Trigonometry Examples

$A = 45$ , $c = 5$ , $a = 4$

Step 1

The Law of Sines produces an ambiguous angle result. This means that there are $2$ angles that will correctly solve the equation. For the first triangle, use the first possible angle value.

Solve for the first triangle.

Step 2

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 3

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

$\frac{\sin (C)}{5} = \frac{\sin (45)}{4}$

Step 4

Solve the equation for $C$ .

Tap for more steps...

Step 4.1

Multiply both sides of the equation by $5$ .

$5 \frac{\sin (C)}{5} = 5 \frac{\sin (45)}{4}$

Step 4.2

Simplify both sides of the equation.

Tap for more steps...

Step 4.2.1

Simplify the left side.

Tap for more steps...

Step 4.2.1.1

Cancel the common factor of $5$ .

Tap for more steps...

Step 4.2.1.1.1

Cancel the common factor.

$5 \frac{\sin (C)}{5} = 5 \frac{\sin (45)}{4}$

Step 4.2.1.1.2

Rewrite the expression.

$\sin (C) = 5 \frac{\sin (45)}{4}$

Step 4.2.2

Simplify the right side.

Tap for more steps...

Step 4.2.2.1

Simplify $5 \frac{\sin (45)}{4}$ .

Tap for more steps...

Step 4.2.2.1.1

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

$\sin (C) = 5 \frac{\frac{\sqrt{2}}{2}}{4}$

Step 4.2.2.1.2

Multiply the numerator by the reciprocal of the denominator.

$\sin (C) = 5 (\frac{\sqrt{2}}{2} \cdot \frac{1}{4})$

Step 4.2.2.1.3

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

Tap for more steps...

Step 4.2.2.1.3.1

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

$\sin (C) = 5 \frac{\sqrt{2}}{2 \cdot 4}$

Step 4.2.2.1.3.2

Multiply $2$ by $4$ .

$\sin (C) = 5 \frac{\sqrt{2}}{8}$

Step 4.2.2.1.4

Combine $5$ and $\frac{\sqrt{2}}{8}$ .

$\sin (C) = \frac{5 \sqrt{2}}{8}$

Step 4.3

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

$C = \arcsin (\frac{5 \sqrt{2}}{8})$

Step 4.4

Simplify the right side.

Tap for more steps...

Step 4.4.1

Evaluate $\arcsin (\frac{5 \sqrt{2}}{8})$ .

$C = 62.11443316$

Step 4.5

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.

$C = 180 - 62.11443316$

Step 4.6

Subtract $62.11443316$ from $180$ .

$C = 117.88556683$

Step 4.7

The solution to the equation $C = 62.11443316$ .

$C = 62.11443316, 117.88556683$

Step 5

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

$45 + 62.11443316 + B = 180$

Step 6

Solve the equation for $B$ .

Tap for more steps...

Step 6.1

Add $45$ and $62.11443316$ .

$107.11443316 + B = 180$

Step 6.2

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

Tap for more steps...

Step 6.2.1

Subtract $107.11443316$ from both sides of the equation.

$B = 180 - 107.11443316$

Step 6.2.2

Subtract $107.11443316$ from $180$ .

$B = 72.88556683$

Step 7

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

Step 8

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

$\frac{\sin (72.88556683)}{b} = \frac{\sin (45)}{4}$

Step 9

Solve the equation for $b$ .

Tap for more steps...

Step 9.1

Factor each term.

Tap for more steps...

Step 9.1.1

Evaluate $\sin (72.88556683)$ .

$\frac{0.95571891}{b} = \frac{\sin (45)}{4}$

Step 9.1.2

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

$\frac{0.95571891}{b} = \frac{\frac{\sqrt{2}}{2}}{4}$

Step 9.1.3

Multiply the numerator by the reciprocal of the denominator.

$\frac{0.95571891}{b} = \frac{\sqrt{2}}{2} \cdot \frac{1}{4}$

Step 9.1.4

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

Tap for more steps...

Step 9.1.4.1

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

$\frac{0.95571891}{b} = \frac{\sqrt{2}}{2 \cdot 4}$

Step 9.1.4.2

Multiply $2$ by $4$ .

$\frac{0.95571891}{b} = \frac{\sqrt{2}}{8}$

Step 9.2

Find the LCD of the terms in the equation.

