Trigonometry Examples

$A = 4915$ , $b = 60$ , $c = 89$

Step 1

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

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

Step 2

Solve the equation.

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

Step 3

Substitute the known values into the equation.

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

Step 4

Simplify the results.

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Step 4.1

Raise $60$ to the power of $2$ .

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

Step 4.2

Raise $89$ to the power of $2$ .

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

Step 4.3

Multiply $- 2 \cdot 60 \cdot 89$ .

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Step 4.3.1

Multiply $- 2$ by $60$ .

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

Step 4.3.2

Multiply $- 120$ by $89$ .

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

Step 4.4

Remove full rotations of $360$ ° until the angle is between $0$ ° and $360$ °.

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

Step 4.5

Evaluate $\cos (235)$ .

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

Step 4.6

Multiply $- 10680$ by $- 0.57357643$ .

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

Step 4.7

Add $3600$ and $7921$ .

$a = \sqrt{11521 + 6125.79634022}$

Step 4.8

Add $11521$ and $6125.79634022$ .

$a = \sqrt{17646.79634022}$

Step 4.9

Evaluate the root.

$a = 132.84124487$

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.

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

Step 6

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

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

Step 7

Solve the equation for $B$ .

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Step 7.1

Multiply both sides of the equation by $60$ .

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

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 $60$ .

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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

$B = 360 + 21.71462472 + 180$

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

$B = 360 + 21.71462472 + 180$

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

$B = 360 + 21.71462472 + 180$

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

$B = 360 + 21.71462472 + 180$

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