Trigonometry Examples

$B = 105$ , $C = 41$ , $b = 12$

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.

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

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

$\frac{\sin (41)}{c} = \frac{\sin (105)}{12}$

Solve the equation for $c$ .

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Factor each term.

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Evaluate $\sin (41)$ .

$\frac{0.65605902}{c} = \frac{\sin (105)}{12}$

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

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

$\frac{0.65605902}{c} = \frac{\sin (75)}{12}$

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

$\frac{0.65605902}{c} = \frac{\sin (30 + 45)}{12}$

Apply the sum of angles identity.

$\frac{0.65605902}{c} = \frac{\sin (30) \cos (45) + \cos (30) \sin (45)}{12}$

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

$\frac{0.65605902}{c} = \frac{\frac{1}{2} \cos (45) + \cos (30) \sin (45)}{12}$

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

$\frac{0.65605902}{c} = \frac{\frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \cos (30) \sin (45)}{12}$

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

$\frac{0.65605902}{c} = \frac{\frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2} \sin (45)}{12}$

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

$\frac{0.65605902}{c} = \frac{\frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}}{12}$

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

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Simplify each term.

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

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

$\frac{0.65605902}{c} = \frac{\frac{\sqrt{2}}{2 \cdot 2} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}}{12}$

Multiply $2$ by $2$ .

$\frac{0.65605902}{c} = \frac{\frac{\sqrt{2}}{4} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}}{12}$

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

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

$\frac{0.65605902}{c} = \frac{\frac{\sqrt{2}}{4} + \frac{\sqrt{3} \sqrt{2}}{2 \cdot 2}}{12}$

Combine using the product rule for radicals.

$\frac{0.65605902}{c} = \frac{\frac{\sqrt{2}}{4} + \frac{\sqrt{3 \cdot 2}}{2 \cdot 2}}{12}$

Multiply $3$ by $2$ .

$\frac{0.65605902}{c} = \frac{\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{2 \cdot 2}}{12}$

Multiply $2$ by $2$ .

$\frac{0.65605902}{c} = \frac{\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}}{12}$

Combine the numerators over the common denominator.

$\frac{0.65605902}{c} = \frac{\frac{\sqrt{2} + \sqrt{6}}{4}}{12}$

Multiply the numerator by the reciprocal of the denominator.

$\frac{0.65605902}{c} = \frac{\sqrt{2} + \sqrt{6}}{4} \cdot \frac{1}{12}$

Multiply $\frac{\sqrt{2} + \sqrt{6}}{4} \cdot \frac{1}{12}$ .

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Multiply $\frac{\sqrt{2} + \sqrt{6}}{4}$ by $\frac{1}{12}$ .

$\frac{0.65605902}{c} = \frac{\sqrt{2} + \sqrt{6}}{4 \cdot 12}$

Multiply $4$ by $12$ .

$\frac{0.65605902}{c} = \frac{\sqrt{2} + \sqrt{6}}{48}$

Find the LCD of the terms in the equation.

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Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$c, 48$

Since $c, 48$ contain both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $1, 48$ then find LCM for the variable part $c^{1}$ .

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

The prime factors for $48$ are $2 \cdot 2 \cdot 2 \cdot 2 \cdot 3$ .

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$48$ has factors of $2$ and $24$ .

$2 \cdot 24$

$24$ has factors of $2$ and $12$ .

$2 \cdot 2 \cdot 12$

$12$ has factors of $2$ and $6$ .

$2 \cdot 2 \cdot 2 \cdot 6$

$6$ has factors of $2$ and $3$ .

$2 \cdot 2 \cdot 2 \cdot 2 \cdot 3$

The LCM of $1, 48$ is $2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 = 48$ .

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Multiply $2$ by $2$ .

$4 \cdot 2 \cdot 2 \cdot 3$

Multiply $4$ by $2$ .

$8 \cdot 2 \cdot 3$

Multiply $8$ by $2$ .

$16 \cdot 3$

Multiply $16$ by $3$ .

$48$

The factor for $c^{1}$ is $c$ itself.

$c^{1} = c$

$c$ occurs $1$ time.

The LCM of $c^{1}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

$c$

The LCM for $c, 48$ is the numeric part $48$ multiplied by the variable part.

$48 c$

Multiply each term in $\frac{0.65605902}{c} = \frac{\sqrt{2} + \sqrt{6}}{48}$ by $48 c$ to eliminate the fractions.

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Multiply each term in $\frac{0.65605902}{c} = \frac{\sqrt{2} + \sqrt{6}}{48}$ by $48 c$ .

$\frac{0.65605902}{c} (48 c) = \frac{\sqrt{2} + \sqrt{6}}{48} (48 c)$

Simplify the left side.

