Trigonometry Examples

 $\begin{matrix} Side & Angle \\ \begin{matrix} b = & 15 \\ c = \\ a = & 9 \end{matrix} & \begin{matrix} A = \\ B = \\ C = & 106 \end{matrix} \end{matrix}$

Step 1

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

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

Step 2

Solve the equation.

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

Step 3

Substitute the known values into the equation.

$c = \sqrt{{(9)}^{2} + {(15)}^{2} - 2 \cdot 9 \cdot 15 \cos (106)}$

Step 4

Simplify the results.

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Step 4.1

Raise $9$ to the power of $2$ .

$c = \sqrt{81 + {(15)}^{2} - 2 \cdot 9 \cdot (15 \cos (106))}$

Step 4.2

Raise $15$ to the power of $2$ .

$c = \sqrt{81 + 225 - 2 \cdot 9 \cdot (15 \cos (106))}$

Step 4.3

Multiply $- 2 \cdot 9 \cdot 15$ .

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Step 4.3.1

Multiply $- 2$ by $9$ .

$c = \sqrt{81 + 225 - 18 \cdot (15 \cos (106))}$

Step 4.3.2

Multiply $- 18$ by $15$ .

$c = \sqrt{81 + 225 - 270 \cos (106)}$

Step 4.4

Evaluate $\cos (106)$ .

$c = \sqrt{81 + 225 - 270 \cdot - 0.27563735}$

Step 4.5

Multiply $- 270$ by $- 0.27563735$ .

$c = \sqrt{81 + 225 + 74.42208607}$

Step 4.6

Add $81$ and $225$ .

$c = \sqrt{306 + 74.42208607}$

Step 4.7

Add $306$ and $74.42208607$ .

$c = \sqrt{380.42208607}$

Step 4.8

Evaluate the root.

$c = 19.50441196$

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 $A$ .

$\frac{\sin (A)}{9} = \frac{\sin (106)}{19.50441196}$

Step 7

Solve the equation for $A$ .

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Step 7.1

Multiply both sides of the equation by $9$ .

$9 \frac{\sin (A)}{9} = 9 \frac{\sin (106)}{19.50441196}$

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 $9$ .

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Step 7.2.1.1.1

Cancel the common factor.

$9 \frac{\sin (A)}{9} = 9 \frac{\sin (106)}{19.50441196}$

Step 7.2.1.1.2

Rewrite the expression.

$\sin (A) = 9 \frac{\sin (106)}{19.50441196}$

Step 7.2.2

Simplify the right side.

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Step 7.2.2.1

Simplify $9 \frac{\sin (106)}{19.50441196}$ .

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Step 7.2.2.1.1

Evaluate $\sin (106)$ .

$\sin (A) = 9 (\frac{0.96126169}{19.50441196})$

Step 7.2.2.1.2

Divide $0.96126169$ by $19.50441196$ .

$\sin (A) = 9 \cdot 0.04928432$

Step 7.2.2.1.3

Multiply $9$ by $0.04928432$ .

$\sin (A) = 0.44355888$

Step 7.3

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

$A = \arcsin (0.44355888)$

Step 7.4

Simplify the right side.

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Step 7.4.1

Evaluate $\arcsin (0.44355888)$ .

$A = 26.33117358$

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.

$A = 180 - 26.33117358$

Step 7.6

Subtract $26.33117358$ from $180$ .

$A = 153.66882641$

Step 7.7

The solution to the equation $A = 26.33117358$ .

$A = 26.33117358, 153.66882641$

Step 7.8

Exclude the invalid angle.

$A = 26.33117358$

Step 8

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

$26.33117358 + 106 + B = 180$

Step 9

Solve the equation for $B$ .

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Step 9.1

Add $26.33117358$ and $106$ .

$132.33117358 + B = 180$

Step 9.2

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

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Step 9.2.1

Subtract $132.33117358$ from both sides of the equation.

$B = 180 - 132.33117358$

Step 9.2.2

Subtract $132.33117358$ from $180$ .

$B = 47.66882641$

Step 10

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

$A = 26.33117358$

$B = 47.66882641$

$C = 106$

$a = 9$

$b = 15$

$c = 19.50441196$