Tap for more steps...

Step 9.2.1

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

$b, 8$

Step 9.2.2

Since $b, 8$ contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $1, 8$ then find LCM for the variable part $b^{1}$ .

Step 9.2.3

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.

Step 9.2.4

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

Not prime

Step 9.2.5

The prime factors for $8$ are $2 \cdot 2 \cdot 2$ .

Tap for more steps...

Step 9.2.5.1

$8$ has factors of $2$ and $4$ .

$2 \cdot 4$

Step 9.2.5.2

$4$ has factors of $2$ and $2$ .

$2 \cdot 2 \cdot 2$

Step 9.2.6

Multiply $2 \cdot 2 \cdot 2$ .

Tap for more steps...

Step 9.2.6.1

Multiply $2$ by $2$ .

$4 \cdot 2$

Step 9.2.6.2

Multiply $4$ by $2$ .

$8$

Step 9.2.7

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

$b^{1} = b$

$b$ occurs $1$ time.

Step 9.2.8

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

$b$

Step 9.2.9

The LCM for $b, 8$ is the numeric part $8$ multiplied by the variable part.

$8 b$

Step 9.3

Multiply each term in $\frac{0.95571891}{b} = \frac{\sqrt{2}}{8}$ by $8 b$ to eliminate the fractions.

Tap for more steps...

Step 9.3.1

Multiply each term in $\frac{0.95571891}{b} = \frac{\sqrt{2}}{8}$ by $8 b$ .

$\frac{0.95571891}{b} (8 b) = \frac{\sqrt{2}}{8} (8 b)$

Step 9.3.2

Simplify the left side.

Tap for more steps...

Step 9.3.2.1

Rewrite using the commutative property of multiplication.

$8 \frac{0.95571891}{b} b = \frac{\sqrt{2}}{8} (8 b)$

Step 9.3.2.2

Multiply $8 \frac{0.95571891}{b}$ .

Tap for more steps...

Step 9.3.2.2.1

Combine $8$ and $\frac{0.95571891}{b}$ .

$\frac{8 \cdot 0.95571891}{b} b = \frac{\sqrt{2}}{8} (8 b)$

Step 9.3.2.2.2

Multiply $8$ by $0.95571891$ .

$\frac{7.64575131}{b} b = \frac{\sqrt{2}}{8} (8 b)$

Step 9.3.2.3

Cancel the common factor of $b$ .

Tap for more steps...

Step 9.3.2.3.1

Cancel the common factor.

$\frac{7.64575131}{b} b = \frac{\sqrt{2}}{8} (8 b)$

Step 9.3.2.3.2

Rewrite the expression.

$7.64575131 = \frac{\sqrt{2}}{8} (8 b)$

Step 9.3.3

Simplify the right side.

Tap for more steps...

Step 9.3.3.1

Cancel the common factor of $8$ .

Tap for more steps...

Step 9.3.3.1.1

Factor $8$ out of $8 b$ .

$7.64575131 = \frac{\sqrt{2}}{8} (8 (b))$

Step 9.3.3.1.2

Cancel the common factor.

$7.64575131 = \frac{\sqrt{2}}{8} (8 b)$

Step 9.3.3.1.3

Rewrite the expression.

$7.64575131 = \sqrt{2} b$

Step 9.4

Solve the equation.

Tap for more steps...

Step 9.4.1

Rewrite the equation as $\sqrt{2} b = 7.64575131$ .

$\sqrt{2} b = 7.64575131$

Step 9.4.2

Divide each term in $\sqrt{2} b = 7.64575131$ by $\sqrt{2}$ and simplify.

Tap for more steps...

Step 9.4.2.1

Divide each term in $\sqrt{2} b = 7.64575131$ by $\sqrt{2}$ .

$\frac{\sqrt{2} b}{\sqrt{2}} = \frac{7.64575131}{\sqrt{2}}$

Step 9.4.2.2

Simplify the left side.

Tap for more steps...

Step 9.4.2.2.1

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

Tap for more steps...

Step 9.4.2.2.1.1

Cancel the common factor.

$\frac{\sqrt{2} b}{\sqrt{2}} = \frac{7.64575131}{\sqrt{2}}$

Step 9.4.2.2.1.2

Divide $b$ by $1$ .

