Trigonometry Examples

$c = 26$ , $b = 53$ , $B = 105$

Step 1

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 2

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

$\frac{\sin (C)}{26} = \frac{\sin (105)}{53}$

Step 3

Solve the equation for $C$ .

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Step 3.1

Multiply both sides of the equation by $26$ .

$26 \frac{\sin (C)}{26} = 26 \frac{\sin (105)}{53}$

Step 3.2

Simplify both sides of the equation.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Cancel the common factor of $26$ .

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Step 3.2.1.1.1

Cancel the common factor.

$26 \frac{\sin (C)}{26} = 26 \frac{\sin (105)}{53}$

Step 3.2.1.1.2

Rewrite the expression.

$\sin (C) = 26 \frac{\sin (105)}{53}$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $26 \frac{\sin (105)}{53}$ .

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Step 3.2.2.1.1

The exact value of $\sin (105)$ is $\frac{\sqrt{2} + \sqrt{6}}{4}$ .

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Step 3.2.2.1.1.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$\sin (C) = 26 \frac{\sin (75)}{53}$

Step 3.2.2.1.1.2

Split $75$ into two angles where the values of the six trigonometric functions are known.

$\sin (C) = 26 \frac{\sin (30 + 45)}{53}$

Step 3.2.2.1.1.3

Apply the sum of angles identity.

$\sin (C) = 26 \frac{\sin (30) \cos (45) + \cos (30) \sin (45)}{53}$

Step 3.2.2.1.1.4

The exact value of $\sin (30)$ is $\frac{1}{2}$ .

$\sin (C) = 26 \frac{\frac{1}{2} \cos (45) + \cos (30) \sin (45)}{53}$

Step 3.2.2.1.1.5

The exact value of $\cos (45)$ is $\frac{\sqrt{2}}{2}$ .

$\sin (C) = 26 \frac{\frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \cos (30) \sin (45)}{53}$

Step 3.2.2.1.1.6

The exact value of $\cos (30)$ is $\frac{\sqrt{3}}{2}$ .

$\sin (C) = 26 \frac{\frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2} \sin (45)}{53}$

Step 3.2.2.1.1.7

The exact value of $\sin (45)$ is $\frac{\sqrt{2}}{2}$ .

$\sin (C) = 26 \frac{\frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}}{53}$

Step 3.2.2.1.1.8

Simplify $\frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}$ .

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Step 3.2.2.1.1.8.1

Simplify each term.

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Step 3.2.2.1.1.8.1.1

Multiply $\frac{1}{2} \cdot \frac{\sqrt{2}}{2}$ .

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Step 3.2.2.1.1.8.1.1.1

Multiply $\frac{1}{2}$ by $\frac{\sqrt{2}}{2}$ .

$\sin (C) = 26 \frac{\frac{\sqrt{2}}{2 \cdot 2} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}}{53}$

Step 3.2.2.1.1.8.1.1.2

Multiply $2$ by $2$ .

$\sin (C) = 26 \frac{\frac{\sqrt{2}}{4} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}}{53}$

Step 3.2.2.1.1.8.1.2

Multiply $\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}$ .

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Step 3.2.2.1.1.8.1.2.1

Multiply $\frac{\sqrt{3}}{2}$ by $\frac{\sqrt{2}}{2}$ .

$\sin (C) = 26 \frac{\frac{\sqrt{2}}{4} + \frac{\sqrt{3} \sqrt{2}}{2 \cdot 2}}{53}$

Step 3.2.2.1.1.8.1.2.2

Combine using the product rule for radicals.

$\sin (C) = 26 \frac{\frac{\sqrt{2}}{4} + \frac{\sqrt{3 \cdot 2}}{2 \cdot 2}}{53}$

Step 3.2.2.1.1.8.1.2.3

Multiply $3$ by $2$ .

$\sin (C) = 26 \frac{\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{2 \cdot 2}}{53}$

Step 3.2.2.1.1.8.1.2.4

Multiply $2$ by $2$ .

$\sin (C) = 26 \frac{\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}}{53}$

Step 3.2.2.1.1.8.2

Combine the numerators over the common denominator.

$\sin (C) = 26 \frac{\frac{\sqrt{2} + \sqrt{6}}{4}}{53}$

Step 3.2.2.1.2

Multiply the numerator by the reciprocal of the denominator.

