Trigonometry Examples

A = 4915

$A = 4915$ ,

b = 60

$b = 60$ ,

c = 89

$c = 89$

Step 1

Use the law of cosines to find the unknown side of the triangle, given the other two sides and the included angle.

a^{2} = b^{2} + c^{2} - 2 b c cos (A)

$a^{2} = b^{2} + c^{2} - 2 b c \cos (A)$

Step 2

Solve the equation.

a = \sqrt{b^{2} + c^{2} - 2 b c cos (A)}

$a = \sqrt{b^{2} + c^{2} - 2 b c \cos (A)}$

Step 3

Substitute the known values into the equation.

a = \sqrt{{(60)}^{2} + {(89)}^{2} - 2 \cdot 60 \cdot 89 cos (4915)}

$a = \sqrt{{(60)}^{2} + {(89)}^{2} - 2 \cdot 60 \cdot 89 \cos (4915)}$

Step 4

Simplify the results.

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Step 4.1

Raise

60

$60$ to the power of

2

$2$ .

a = \sqrt{3600 + {(89)}^{2} - 2 \cdot 60 \cdot (89 cos (4915))}

$a = \sqrt{3600 + {(89)}^{2} - 2 \cdot 60 \cdot (89 \cos (4915))}$

Step 4.2

Raise

89

$89$ to the power of

2

$2$ .

a = \sqrt{3600 + 7921 - 2 \cdot 60 \cdot (89 cos (4915))}

$a = \sqrt{3600 + 7921 - 2 \cdot 60 \cdot (89 \cos (4915))}$

Step 4.3

Multiply

- 2 \cdot 60 \cdot 89

$- 2 \cdot 60 \cdot 89$ .

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Step 4.3.1

Multiply

- 2

$- 2$ by

60

$60$ .

a = \sqrt{3600 + 7921 - 120 \cdot (89 cos (4915))}

$a = \sqrt{3600 + 7921 - 120 \cdot (89 \cos (4915))}$

Step 4.3.2

Multiply

- 120

$- 120$ by

89

$89$ .

a = \sqrt{3600 + 7921 - 10680 cos (4915)}

$a = \sqrt{3600 + 7921 - 10680 \cos (4915)}$

a = \sqrt{3600 + 7921 - 10680 cos (4915)}

$a = \sqrt{3600 + 7921 - 10680 \cos (4915)}$

Step 4.4

Remove full rotations of

360

$360$ ° until the angle is between

0

$0$ ° and

360

$360$ °.

a = \sqrt{3600 + 7921 - 10680 cos (235)}

$a = \sqrt{3600 + 7921 - 10680 \cos (235)}$

Step 4.5

Evaluate

cos (235)

$\cos (235)$ .

a = \sqrt{3600 + 7921 - 10680 \cdot - 0.57357643}

$a = \sqrt{3600 + 7921 - 10680 \cdot - 0.57357643}$

Step 4.6

Multiply

- 10680

$- 10680$ by

- 0.57357643

$- 0.57357643$ .

a = \sqrt{3600 + 7921 + 6125.79634022}

$a = \sqrt{3600 + 7921 + 6125.79634022}$

Step 4.7

Add

3600

$3600$ and

7921

$7921$ .

a = \sqrt{11521 + 6125.79634022}

$a = \sqrt{11521 + 6125.79634022}$

Step 4.8

Add

11521

$11521$ and

6125.79634022

$6125.79634022$ .

a = \sqrt{17646.79634022}

$a = \sqrt{17646.79634022}$

Step 4.9

Evaluate the root.

a = 132.84124487

$a = 132.84124487$

a = 132.84124487

$a = 132.84124487$

Step 5

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 6

Substitute the known values into the law of sines to find

B

$B$ .

\frac{sin (B)}{60} = \frac{sin (4915)}{132.84124487}

$\frac{\sin (B)}{60} = \frac{\sin (4915)}{132.84124487}$

Step 7

Solve the equation for

B

$B$ .

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Step 7.1

Multiply both sides of the equation by

60

$60$ .

60 \frac{sin (B)}{60} = 60 \frac{sin (4915)}{132.84124487}

$60 \frac{\sin (B)}{60} = 60 \frac{\sin (4915)}{132.84124487}$

Step 7.2

Simplify both sides of the equation.

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Step 7.2.1

Simplify the left side.

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Step 7.2.1.1

Cancel the common factor of

60

$60$ .

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Step 7.2.1.1.1

Cancel the common factor.

$60 \frac{\sin (B)}{60} = 60 \frac{\sin (4915)}{132.84124487}$

Step 7.2.1.1.2

Rewrite the expression.

