Trigonometry Examples

a = 7

$a = 7$ ,

b = 6

$b = 6$ ,

B = 25

$B = 25$

Step 1

The Law of Sines produces an ambiguous angle result. This means that there are

2

$2$ angles that will correctly solve the equation. For the first triangle, use the first possible angle value.

Solve for the first triangle.

Step 2

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}

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

Step 3

Substitute the known values into the law of sines to find

A

$A$ .

\frac{sin (A)}{7} = \frac{sin (25)}{6}

$\frac{\sin (A)}{7} = \frac{\sin (25)}{6}$

Step 4

Solve the equation for

A

$A$ .

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Step 4.1

Multiply both sides of the equation by

7

$7$ .

7 \frac{sin (A)}{7} = 7 \frac{sin (25)}{6}

$7 \frac{\sin (A)}{7} = 7 \frac{\sin (25)}{6}$

Step 4.2

Simplify both sides of the equation.

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Step 4.2.1

Simplify the left side.

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Step 4.2.1.1

Cancel the common factor of

7

$7$ .

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Step 4.2.1.1.1

Cancel the common factor.

$7 \frac{\sin (A)}{7} = 7 \frac{\sin (25)}{6}$

Step 4.2.1.1.2

Rewrite the expression.

$\sin (A) = 7 \frac{\sin (25)}{6}$

Step 4.2.2

Simplify the right side.

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Step 4.2.2.1

Simplify

$7 \frac{\sin (25)}{6}$ .

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Step 4.2.2.1.1

Evaluate

$\sin (25)$ .

$\sin (A) = 7 (\frac{0.42261826}{6})$

Step 4.2.2.1.2

Divide

$0.42261826$ by

$6$ .

$\sin (A) = 7 \cdot 0.07043637$

Step 4.2.2.1.3

Multiply

$7$ by

$0.07043637$ .

$\sin (A) = 0.49305463$

Step 4.3

Take the inverse sine of both sides of the equation to extract

$A$ from inside the sine.

$A = \arcsin (0.49305463)$

Step 4.4

Simplify the right side.

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Step 4.4.1

Evaluate

$\arcsin (0.49305463)$ .

$A = 29.54155262$

Step 4.5

The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from

$180$ to find the solution in the second quadrant.

$A = 180 - 29.54155262$

Step 4.6

Subtract

$29.54155262$ from

$180$ .

$A = 150.45844737$

Step 4.7

The solution to the equation

$A = 29.54155262$ .

$A = 29.54155262, 150.45844737$

Step 5

The sum of all the angles in a triangle is

$180$ degrees.

$29.54155262 + C + 25 = 180$

Step 6

Solve the equation for

$C$ .

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Step 6.1

Add

$29.54155262$ and

$25$ .

$C + 54.54155262 = 180$

Step 6.2

Move all terms not containing

$C$ to the right side of the equation.

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Step 6.2.1

Subtract

$54.54155262$ from both sides of the equation.

$C = 180 - 54.54155262$

Step 6.2.2

Subtract

$54.54155262$ from

$180$ .

$C = 125.45844737$

Step 7

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

Step 8

Substitute the known values into the law of sines to find

$c$ .

$\frac{\sin (125.45844737)}{c} = \frac{\sin (29.54155262)}{7}$

Step 9

Solve the equation for

$c$ .

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Step 9.1

Factor each term.

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Step 9.1.1

Evaluate

$\sin (125.45844737)$ .

$\frac{0.81453644}{c} = \frac{\sin (29.54155262)}{7}$

Step 9.1.2

Evaluate

$\sin (29.54155262)$ .

$\frac{0.81453644}{c} = \frac{0.49305463}{7}$

Step 9.1.3

Divide

$0.49305463$ by

$7$ .

$\frac{0.81453644}{c} = 0.07043637$

Step 9.2

Find the LCD of the terms in the equation.

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Step 9.2.1

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

$c, 1$

Step 9.2.2

The LCM of one and any expression is the expression.

$c$

Step 9.3

Multiply each term in

$\frac{0.81453644}{c} = 0.07043637$ by

$c$ to eliminate the fractions.

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Step 9.3.1

Multiply each term in

$\frac{0.81453644}{c} = 0.07043637$ by

$c$ .

$\frac{0.81453644}{c} c = 0.07043637 c$

Step 9.3.2

Simplify the left side.

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Step 9.3.2.1

Cancel the common factor of

$c$ .

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Step 9.3.2.1.1

Cancel the common factor.

$\frac{0.81453644}{c} c = 0.07043637 c$

Step 9.3.2.1.2

Rewrite the expression.

$0.81453644 = 0.07043637 c$

Step 9.4

Solve the equation.

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Step 9.4.1

Rewrite the equation as

$0.07043637 c = 0.81453644$ .

$0.07043637 c = 0.81453644$

Step 9.4.2

Divide each term in

$0.07043637 c = 0.81453644$ by

$0.07043637$ and simplify.

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Step 9.4.2.1

Divide each term in

$0.07043637 c = 0.81453644$ by

$0.07043637$ .

$\frac{0.07043637 c}{0.07043637} = \frac{0.81453644}{0.07043637}$

Step 9.4.2.2

Simplify the left side.

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Step 9.4.2.2.1

Cancel the common factor of

$0.07043637$ .

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Step 9.4.2.2.1.1

Cancel the common factor.

$\frac{0.07043637 c}{0.07043637} = \frac{0.81453644}{0.07043637}$

Step 9.4.2.2.1.2

Divide

$c$ by

$1$ .

$c = \frac{0.81453644}{0.07043637}$

Step 9.4.2.3

Simplify the right side.

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Step 9.4.2.3.1

Divide

$0.81453644$ by

$0.07043637$ .