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Rewrite using the commutative property of multiplication.

$48 \frac{0.65605902}{c} c = \frac{\sqrt{2} + \sqrt{6}}{48} (48 c)$

Multiply $48 \frac{0.65605902}{c}$ .

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Combine $48$ and $\frac{0.65605902}{c}$ .

$\frac{48 \cdot 0.65605902}{c} c = \frac{\sqrt{2} + \sqrt{6}}{48} (48 c)$

Multiply $48$ by $0.65605902$ .

$\frac{31.49083339}{c} c = \frac{\sqrt{2} + \sqrt{6}}{48} (48 c)$

Cancel the common factor of $c$ .

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Cancel the common factor.

$\frac{31.49083339}{c} c = \frac{\sqrt{2} + \sqrt{6}}{48} (48 c)$

Rewrite the expression.

$31.49083339 = \frac{\sqrt{2} + \sqrt{6}}{48} (48 c)$

Simplify the right side.

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Cancel the common factor of $48$ .

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Factor $48$ out of $48 c$ .

$31.49083339 = \frac{\sqrt{2} + \sqrt{6}}{48} (48 (c))$

Cancel the common factor.

$31.49083339 = \frac{\sqrt{2} + \sqrt{6}}{48} (48 c)$

Rewrite the expression.

$31.49083339 = (\sqrt{2} + \sqrt{6}) c$

Apply the distributive property.

$31.49083339 = \sqrt{2} c + \sqrt{6} c$

Solve the equation.

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Rewrite the equation as $\sqrt{2} c + \sqrt{6} c = 31.49083339$ .

$\sqrt{2} c + \sqrt{6} c = 31.49083339$

Factor $c$ out of $\sqrt{2} c + \sqrt{6} c$ .

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Factor $c$ out of $\sqrt{2} c$ .

$c \sqrt{2} + \sqrt{6} c = 31.49083339$

Factor $c$ out of $\sqrt{6} c$ .

$c \sqrt{2} + c \sqrt{6} = 31.49083339$

Factor $c$ out of $c \sqrt{2} + c \sqrt{6}$ .

$c (\sqrt{2} + \sqrt{6}) = 31.49083339$

Divide each term in $c (\sqrt{2} + \sqrt{6}) = 31.49083339$ by $\sqrt{2} + \sqrt{6}$ and simplify.

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Divide each term in $c (\sqrt{2} + \sqrt{6}) = 31.49083339$ by $\sqrt{2} + \sqrt{6}$ .

$\frac{c (\sqrt{2} + \sqrt{6})}{\sqrt{2} + \sqrt{6}} = \frac{31.49083339}{\sqrt{2} + \sqrt{6}}$

Simplify the left side.

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Cancel the common factor of $\sqrt{2} + \sqrt{6}$ .

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Cancel the common factor.

$\frac{c (\sqrt{2} + \sqrt{6})}{\sqrt{2} + \sqrt{6}} = \frac{31.49083339}{\sqrt{2} + \sqrt{6}}$

Divide $c$ by $1$ .

$c = \frac{31.49083339}{\sqrt{2} + \sqrt{6}}$

Simplify the right side.

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Multiply $\frac{31.49083339}{\sqrt{2} + \sqrt{6}}$ by $\frac{\sqrt{2} - \sqrt{6}}{\sqrt{2} - \sqrt{6}}$ .

$c = \frac{31.49083339}{\sqrt{2} + \sqrt{6}} \cdot \frac{\sqrt{2} - \sqrt{6}}{\sqrt{2} - \sqrt{6}}$

Multiply $\frac{31.49083339}{\sqrt{2} + \sqrt{6}}$ by $\frac{\sqrt{2} - \sqrt{6}}{\sqrt{2} - \sqrt{6}}$ .

$c = \frac{31.49083339 (\sqrt{2} - \sqrt{6})}{(\sqrt{2} + \sqrt{6}) (\sqrt{2} - \sqrt{6})}$

Expand the denominator using the FOIL method.

$c = \frac{31.49083339 (\sqrt{2} - \sqrt{6})}{{\sqrt{2}}^{2} - \sqrt{12} + \sqrt{12} - {\sqrt{6}}^{2}}$

Simplify.

$c = \frac{31.49083339 (\sqrt{2} - \sqrt{6})}{- 4}$

Multiply $31.49083339$ by $\sqrt{2} - \sqrt{6}$ .

$c = \frac{- 32.60170971}{- 4}$

Divide $- 32.60170971$ by $- 4$ .

$c = 8.15042742$

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

$A + 41 + 105 = 180$

Solve the equation for $A$ .

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Add $41$ and $105$ .

$A + 146 = 180$

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

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Subtract $146$ from both sides of the equation.