$b = \frac{7.64575131}{\sqrt{2}}$

Step 9.4.2.3

Simplify the right side.

Tap for more steps...

Step 9.4.2.3.1

Multiply $\frac{7.64575131}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$b = \frac{7.64575131}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 9.4.2.3.2

Combine and simplify the denominator.

Tap for more steps...

Step 9.4.2.3.2.1

Multiply $\frac{7.64575131}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$b = \frac{7.64575131 \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 9.4.2.3.2.2

Raise $\sqrt{2}$ to the power of $1$ .

$b = \frac{7.64575131 \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 9.4.2.3.2.3

Raise $\sqrt{2}$ to the power of $1$ .

$b = \frac{7.64575131 \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 9.4.2.3.2.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$b = \frac{7.64575131 \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 9.4.2.3.2.5

Add $1$ and $1$ .

$b = \frac{7.64575131 \sqrt{2}}{{\sqrt{2}}^{2}}$

Step 9.4.2.3.2.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

Tap for more steps...

Step 9.4.2.3.2.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$b = \frac{7.64575131 \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 9.4.2.3.2.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$b = \frac{7.64575131 \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 9.4.2.3.2.6.3

Combine $\frac{1}{2}$ and $2$ .

$b = \frac{7.64575131 \sqrt{2}}{2^{\frac{2}{2}}}$

Step 9.4.2.3.2.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 9.4.2.3.2.6.4.1

Cancel the common factor.

$b = \frac{7.64575131 \sqrt{2}}{2^{\frac{2}{2}}}$

Step 9.4.2.3.2.6.4.2

Rewrite the expression.

$b = \frac{7.64575131 \sqrt{2}}{2^{1}}$

Step 9.4.2.3.2.6.5

Evaluate the exponent.

$b = \frac{7.64575131 \sqrt{2}}{2}$

Step 9.4.2.3.3

Multiply $7.64575131$ by $\sqrt{2}$ .

$b = \frac{10.81272519}{2}$

Step 9.4.2.3.4

Divide $10.81272519$ by $2$ .

$b = 5.40636259$

Step 10

For the second triangle, use the second possible angle value.

Solve for the second 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 $C$ .

$\frac{\sin (C)}{5} = \frac{\sin (45)}{4}$

Step 13

Solve the equation for $C$ .

Tap for more steps...

Step 13.1

Multiply both sides of the equation by $5$ .

$5 \frac{\sin (C)}{5} = 5 \frac{\sin (45)}{4}$

Step 13.2

Simplify both sides of the equation.

Tap for more steps...

Step 13.2.1

Simplify the left side.

Tap for more steps...

Step 13.2.1.1

Cancel the common factor of $5$ .

Tap for more steps...

Step 13.2.1.1.1

Cancel the common factor.

$5 \frac{\sin (C)}{5} = 5 \frac{\sin (45)}{4}$

Step 13.2.1.1.2

Rewrite the expression.

$\sin (C) = 5 \frac{\sin (45)}{4}$

Step 13.2.2

Simplify the right side.

Tap for more steps...

Step 13.2.2.1

Simplify $5 \frac{\sin (45)}{4}$ .

Tap for more steps...

Step 13.2.2.1.1

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

$\sin (C) = 5 \frac{\frac{\sqrt{2}}{2}}{4}$

Step 13.2.2.1.2

Multiply the numerator by the reciprocal of the denominator.

$\sin (C) = 5 (\frac{\sqrt{2}}{2} \cdot \frac{1}{4})$

Step 13.2.2.1.3

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

Tap for more steps...

Step 13.2.2.1.3.1

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

$\sin (C) = 5 \frac{\sqrt{2}}{2 \cdot 4}$

Step 13.2.2.1.3.2

Multiply $2$ by $4$ .

$\sin (C) = 5 \frac{\sqrt{2}}{8}$

Step 13.2.2.1.4

Combine $5$ and $\frac{\sqrt{2}}{8}$ .

$\sin (C) = \frac{5 \sqrt{2}}{8}$

Step 13.3

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

$C = \arcsin (\frac{5 \sqrt{2}}{8})$

Step 13.4

Simplify the right side.

Tap for more steps...

Step 13.4.1

Evaluate $\arcsin (\frac{5 \sqrt{2}}{8})$ .