$\sin (B) = 60 \frac{\sin (4915)}{132.84124487}$

Step 7.2.2

Simplify the right side.

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Step 7.2.2.1

Simplify

$60 \frac{\sin (4915)}{132.84124487}$ .

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Step 7.2.2.1.1

Simplify the numerator.

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Step 7.2.2.1.1.1

Remove full rotations of

$360$ ° until the angle is between

$0$ ° and

$360$ °.

$\sin (B) = 60 \frac{\sin (235)}{132.84124487}$

Step 7.2.2.1.1.2

Evaluate

$\sin (235)$ .

$\sin (B) = 60 (\frac{- 0.81915204}{132.84124487})$

Step 7.2.2.1.2

Simplify the expression.

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Step 7.2.2.1.2.1

Divide

$- 0.81915204$ by

$132.84124487$ .

$\sin (B) = 60 \cdot - 0.00616639$

Step 7.2.2.1.2.2

Multiply

$60$ by

$- 0.00616639$ .

$\sin (B) = - 0.3699839$

Step 7.3

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

$B$ from inside the sine.

$B = \arcsin (- 0.3699839)$

Step 7.4

Simplify the right side.

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Step 7.4.1

Evaluate

$\arcsin (- 0.3699839)$ .

$B = - 21.71462472$

Step 7.5

The sine function is negative in the third and fourth quadrants. To find the second solution, subtract the solution from

$360$ , to find a reference angle. Next, add this reference angle to

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

$B = 360 + 21.71462472 + 180$

Step 7.6

Simplify the expression to find the second solution.

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Step 7.6.1

Subtract

$360 °$ from

$360 + 21.71462472 + 180 °$ .

$B = 360 + 21.71462472 + 180 ° - 360 °$

Step 7.6.2

The resulting angle of

$201.71462472 °$ is positive, less than

$360 °$ , and coterminal with

$360 + 21.71462472 + 180$ .

$B = 201.71462472 °$

Step 7.7

The solution to the equation

$B = - 21.71462472$ .

$B = - 21.71462472, 201.71462472$

Step 7.8

The triangle is invalid.

Invalid triangle

Step 8

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

Step 9

Substitute the known values into the law of sines to find

$B$ .

$\frac{\sin (B)}{60} = \frac{\sin (4915)}{132.84124487}$

Step 10

Solve the equation for

$B$ .

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Step 10.1

Multiply both sides of the equation by

$60$ .

$60 \frac{\sin (B)}{60} = 60 \frac{\sin (4915)}{132.84124487}$

Step 10.2

Simplify both sides of the equation.

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Step 10.2.1

Simplify the left side.

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Step 10.2.1.1

Cancel the common factor of

$60$ .

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Step 10.2.1.1.1

Cancel the common factor.

$60 \frac{\sin (B)}{60} = 60 \frac{\sin (4915)}{132.84124487}$

Step 10.2.1.1.2

Rewrite the expression.

$\sin (B) = 60 \frac{\sin (4915)}{132.84124487}$

Step 10.2.2

Simplify the right side.

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Step 10.2.2.1

Simplify

$60 \frac{\sin (4915)}{132.84124487}$ .

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Step 10.2.2.1.1

Simplify the numerator.

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Step 10.2.2.1.1.1

Remove full rotations of

$360$ ° until the angle is between

$0$ ° and

$360$ °.

$\sin (B) = 60 \frac{\sin (235)}{132.84124487}$

Step 10.2.2.1.1.2

Evaluate

$\sin (235)$ .

$\sin (B) = 60 (\frac{- 0.81915204}{132.84124487})$

Step 10.2.2.1.2

Simplify the expression.

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Step 10.2.2.1.2.1

Divide

$- 0.81915204$ by

$132.84124487$ .

$\sin (B) = 60 \cdot - 0.00616639$

Step 10.2.2.1.2.2

Multiply

$60$ by

$- 0.00616639$ .

$\sin (B) = - 0.3699839$

Step 10.3

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

$B$ from inside the sine.

$B = \arcsin (- 0.3699839)$

Step 10.4

Simplify the right side.

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Step 10.4.1

Evaluate

$\arcsin (- 0.3699839)$ .

$B = - 21.71462472$

Step 10.5

The sine function is negative in the third and fourth quadrants. To find the second solution, subtract the solution from

$360$ , to find a reference angle. Next, add this reference angle to

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

$B = 360 + 21.71462472 + 180$

Step 10.6

Simplify the expression to find the second solution.