$A = 180 - 146$

Subtract $146$ from $180$ .

$A = 34$

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

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

$\frac{\sin (34)}{a} = \frac{\sin (105)}{12}$

Solve the equation for $a$ .

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Factor each term.

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Evaluate $\sin (34)$ .

$\frac{0.5591929}{a} = \frac{\sin (105)}{12}$

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

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

$\frac{0.5591929}{a} = \frac{\sin (75)}{12}$

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

$\frac{0.5591929}{a} = \frac{\sin (30 + 45)}{12}$

Apply the sum of angles identity.

$\frac{0.5591929}{a} = \frac{\sin (30) \cos (45) + \cos (30) \sin (45)}{12}$

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

$\frac{0.5591929}{a} = \frac{\frac{1}{2} \cos (45) + \cos (30) \sin (45)}{12}$

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

$\frac{0.5591929}{a} = \frac{\frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \cos (30) \sin (45)}{12}$

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

$\frac{0.5591929}{a} = \frac{\frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2} \sin (45)}{12}$

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

$\frac{0.5591929}{a} = \frac{\frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}}{12}$

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

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Simplify each term.

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

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

$\frac{0.5591929}{a} = \frac{\frac{\sqrt{2}}{2 \cdot 2} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}}{12}$

Multiply $2$ by $2$ .

$\frac{0.5591929}{a} = \frac{\frac{\sqrt{2}}{4} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}}{12}$

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

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

$\frac{0.5591929}{a} = \frac{\frac{\sqrt{2}}{4} + \frac{\sqrt{3} \sqrt{2}}{2 \cdot 2}}{12}$

Combine using the product rule for radicals.

$\frac{0.5591929}{a} = \frac{\frac{\sqrt{2}}{4} + \frac{\sqrt{3 \cdot 2}}{2 \cdot 2}}{12}$

Multiply $3$ by $2$ .

$\frac{0.5591929}{a} = \frac{\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{2 \cdot 2}}{12}$

Multiply $2$ by $2$ .

$\frac{0.5591929}{a} = \frac{\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}}{12}$

Combine the numerators over the common denominator.

$\frac{0.5591929}{a} = \frac{\frac{\sqrt{2} + \sqrt{6}}{4}}{12}$

Multiply the numerator by the reciprocal of the denominator.

$\frac{0.5591929}{a} = \frac{\sqrt{2} + \sqrt{6}}{4} \cdot \frac{1}{12}$

Multiply $\frac{\sqrt{2} + \sqrt{6}}{4} \cdot \frac{1}{12}$ .

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Multiply $\frac{\sqrt{2} + \sqrt{6}}{4}$ by $\frac{1}{12}$ .

$\frac{0.5591929}{a} = \frac{\sqrt{2} + \sqrt{6}}{4 \cdot 12}$

Multiply $4$ by $12$ .

$\frac{0.5591929}{a} = \frac{\sqrt{2} + \sqrt{6}}{48}$

Find the LCD of the terms in the equation.

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Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$a, 48$

Since $a, 48$ contain both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $1, 48$ then find LCM for the variable part $a^{1}$ .

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

The prime factors for $48$ are $2 \cdot 2 \cdot 2 \cdot 2 \cdot 3$ .

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$48$ has factors of $2$ and $24$ .

$2 \cdot 24$

$24$ has factors of $2$ and $12$ .

$2 \cdot 2 \cdot 12$

$12$ has factors of $2$ and $6$ .

$2 \cdot 2 \cdot 2 \cdot 6$

$6$ has factors of $2$ and $3$ .

$2 \cdot 2 \cdot 2 \cdot 2 \cdot 3$

The LCM of $1, 48$ is $2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 = 48$ .

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Multiply $2$ by $2$ .

$4 \cdot 2 \cdot 2 \cdot 3$

Multiply $4$ by $2$ .

$8 \cdot 2 \cdot 3$

Multiply $8$ by $2$ .

$16 \cdot 3$

Multiply $16$ by $3$ .

$48$

The factor for $a^{1}$ is $a$ itself.

$a^{1} = a$

$a$ occurs $1$ time.

The LCM of $a^{1}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

$a$

The LCM for $a, 48$ is the numeric part $48$ multiplied by the variable part.

$48 a$

Multiply each term in $\frac{0.5591929}{a} = \frac{\sqrt{2} + \sqrt{6}}{48}$ by $48 a$ to eliminate the fractions.

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Multiply each term in $\frac{0.5591929}{a} = \frac{\sqrt{2} + \sqrt{6}}{48}$ by $48 a$ .

$\frac{0.5591929}{a} (48 a) = \frac{\sqrt{2} + \sqrt{6}}{48} (48 a)$

Simplify the left side.