$C = 62.11443316$

Step 13.5

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.

$C = 180 - 62.11443316$

Step 13.6

Subtract $62.11443316$ from $180$ .

$C = 117.88556683$

Step 13.7

The solution to the equation $C = 62.11443316$ .

$C = 62.11443316, 117.88556683$

Step 14

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

$45 + 117.88556683 + B = 180$

Step 15

Solve the equation for $B$ .

Tap for more steps...

Step 15.1

Add $45$ and $117.88556683$ .

$162.88556683 + B = 180$

Step 15.2

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

Tap for more steps...

Step 15.2.1

Subtract $162.88556683$ from both sides of the equation.

$B = 180 - 162.88556683$

Step 15.2.2

Subtract $162.88556683$ from $180$ .

$B = 17.11443316$

Step 16

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

Step 17

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

$\frac{\sin (17.11443316)}{b} = \frac{\sin (45)}{4}$

Step 18

Solve the equation for $b$ .

Tap for more steps...

Step 18.1

Factor each term.

Tap for more steps...

Step 18.1.1

Evaluate $\sin (17.11443316)$ .

$\frac{0.29428108}{b} = \frac{\sin (45)}{4}$

Step 18.1.2

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

$\frac{0.29428108}{b} = \frac{\frac{\sqrt{2}}{2}}{4}$

Step 18.1.3

Multiply the numerator by the reciprocal of the denominator.

$\frac{0.29428108}{b} = \frac{\sqrt{2}}{2} \cdot \frac{1}{4}$

Step 18.1.4

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

Tap for more steps...

Step 18.1.4.1

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

$\frac{0.29428108}{b} = \frac{\sqrt{2}}{2 \cdot 4}$

Step 18.1.4.2

Multiply $2$ by $4$ .

$\frac{0.29428108}{b} = \frac{\sqrt{2}}{8}$

Step 18.2

Find the LCD of the terms in the equation.

Tap for more steps...

Step 18.2.1

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

$b, 8$

Step 18.2.2

Since $b, 8$ contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $1, 8$ then find LCM for the variable part $b^{1}$ .

Step 18.2.3

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.

Step 18.2.4

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

Not prime

Step 18.2.5

The prime factors for $8$ are $2 \cdot 2 \cdot 2$ .

Tap for more steps...

Step 18.2.5.1

$8$ has factors of $2$ and $4$ .

$2 \cdot 4$

Step 18.2.5.2

$4$ has factors of $2$ and $2$ .

$2 \cdot 2 \cdot 2$

Step 18.2.6

Multiply $2 \cdot 2 \cdot 2$ .

Tap for more steps...

Step 18.2.6.1

Multiply $2$ by $2$ .

$4 \cdot 2$

Step 18.2.6.2

Multiply $4$ by $2$ .

$8$

Step 18.2.7

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

$b^{1} = b$

$b$ occurs $1$ time.

Step 18.2.8

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

$b$

Step 18.2.9

The LCM for $b, 8$ is the numeric part $8$ multiplied by the variable part.

$8 b$

Step 18.3

Multiply each term in $\frac{0.29428108}{b} = \frac{\sqrt{2}}{8}$ by $8 b$ to eliminate the fractions.

Tap for more steps...

Step 18.3.1

Multiply each term in $\frac{0.29428108}{b} = \frac{\sqrt{2}}{8}$ by $8 b$ .

$\frac{0.29428108}{b} (8 b) = \frac{\sqrt{2}}{8} (8 b)$

Step 18.3.2

Simplify the left side.

Tap for more steps...

Step 18.3.2.1

Rewrite using the commutative property of multiplication.

$8 \frac{0.29428108}{b} b = \frac{\sqrt{2}}{8} (8 b)$

Step 18.3.2.2

Multiply $8 \frac{0.29428108}{b}$ .

Tap for more steps...

Step 18.3.2.2.1

Combine $8$ and $\frac{0.29428108}{b}$ .

$\frac{8 \cdot 0.29428108}{b} b = \frac{\sqrt{2}}{8} (8 b)$

Step 18.3.2.2.2

Multiply $8$ by $0.29428108$ .

$\frac{2.35424868}{b} b = \frac{\sqrt{2}}{8} (8 b)$

Step 18.3.2.3

Cancel the common factor of $b$ .

