.

Tap for more steps...

Step 8.4.2.1

Divide each term in $0.10441029 b = 0.5591929$ by $0.10441029$ .

$\frac{0.10441029 b}{0.10441029} = \frac{0.5591929}{0.10441029}$

Step 8.4.2.2

Simplify the left side.

Tap for more steps...

Step 8.4.2.2.1

Cancel the common factor of $0.10441029$ .

Tap for more steps...

Step 8.4.2.2.1.1

Cancel the common factor.

$\frac{0.10441029 b}{0.10441029} = \frac{0.5591929}{0.10441029}$

Step 8.4.2.2.1.2

Divide $b$ by $1$ .

$b = \frac{0.5591929}{0.10441029}$

Step 8.4.2.3

Simplify the right side.

Tap for more steps...

Step 8.4.2.3.1

Divide $0.5591929$ by $0.10441029$ .

$b = 5.35572592$

Step 9

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

$A = 76$

$B = 34$

$C = 70$

$a = 9.2931043$

$b = 5.35572592$

$c = 9$

Enter YOUR Problem