$c = 11.56414458$

Step 10

For the second triangle, use the second possible angle value.

Solve for the second triangle.

Step 11

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

Step 12

Substitute the known values into the law of sines to find

$A$ .

$\frac{\sin (A)}{7} = \frac{\sin (25)}{6}$

Step 13

Solve the equation for

$A$ .

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Step 13.1

Multiply both sides of the equation by

$7$ .

$7 \frac{\sin (A)}{7} = 7 \frac{\sin (25)}{6}$

Step 13.2

Simplify both sides of the equation.

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Step 13.2.1

Simplify the left side.

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Step 13.2.1.1

Cancel the common factor of

$7$ .

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Step 13.2.1.1.1

Cancel the common factor.

$7 \frac{\sin (A)}{7} = 7 \frac{\sin (25)}{6}$

Step 13.2.1.1.2

Rewrite the expression.

$\sin (A) = 7 \frac{\sin (25)}{6}$

Step 13.2.2

Simplify the right side.

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Step 13.2.2.1

Simplify

$7 \frac{\sin (25)}{6}$ .

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Step 13.2.2.1.1

Evaluate

$\sin (25)$ .

$\sin (A) = 7 (\frac{0.42261826}{6})$

Step 13.2.2.1.2

Divide

$0.42261826$ by

$6$ .

$\sin (A) = 7 \cdot 0.07043637$

Step 13.2.2.1.3

Multiply

$7$ by

$0.07043637$ .

$\sin (A) = 0.49305463$

Step 13.3

Take the inverse sine of both sides of the equation to extract

$A$ from inside the sine.

$A = \arcsin (0.49305463)$

Step 13.4

Simplify the right side.

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Step 13.4.1

Evaluate

$\arcsin (0.49305463)$ .

$A = 29.54155262$

Step 13.5

The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from

$180$ to find the solution in the second quadrant.

$A = 180 - 29.54155262$

Step 13.6

Subtract

$29.54155262$ from

$180$ .

$A = 150.45844737$

Step 13.7

The solution to the equation

$A = 29.54155262$ .

$A = 29.54155262, 150.45844737$

Step 14

The sum of all the angles in a triangle is

$180$ degrees.

$150.45844737 + C + 25 = 180$

Step 15

Solve the equation for

$C$ .

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Step 15.1

Add

$150.45844737$ and

$25$ .

$C + 175.45844737 = 180$

Step 15.2

Move all terms not containing

$C$ to the right side of the equation.

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Step 15.2.1

Subtract

$175.45844737$ from both sides of the equation.

$C = 180 - 175.45844737$

Step 15.2.2

Subtract

$175.45844737$ from

$180$ .

$C = 4.54155262$

Step 16

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

Step 17

Substitute the known values into the law of sines to find

$c$ .

$\frac{\sin (4.54155262)}{c} = \frac{\sin (150.45844737)}{7}$

Step 18

Solve the equation for

$c$ .

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Step 18.1

Factor each term.

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Step 18.1.1

Evaluate

$\sin (4.54155262)$ .

$\frac{0.07918206}{c} = \frac{\sin (150.45844737)}{7}$

Step 18.1.2

Evaluate

$\sin (150.45844737)$ .

$\frac{0.07918206}{c} = \frac{0.49305463}{7}$

Step 18.1.3

Divide

$0.49305463$ by

$7$ .

$\frac{0.07918206}{c} = 0.07043637$

Step 18.2

Find the LCD of the terms in the equation.

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Step 18.2.1

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

$c, 1$

Step 18.2.2

The LCM of one and any expression is the expression.

$c$

Step 18.3

Multiply each term in

$\frac{0.07918206}{c} = 0.07043637$ by

$c$ to eliminate the fractions.

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Step 18.3.1

Multiply each term in

$\frac{0.07918206}{c} = 0.07043637$ by

$c$ .

$\frac{0.07918206}{c} c = 0.07043637 c$

Step 18.3.2

Simplify the left side.

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Step 18.3.2.1

Cancel the common factor of

$c$ .

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Step 18.3.2.1.1

Cancel the common factor.

$\frac{0.07918206}{c} c = 0.07043637 c$

Step 18.3.2.1.2

Rewrite the expression.

$0.07918206 = 0.07043637 c$

Step 18.4

Solve the equation.

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Step 18.4.1

Rewrite the equation as

$0.07043637 c = 0.07918206$ .

$0.07043637 c = 0.07918206$

Step 18.4.2

Divide each term in

$0.07043637 c = 0.07918206$ by

$0.07043637$ and simplify.

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Step 18.4.2.1

Divide each term in

$0.07043637 c = 0.07918206$ by

$0.07043637$ .

$\frac{0.07043637 c}{0.07043637} = \frac{0.07918206}{0.07043637}$

Step 18.4.2.2

Simplify the left side.

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Step 18.4.2.2.1

Cancel the common factor of

$0.07043637$ .

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Step 18.4.2.2.1.1

Cancel the common factor.

$\frac{0.07043637 c}{0.07043637} = \frac{0.07918206}{0.07043637}$

Step 18.4.2.2.1.2

Divide

$c$ by

$1$ .

$c = \frac{0.07918206}{0.07043637}$

Step 18.4.2.3

Simplify the right side.

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Step 18.4.2.3.1

Divide

$0.07918206$ by

$0.07043637$ .

$c = 1.12416442$

Step 19

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

First Triangle Combination:

$A = 29.54155262$

$B = 25$

$C = 125.45844737$

$a = 7$

$b = 6$

$c = 11.56414458$

Second Triangle Combination:

$A = 150.45844737$

$B = 25$

$C = 4.54155262$

$a = 7$

$b = 6$

$c = 1.12416442$