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Step 10.6.1

Subtract

$360 °$ from

$360 + 21.71462472 + 180 °$ .

$B = 360 + 21.71462472 + 180 ° - 360 °$

Step 10.6.2

The resulting angle of

$201.71462472 °$ is positive, less than

$360 °$ , and coterminal with

$360 + 21.71462472 + 180$ .

$B = 201.71462472 °$

Step 10.7

The solution to the equation

$B = - 21.71462472$ .

$B = - 21.71462472, 201.71462472$

Step 10.8

The triangle is invalid.

Invalid 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

$B$ .

$\frac{\sin (B)}{60} = \frac{\sin (4915)}{132.84124487}$

Step 13

Solve the equation for

$B$ .

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Step 13.1

Multiply both sides of the equation by

$60$ .

$60 \frac{\sin (B)}{60} = 60 \frac{\sin (4915)}{132.84124487}$

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

$60$ .

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Step 13.2.1.1.1

Cancel the common factor.

$60 \frac{\sin (B)}{60} = 60 \frac{\sin (4915)}{132.84124487}$

Step 13.2.1.1.2

Rewrite the expression.

$\sin (B) = 60 \frac{\sin (4915)}{132.84124487}$

Step 13.2.2

Simplify the right side.

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Step 13.2.2.1

Simplify

$60 \frac{\sin (4915)}{132.84124487}$ .

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Step 13.2.2.1.1

Simplify the numerator.

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Step 13.2.2.1.1.1

Remove full rotations of

$360$ ° until the angle is between

$0$ ° and

$360$ °.

$\sin (B) = 60 \frac{\sin (235)}{132.84124487}$

Step 13.2.2.1.1.2

Evaluate

$\sin (235)$ .

$\sin (B) = 60 (\frac{- 0.81915204}{132.84124487})$

Step 13.2.2.1.2

Simplify the expression.

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Step 13.2.2.1.2.1

Divide

$- 0.81915204$ by

$132.84124487$ .

$\sin (B) = 60 \cdot - 0.00616639$

Step 13.2.2.1.2.2

Multiply

$60$ by

$- 0.00616639$ .

$\sin (B) = - 0.3699839$

Step 13.3

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

$B$ from inside the sine.

$B = \arcsin (- 0.3699839)$

Step 13.4

Simplify the right side.

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Step 13.4.1

Evaluate

$\arcsin (- 0.3699839)$ .

$B = - 21.71462472$

Step 13.5

The sine function is negative in the third and fourth quadrants. To find the second solution, subtract the solution from

$360$ , to find a reference angle. Next, add this reference angle to

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

$B = 360 + 21.71462472 + 180$

Step 13.6

Simplify the expression to find the second solution.

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Step 13.6.1

Subtract

$360 °$ from

$360 + 21.71462472 + 180 °$ .

$B = 360 + 21.71462472 + 180 ° - 360 °$

Step 13.6.2

The resulting angle of

$201.71462472 °$ is positive, less than

$360 °$ , and coterminal with

$360 + 21.71462472 + 180$ .

$B = 201.71462472 °$

Step 13.7

The solution to the equation

$B = - 21.71462472$ .

$B = - 21.71462472, 201.71462472$

Step 13.8

The triangle is invalid.

Invalid triangle

Step 14

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

Step 15

Substitute the known values into the law of sines to find

$B$ .

$\frac{\sin (B)}{60} = \frac{\sin (4915)}{132.84124487}$

Step 16

Solve the equation for

$B$ .

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Step 16.1

Multiply both sides of the equation by

$60$ .

$60 \frac{\sin (B)}{60} = 60 \frac{\sin (4915)}{132.84124487}$

Step 16.2

Simplify both sides of the equation.

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Step 16.2.1

Simplify the left side.

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Step 16.2.1.1

Cancel the common factor of

$60$ .

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Step 16.2.1.1.1

Cancel the common factor.

$60 \frac{\sin (B)}{60} = 60 \frac{\sin (4915)}{132.84124487}$

Step 16.2.1.1.2

Rewrite the expression.

$\sin (B) = 60 \frac{\sin (4915)}{132.84124487}$

Step 16.2.2

Simplify the right side.

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Step 16.2.2.1

Simplify

$60 \frac{\sin (4915)}{132.84124487}$ .

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Step 16.2.2.1.1

Simplify the numerator.

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Step 16.2.2.1.1.1

Remove full rotations of

$360$ ° until the angle is between

$0$ ° and

$360$ °.

$\sin (B) = 60 \frac{\sin (235)}{132.84124487}$

Step 16.2.2.1.1.2

Evaluate

$\sin (235)$ .