Tap for more steps...

Step 18.3.2.3.1

Cancel the common factor.

$\frac{2.35424868}{b} b = \frac{\sqrt{2}}{8} (8 b)$

Step 18.3.2.3.2

Rewrite the expression.

$2.35424868 = \frac{\sqrt{2}}{8} (8 b)$

Step 18.3.3

Simplify the right side.

Tap for more steps...

Step 18.3.3.1

Cancel the common factor of $8$ .

Tap for more steps...

Step 18.3.3.1.1

Factor $8$ out of $8 b$ .

$2.35424868 = \frac{\sqrt{2}}{8} (8 (b))$

Step 18.3.3.1.2

Cancel the common factor.

$2.35424868 = \frac{\sqrt{2}}{8} (8 b)$

Step 18.3.3.1.3

Rewrite the expression.

$2.35424868 = \sqrt{2} b$

Step 18.4

Solve the equation.

Tap for more steps...

Step 18.4.1

Rewrite the equation as $\sqrt{2} b = 2.35424868$ .

$\sqrt{2} b = 2.35424868$

Step 18.4.2

Divide each term in $\sqrt{2} b = 2.35424868$ by $\sqrt{2}$ and simplify.

Tap for more steps...

Step 18.4.2.1

Divide each term in $\sqrt{2} b = 2.35424868$ by $\sqrt{2}$ .

$\frac{\sqrt{2} b}{\sqrt{2}} = \frac{2.35424868}{\sqrt{2}}$

Step 18.4.2.2

Simplify the left side.

Tap for more steps...

Step 18.4.2.2.1

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

Tap for more steps...

Step 18.4.2.2.1.1

Cancel the common factor.

$\frac{\sqrt{2} b}{\sqrt{2}} = \frac{2.35424868}{\sqrt{2}}$

Step 18.4.2.2.1.2

Divide $b$ by $1$ .

$b = \frac{2.35424868}{\sqrt{2}}$

Step 18.4.2.3

Simplify the right side.

Tap for more steps...

Step 18.4.2.3.1

Multiply $\frac{2.35424868}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$b = \frac{2.35424868}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 18.4.2.3.2

Combine and simplify the denominator.

Tap for more steps...

Step 18.4.2.3.2.1

Multiply $\frac{2.35424868}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$b = \frac{2.35424868 \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 18.4.2.3.2.2

Raise $\sqrt{2}$ to the power of $1$ .

$b = \frac{2.35424868 \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 18.4.2.3.2.3

Raise $\sqrt{2}$ to the power of $1$ .

$b = \frac{2.35424868 \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 18.4.2.3.2.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$b = \frac{2.35424868 \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 18.4.2.3.2.5

Add $1$ and $1$ .

$b = \frac{2.35424868 \sqrt{2}}{{\sqrt{2}}^{2}}$

Step 18.4.2.3.2.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

Tap for more steps...

Step 18.4.2.3.2.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$b = \frac{2.35424868 \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 18.4.2.3.2.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$b = \frac{2.35424868 \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 18.4.2.3.2.6.3

Combine $\frac{1}{2}$ and $2$ .

$b = \frac{2.35424868 \sqrt{2}}{2^{\frac{2}{2}}}$

Step 18.4.2.3.2.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 18.4.2.3.2.6.4.1

Cancel the common factor.

$b = \frac{2.35424868 \sqrt{2}}{2^{\frac{2}{2}}}$

Step 18.4.2.3.2.6.4.2

Rewrite the expression.

$b = \frac{2.35424868 \sqrt{2}}{2^{1}}$

Step 18.4.2.3.2.6.5

Evaluate the exponent.

$b = \frac{2.35424868 \sqrt{2}}{2}$

Step 18.4.2.3.3

Multiply $2.35424868$ by $\sqrt{2}$ .

$b = \frac{3.32941042}{2}$

Step 18.4.2.3.4

Divide $3.32941042$ by $2$ .

$b = 1.66470521$

Step 19

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

First Triangle Combination:

$A = 45$

$B = 72.88556683$

$C = 62.11443316$

$a = 4$

$b = 5.40636259$

$c = 5$

Second Triangle Combination:

$A = 45$

$B = 17.11443316$

$C = 117.88556683$

$a = 4$

$b = 1.66470521$

$c = 5$