Trigonometry Examples

 $\begin{matrix} Side & Angle \\ \begin{matrix} b = & 4 \\ c = \\ a = \end{matrix} & \begin{matrix} A = & 75 \\ B = & 20 \\ C = \end{matrix} \end{matrix}$

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 $a$ .

$\frac{\sin (75)}{a} = \frac{\sin (20)}{4}$

Step 3

Solve the equation for $a$ .

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Step 3.1

Factor each term.

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Step 3.1.1

The exact value of $\sin (75)$ is $\frac{\sqrt{2} + \sqrt{6}}{4}$ .

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Step 3.1.1.1

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

$\frac{\sin (30 + 45)}{a} = \frac{\sin (20)}{4}$

Step 3.1.1.2

Apply the sum of angles identity.

$\frac{\sin (30) \cos (45) + \cos (30) \sin (45)}{a} = \frac{\sin (20)}{4}$

Step 3.1.1.3

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

$\frac{\frac{1}{2} \cos (45) + \cos (30) \sin (45)}{a} = \frac{\sin (20)}{4}$

Step 3.1.1.4

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

$\frac{\frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \cos (30) \sin (45)}{a} = \frac{\sin (20)}{4}$

Step 3.1.1.5

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

$\frac{\frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2} \sin (45)}{a} = \frac{\sin (20)}{4}$

Step 3.1.1.6

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

$\frac{\frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}}{a} = \frac{\sin (20)}{4}$

Step 3.1.1.7

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

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Step 3.1.1.7.1

Simplify each term.

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Step 3.1.1.7.1.1

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

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Step 3.1.1.7.1.1.1

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

$\frac{\frac{\sqrt{2}}{2 \cdot 2} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}}{a} = \frac{\sin (20)}{4}$

Step 3.1.1.7.1.1.2

Multiply $2$ by $2$ .

$\frac{\frac{\sqrt{2}}{4} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}}{a} = \frac{\sin (20)}{4}$

Step 3.1.1.7.1.2

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

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Step 3.1.1.7.1.2.1

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

$\frac{\frac{\sqrt{2}}{4} + \frac{\sqrt{3} \sqrt{2}}{2 \cdot 2}}{a} = \frac{\sin (20)}{4}$

Step 3.1.1.7.1.2.2

Combine using the product rule for radicals.

$\frac{\frac{\sqrt{2}}{4} + \frac{\sqrt{3 \cdot 2}}{2 \cdot 2}}{a} = \frac{\sin (20)}{4}$

Step 3.1.1.7.1.2.3

Multiply $3$ by $2$ .

$\frac{\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{2 \cdot 2}}{a} = \frac{\sin (20)}{4}$

Step 3.1.1.7.1.2.4

Multiply $2$ by $2$ .

$\frac{\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}}{a} = \frac{\sin (20)}{4}$

Step 3.1.1.7.2

Combine the numerators over the common denominator.

$\frac{\frac{\sqrt{2} + \sqrt{6}}{4}}{a} = \frac{\sin (20)}{4}$

Step 3.1.2

Multiply the numerator by the reciprocal of the denominator.

$\frac{\sqrt{2} + \sqrt{6}}{4} \cdot \frac{1}{a} = \frac{\sin (20)}{4}$

Step 3.1.3

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

$\frac{\sqrt{2} + \sqrt{6}}{4 a} = \frac{\sin (20)}{4}$

Step 3.1.4

Evaluate $\sin (20)$ .

$\frac{\sqrt{2} + \sqrt{6}}{4 a} = \frac{0.34202014}{4}$

Step 3.1.5

Divide $0.34202014$ by $4$ .

$\frac{\sqrt{2} + \sqrt{6}}{4 a} = 0.08550503$

Step 3.2

Find the LCD of the terms in the equation.

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Step 3.2.1

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

$4 a, 1$

Step 3.2.2

The LCM of one and any expression is the expression.

$4 a$

Step 3.3

Multiply each term in $\frac{\sqrt{2} + \sqrt{6}}{4 a} = 0.08550503$ by $4 a$ to eliminate the fractions.

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Step 3.3.1

Multiply each term in $\frac{\sqrt{2} + \sqrt{6}}{4 a} = 0.08550503$ by $4 a$ .

$\frac{\sqrt{2} + \sqrt{6}}{4 a} (4 a) = 0.08550503 (4 a)$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Rewrite using the commutative property of multiplication.

$4 \frac{\sqrt{2} + \sqrt{6}}{4 a} a = 0.08550503 (4 a)$

Step 3.3.2.2

Cancel the common factor of $4$ .