$\sin (C) = 26 (\frac{\sqrt{2} + \sqrt{6}}{4} \cdot \frac{1}{53})$

Step 3.2.2.1.3

Multiply $\frac{\sqrt{2} + \sqrt{6}}{4} \cdot \frac{1}{53}$ .

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Step 3.2.2.1.3.1

Multiply $\frac{\sqrt{2} + \sqrt{6}}{4}$ by $\frac{1}{53}$ .

$\sin (C) = 26 \frac{\sqrt{2} + \sqrt{6}}{4 \cdot 53}$

Step 3.2.2.1.3.2

Multiply $4$ by $53$ .

$\sin (C) = 26 \frac{\sqrt{2} + \sqrt{6}}{212}$

Step 3.2.2.1.4

Cancel the common factor of $2$ .

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Step 3.2.2.1.4.1

Factor $2$ out of $26$ .

$\sin (C) = 2 (13) \frac{\sqrt{2} + \sqrt{6}}{212}$

Step 3.2.2.1.4.2

Factor $2$ out of $212$ .

$\sin (C) = 2 \cdot 13 \frac{\sqrt{2} + \sqrt{6}}{2 \cdot 106}$

Step 3.2.2.1.4.3

Cancel the common factor.

$\sin (C) = 2 \cdot 13 \frac{\sqrt{2} + \sqrt{6}}{2 \cdot 106}$

Step 3.2.2.1.4.4

Rewrite the expression.

$\sin (C) = 13 \frac{\sqrt{2} + \sqrt{6}}{106}$

Step 3.2.2.1.5

Combine $13$ and $\frac{\sqrt{2} + \sqrt{6}}{106}$ .

$\sin (C) = \frac{13 (\sqrt{2} + \sqrt{6})}{106}$

Step 3.3

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

$C = \arcsin (\frac{13 (\sqrt{2} + \sqrt{6})}{106})$

Step 3.4

Simplify the right side.

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Step 3.4.1

Evaluate $\arcsin (\frac{13 (\sqrt{2} + \sqrt{6})}{106})$ .

$C = 28.28452639$

Step 3.5

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.

$C = 180 - 28.28452639$

Step 3.6

Subtract $28.28452639$ from $180$ .

$C = 151.7154736$

Step 3.7

The solution to the equation $C = 28.28452639$ .

$C = 28.28452639, 151.7154736$

Step 3.8

Exclude the invalid angle.

$C = 28.28452639$

Step 4

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

$A + 28.28452639 + 105 = 180$

Step 5

Solve the equation for $A$ .

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Step 5.1

Add $28.28452639$ and $105$ .

$A + 133.28452639 = 180$

Step 5.2

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

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Step 5.2.1

Subtract $133.28452639$ from both sides of the equation.

$A = 180 - 133.28452639$

Step 5.2.2

Subtract $133.28452639$ from $180$ .

$A = 46.7154736$

Step 6

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 7

Solve the equation.

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

Step 8

Substitute the known values into the equation.

$a = \sqrt{{(53)}^{2} + {(26)}^{2} - 2 \cdot 53 \cdot 26 \cos (46.7154736)}$

Step 9

Simplify the results.

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Step 9.1

Raise $53$ to the power of $2$ .

$a = \sqrt{2809 + {(26)}^{2} - 2 \cdot 53 \cdot (26 \cos (46.7154736))}$

Step 9.2

Raise $26$ to the power of $2$ .

$a = \sqrt{2809 + 676 - 2 \cdot 53 \cdot (26 \cos (46.7154736))}$

Step 9.3

Multiply $- 2 \cdot 53 \cdot 26$ .

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Step 9.3.1

Multiply $- 2$ by $53$ .

$a = \sqrt{2809 + 676 - 106 \cdot (26 \cos (46.7154736))}$

Step 9.3.2

Multiply $- 106$ by $26$ .

$a = \sqrt{2809 + 676 - 2756 \cos (46.7154736)}$

Step 9.4

Add $2809$ and $676$ .

$a = \sqrt{3485 - 2756 \cos (46.7154736)}$

Step 10

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

$A = 46.7154736$

$B = 105$

$C = 28.28452639$

$a = \sqrt{3485 - 2756 \cos (46.7154736)}$

$b = 53$

$c = 26$