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Rewrite using the commutative property of multiplication.

$48 \frac{0.5591929}{a} a = \frac{\sqrt{2} + \sqrt{6}}{48} (48 a)$

Multiply $48 \frac{0.5591929}{a}$ .

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Combine $48$ and $\frac{0.5591929}{a}$ .

$\frac{48 \cdot 0.5591929}{a} a = \frac{\sqrt{2} + \sqrt{6}}{48} (48 a)$

Multiply $48$ by $0.5591929$ .

$\frac{26.84125936}{a} a = \frac{\sqrt{2} + \sqrt{6}}{48} (48 a)$

Cancel the common factor of $a$ .

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Cancel the common factor.

$\frac{26.84125936}{a} a = \frac{\sqrt{2} + \sqrt{6}}{48} (48 a)$

Rewrite the expression.

$26.84125936 = \frac{\sqrt{2} + \sqrt{6}}{48} (48 a)$

Simplify the right side.

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Cancel the common factor of $48$ .

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Factor $48$ out of $48 a$ .

$26.84125936 = \frac{\sqrt{2} + \sqrt{6}}{48} (48 (a))$

Cancel the common factor.

$26.84125936 = \frac{\sqrt{2} + \sqrt{6}}{48} (48 a)$

Rewrite the expression.

$26.84125936 = (\sqrt{2} + \sqrt{6}) a$

Apply the distributive property.

$26.84125936 = \sqrt{2} a + \sqrt{6} a$

Solve the equation.

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Rewrite the equation as $\sqrt{2} a + \sqrt{6} a = 26.84125936$ .

$\sqrt{2} a + \sqrt{6} a = 26.84125936$

Factor $a$ out of $\sqrt{2} a + \sqrt{6} a$ .

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Factor $a$ out of $\sqrt{2} a$ .

$a \sqrt{2} + \sqrt{6} a = 26.84125936$

Factor $a$ out of $\sqrt{6} a$ .

$a \sqrt{2} + a \sqrt{6} = 26.84125936$

Factor $a$ out of $a \sqrt{2} + a \sqrt{6}$ .

$a (\sqrt{2} + \sqrt{6}) = 26.84125936$

Divide each term in $a (\sqrt{2} + \sqrt{6}) = 26.84125936$ by $\sqrt{2} + \sqrt{6}$ and simplify.

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Divide each term in $a (\sqrt{2} + \sqrt{6}) = 26.84125936$ by $\sqrt{2} + \sqrt{6}$ .

$\frac{a (\sqrt{2} + \sqrt{6})}{\sqrt{2} + \sqrt{6}} = \frac{26.84125936}{\sqrt{2} + \sqrt{6}}$

Simplify the left side.

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Cancel the common factor of $\sqrt{2} + \sqrt{6}$ .

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Cancel the common factor.

$\frac{a (\sqrt{2} + \sqrt{6})}{\sqrt{2} + \sqrt{6}} = \frac{26.84125936}{\sqrt{2} + \sqrt{6}}$

Divide $a$ by $1$ .

$a = \frac{26.84125936}{\sqrt{2} + \sqrt{6}}$

Simplify the right side.

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Multiply $\frac{26.84125936}{\sqrt{2} + \sqrt{6}}$ by $\frac{\sqrt{2} - \sqrt{6}}{\sqrt{2} - \sqrt{6}}$ .

$a = \frac{26.84125936}{\sqrt{2} + \sqrt{6}} \cdot \frac{\sqrt{2} - \sqrt{6}}{\sqrt{2} - \sqrt{6}}$

Multiply $\frac{26.84125936}{\sqrt{2} + \sqrt{6}}$ by $\frac{\sqrt{2} - \sqrt{6}}{\sqrt{2} - \sqrt{6}}$ .

$a = \frac{26.84125936 (\sqrt{2} - \sqrt{6})}{(\sqrt{2} + \sqrt{6}) (\sqrt{2} - \sqrt{6})}$

Expand the denominator using the FOIL method.

$a = \frac{26.84125936 (\sqrt{2} - \sqrt{6})}{{\sqrt{2}}^{2} - \sqrt{12} + \sqrt{12} - {\sqrt{6}}^{2}}$

Simplify.

$a = \frac{26.84125936 (\sqrt{2} - \sqrt{6})}{- 4}$

Multiply $26.84125936$ by $\sqrt{2} - \sqrt{6}$ .

$a = \frac{- 27.78811647}{- 4}$

Divide $- 27.78811647$ by $- 4$ .

$a = 6.94702911$

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

$A = 34$

$B = 105$

$C = 41$

$a = 6.94702911$

$b = 12$

$c = 8.15042742$