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Step 3.3.2.2.1

Factor $4$ out of $4 a$ .

$4 \frac{\sqrt{2} + \sqrt{6}}{4 (a)} a = 0.08550503 (4 a)$

Step 3.3.2.2.2

Cancel the common factor.

$4 \frac{\sqrt{2} + \sqrt{6}}{4 a} a = 0.08550503 (4 a)$

Step 3.3.2.2.3

Rewrite the expression.

$\frac{\sqrt{2} + \sqrt{6}}{a} a = 0.08550503 (4 a)$

Step 3.3.2.3

Cancel the common factor of $a$ .

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Step 3.3.2.3.1

Cancel the common factor.

$\frac{\sqrt{2} + \sqrt{6}}{a} a = 0.08550503 (4 a)$

Step 3.3.2.3.2

Rewrite the expression.

$\sqrt{2} + \sqrt{6} = 0.08550503 (4 a)$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Multiply $4$ by $0.08550503$ .

$\sqrt{2} + \sqrt{6} = 0.34202014 a$

Step 3.4

Solve the equation.

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Step 3.4.1

Rewrite the equation as $0.34202014 a = \sqrt{2} + \sqrt{6}$ .

$0.34202014 a = \sqrt{2} + \sqrt{6}$

Step 3.4.2

Divide each term in $0.34202014 a = \sqrt{2} + \sqrt{6}$ by $0.34202014$ and simplify.

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Step 3.4.2.1

Divide each term in $0.34202014 a = \sqrt{2} + \sqrt{6}$ by $0.34202014$ .

$\frac{0.34202014 a}{0.34202014} = \frac{\sqrt{2}}{0.34202014} + \frac{\sqrt{6}}{0.34202014}$

Step 3.4.2.2

Simplify the left side.

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Step 3.4.2.2.1

Cancel the common factor of $0.34202014$ .

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Step 3.4.2.2.1.1

Cancel the common factor.

$\frac{0.34202014 a}{0.34202014} = \frac{\sqrt{2}}{0.34202014} + \frac{\sqrt{6}}{0.34202014}$

Step 3.4.2.2.1.2

Divide $a$ by $1$ .

$a = \frac{\sqrt{2}}{0.34202014} + \frac{\sqrt{6}}{0.34202014}$

Step 3.4.2.3

Simplify the right side.

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Step 3.4.2.3.1

Simplify each term.

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Step 3.4.2.3.1.1

Evaluate the root.

$a = \frac{1.41421356}{0.34202014} + \frac{\sqrt{6}}{0.34202014}$

Step 3.4.2.3.1.2

Divide $1.41421356$ by $0.34202014$ .

$a = 4.13488383 + \frac{\sqrt{6}}{0.34202014}$

Step 3.4.2.3.1.3

Evaluate the root.

$a = 4.13488383 + \frac{2.44948974}{0.34202014}$

Step 3.4.2.3.1.4

Divide $2.44948974$ by $0.34202014$ .

$a = 4.13488383 + 7.16182888$

Step 3.4.2.3.2

Add $4.13488383$ and $7.16182888$ .

$a = 11.29671272$

Step 4

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

$75 + C + 20 = 180$

Step 5

Solve the equation for $C$ .

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Step 5.1

Add $75$ and $20$ .

$C + 95 = 180$

Step 5.2

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

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Step 5.2.1

Subtract $95$ from both sides of the equation.

$C = 180 - 95$

Step 5.2.2

Subtract $95$ from $180$ .

$C = 85$

Step 6

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

Step 7

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

$\frac{\sin (85)}{c} = \frac{\sin (75)}{11.29671272}$

Step 8

Solve the equation for $c$ .

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Step 8.1

Factor each term.

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Step 8.1.1

Evaluate $\sin (85)$ .

$\frac{0.99619469}{c} = \frac{\sin (75)}{11.29671272}$

Step 8.1.2

The exact value of $\sin (75)$ is $\frac{\sqrt{2} + \sqrt{6}}{4}$ .

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Step 8.1.2.1

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

$\frac{0.99619469}{c} = \frac{\sin (30 + 45)}{11.29671272}$

Step 8.1.2.2

Apply the sum of angles identity.