$\sin (B) = 60 (\frac{- 0.81915204}{132.84124487})$

Step 16.2.2.1.2

Simplify the expression.

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Step 16.2.2.1.2.1

Divide

$- 0.81915204$ by

$132.84124487$ .

$\sin (B) = 60 \cdot - 0.00616639$

Step 16.2.2.1.2.2

Multiply

$60$ by

$- 0.00616639$ .

$\sin (B) = - 0.3699839$

Step 16.3

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

$B$ from inside the sine.

$B = \arcsin (- 0.3699839)$

Step 16.4

Simplify the right side.

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Step 16.4.1

Evaluate

$\arcsin (- 0.3699839)$ .

$B = - 21.71462472$

Step 16.5

The sine function is negative in the third and fourth quadrants. To find the second solution, subtract the solution from

$360$ , to find a reference angle. Next, add this reference angle to

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

$B = 360 + 21.71462472 + 180$

Step 16.6

Simplify the expression to find the second solution.

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Step 16.6.1

Subtract

$360 °$ from

$360 + 21.71462472 + 180 °$ .

$B = 360 + 21.71462472 + 180 ° - 360 °$

Step 16.6.2

The resulting angle of

$201.71462472 °$ is positive, less than

$360 °$ , and coterminal with

$360 + 21.71462472 + 180$ .

$B = 201.71462472 °$

Step 16.7

The solution to the equation

$B = - 21.71462472$ .

$B = - 21.71462472, 201.71462472$

Step 16.8

The triangle is invalid.

Invalid triangle

Step 17

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

Step 18

Substitute the known values into the law of sines to find

$B$ .

$\frac{\sin (B)}{60} = \frac{\sin (4915)}{132.84124487}$

Step 19

Solve the equation for

$B$ .

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Step 19.1

Multiply both sides of the equation by

$60$ .

$60 \frac{\sin (B)}{60} = 60 \frac{\sin (4915)}{132.84124487}$

Step 19.2

Simplify both sides of the equation.

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Step 19.2.1

Simplify the left side.

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Step 19.2.1.1

Cancel the common factor of

$60$ .

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Step 19.2.1.1.1

Cancel the common factor.

$60 \frac{\sin (B)}{60} = 60 \frac{\sin (4915)}{132.84124487}$

Step 19.2.1.1.2

Rewrite the expression.

$\sin (B) = 60 \frac{\sin (4915)}{132.84124487}$

Step 19.2.2

Simplify the right side.

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Step 19.2.2.1

Simplify

$60 \frac{\sin (4915)}{132.84124487}$ .

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Step 19.2.2.1.1

Simplify the numerator.

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Step 19.2.2.1.1.1

Remove full rotations of

$360$ ° until the angle is between

$0$ ° and

$360$ °.

$\sin (B) = 60 \frac{\sin (235)}{132.84124487}$

Step 19.2.2.1.1.2

Evaluate

$\sin (235)$ .

$\sin (B) = 60 (\frac{- 0.81915204}{132.84124487})$

Step 19.2.2.1.2

Simplify the expression.

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Step 19.2.2.1.2.1

Divide

$- 0.81915204$ by

$132.84124487$ .

$\sin (B) = 60 \cdot - 0.00616639$

Step 19.2.2.1.2.2

Multiply

$60$ by

$- 0.00616639$ .

$\sin (B) = - 0.3699839$

Step 19.3

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

$B$ from inside the sine.

$B = \arcsin (- 0.3699839)$

Step 19.4

Simplify the right side.

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Step 19.4.1

Evaluate

$\arcsin (- 0.3699839)$ .

$B = - 21.71462472$

Step 19.5

The sine function is negative in the third and fourth quadrants. To find the second solution, subtract the solution from

$360$ , to find a reference angle. Next, add this reference angle to

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

$B = 360 + 21.71462472 + 180$

Step 19.6

Simplify the expression to find the second solution.

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Step 19.6.1

Subtract

$360 °$ from

$360 + 21.71462472 + 180 °$ .

$B = 360 + 21.71462472 + 180 ° - 360 °$

Step 19.6.2

The resulting angle of

$201.71462472 °$ is positive, less than

$360 °$ , and coterminal with

$360 + 21.71462472 + 180$ .

$B = 201.71462472 °$

Step 19.7

The solution to the equation

$B = - 21.71462472$ .

$B = - 21.71462472, 201.71462472$

Step 19.8

The triangle is invalid.

Invalid triangle

Step 20

There are not enough parameters given to solve the triangle.

Unknown triangle