Trigonometry Examples

$b = 4$ , $c = 1$ , $A = 100$

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.

$a = \sqrt{{(4)}^{2} + {(1)}^{2} - 2 \cdot 4 \cdot 1 \cos (100)}$

Step 4

Simplify the results.

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Step 4.1

Raise $4$ to the power of $2$ .

$a = \sqrt{16 + {(1)}^{2} - 2 \cdot 4 \cdot (1 \cos (100))}$

Step 4.2

One to any power is one.

$a = \sqrt{16 + 1 - 2 \cdot 4 \cdot (1 \cos (100))}$

Step 4.3

Multiply $- 2 \cdot 4 \cdot 1$ .

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Step 4.3.1

Multiply $- 2$ by $4$ .

$a = \sqrt{16 + 1 - 8 \cdot (1 \cos (100))}$

Step 4.3.2

Multiply $- 8$ by $1$ .

$a = \sqrt{16 + 1 - 8 \cos (100)}$

Step 4.4

Evaluate $\cos (100)$ .

$a = \sqrt{16 + 1 - 8 \cdot - 0.17364817}$

Step 4.5

Multiply $- 8$ by $- 0.17364817$ .

$a = \sqrt{16 + 1 + 1.38918542}$

Step 4.6

Add $16$ and $1$ .

$a = \sqrt{17 + 1.38918542}$

Step 4.7

Add $17$ and $1.38918542$ .

$a = \sqrt{18.38918542}$

Step 4.8

Evaluate the root.

$a = 4.28826135$

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 $B$ .

$\frac{\sin (B)}{4} = \frac{\sin (100)}{4.28826135}$

Step 7

Solve the equation for $B$ .

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Step 7.1

Multiply both sides of the equation by $4$ .

$4 \frac{\sin (B)}{4} = 4 \frac{\sin (100)}{4.28826135}$

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 $4$ .

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Step 7.2.1.1.1

Cancel the common factor.

$4 \frac{\sin (B)}{4} = 4 \frac{\sin (100)}{4.28826135}$

Step 7.2.1.1.2

Rewrite the expression.

$\sin (B) = 4 \frac{\sin (100)}{4.28826135}$

Step 7.2.2

Simplify the right side.

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Step 7.2.2.1

Simplify $4 \frac{\sin (100)}{4.28826135}$ .

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Step 7.2.2.1.1

Evaluate $\sin (100)$ .

$\sin (B) = 4 (\frac{0.98480775}{4.28826135})$

Step 7.2.2.1.2

Divide $0.98480775$ by $4.28826135$ .

$\sin (B) = 4 \cdot 0.22965198$

Step 7.2.2.1.3

Multiply $4$ by $0.22965198$ .

$\sin (B) = 0.91860795$

Step 7.3

Take the inverse sine of both sides of the equation to extract $B$ from inside the sine.

$B = \arcsin (0.91860795)$

Step 7.4

Simplify the right side.

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Step 7.4.1

Evaluate $\arcsin (0.91860795)$ .

$B = 66.72341618$

Step 7.5

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.

$B = 180 - 66.72341618$

Step 7.6

Subtract $66.72341618$ from $180$ .

$B = 113.27658381$

Step 7.7

The solution to the equation $B = 66.72341618$ .

$B = 66.72341618, 113.27658381$

Step 7.8

Exclude the invalid angle.

$B = 66.72341618$

Step 8

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

$100 + C + 66.72341618 = 180$

Step 9

Solve the equation for $C$ .

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Step 9.1

Add $100$ and $66.72341618$ .

$C + 166.72341618 = 180$

Step 9.2

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

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Step 9.2.1

Subtract $166.72341618$ from both sides of the equation.

$C = 180 - 166.72341618$

Step 9.2.2

Subtract $166.72341618$ from $180$ .

$C = 13.27658381$

Step 10

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

$A = 100$

$B = 66.72341618$

$C = 13.27658381$

$a = 4.28826135$

$b = 4$

$c = 1$