$\frac{0.99619469}{c} = \frac{\sin (30) \cos (45) + \cos (30) \sin (45)}{11.29671272}$

Step 8.1.2.3

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

$\frac{0.99619469}{c} = \frac{\frac{1}{2} \cos (45) + \cos (30) \sin (45)}{11.29671272}$

Step 8.1.2.4

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

$\frac{0.99619469}{c} = \frac{\frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \cos (30) \sin (45)}{11.29671272}$

Step 8.1.2.5

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

$\frac{0.99619469}{c} = \frac{\frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2} \sin (45)}{11.29671272}$

Step 8.1.2.6

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

$\frac{0.99619469}{c} = \frac{\frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}}{11.29671272}$

Step 8.1.2.7

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

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Step 8.1.2.7.1

Simplify each term.

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Step 8.1.2.7.1.1

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

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Step 8.1.2.7.1.1.1

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

$\frac{0.99619469}{c} = \frac{\frac{\sqrt{2}}{2 \cdot 2} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}}{11.29671272}$

Step 8.1.2.7.1.1.2

Multiply $2$ by $2$ .

$\frac{0.99619469}{c} = \frac{\frac{\sqrt{2}}{4} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}}{11.29671272}$

Step 8.1.2.7.1.2

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

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Step 8.1.2.7.1.2.1

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

$\frac{0.99619469}{c} = \frac{\frac{\sqrt{2}}{4} + \frac{\sqrt{3} \sqrt{2}}{2 \cdot 2}}{11.29671272}$

Step 8.1.2.7.1.2.2

Combine using the product rule for radicals.

$\frac{0.99619469}{c} = \frac{\frac{\sqrt{2}}{4} + \frac{\sqrt{3 \cdot 2}}{2 \cdot 2}}{11.29671272}$

Step 8.1.2.7.1.2.3

Multiply $3$ by $2$ .

$\frac{0.99619469}{c} = \frac{\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{2 \cdot 2}}{11.29671272}$

Step 8.1.2.7.1.2.4

Multiply $2$ by $2$ .

$\frac{0.99619469}{c} = \frac{\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}}{11.29671272}$

Step 8.1.2.7.2

Combine the numerators over the common denominator.

$\frac{0.99619469}{c} = \frac{\frac{\sqrt{2} + \sqrt{6}}{4}}{11.29671272}$

Step 8.1.3

Multiply the numerator by the reciprocal of the denominator.

$\frac{0.99619469}{c} = \frac{\sqrt{2} + \sqrt{6}}{4} \cdot \frac{1}{11.29671272}$

Step 8.1.4

Divide $1$ by $11.29671272$ .

$\frac{0.99619469}{c} = \frac{\sqrt{2} + \sqrt{6}}{4} \cdot 0.08852132$

Step 8.1.5

Multiply $\frac{\sqrt{2} + \sqrt{6}}{4} \cdot 0.08852132$ .

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Step 8.1.5.1

Combine $\frac{\sqrt{2} + \sqrt{6}}{4}$ and $0.08852132$ .

$\frac{0.99619469}{c} = \frac{(\sqrt{2} + \sqrt{6}) \cdot 0.08852132}{4}$

Step 8.1.5.2

Multiply $\sqrt{2} + \sqrt{6}$ by $0.08852132$ .

$\frac{0.99619469}{c} = \frac{0.34202014}{4}$

Step 8.1.6

Divide $0.34202014$ by $4$ .

$\frac{0.99619469}{c} = 0.08550503$

Step 8.2

Find the LCD of the terms in the equation.

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Step 8.2.1

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

$c, 1$

Step 8.2.2

The LCM of one and any expression is the expression.

$c$

Step 8.3

Multiply each term in $\frac{0.99619469}{c} = 0.08550503$ by $c$ to eliminate the fractions.

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Step 8.3.1

Multiply each term in $\frac{0.99619469}{c} = 0.08550503$ by $c$ .

$\frac{0.99619469}{c} c = 0.08550503 c$

Step 8.3.2

Simplify the left side.

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Step 8.3.2.1

Cancel the common factor of $c$ .

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Step 8.3.2.1.1

Cancel the common factor.

$\frac{0.99619469}{c} c = 0.08550503 c$

Step 8.3.2.1.2

Rewrite the expression.

$0.99619469 = 0.08550503 c$

Step 8.4

Solve the equation.

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Step 8.4.1

Rewrite the equation as $0.08550503 c = 0.99619469$ .

$0.08550503 c = 0.99619469$

Step 8.4.2

Divide each term in $0.08550503 c = 0.99619469$ by $0.08550503$ and simplify.

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Step 8.4.2.1

Divide each term in $0.08550503 c = 0.99619469$ by $0.08550503$ .

$\frac{0.08550503 c}{0.08550503} = \frac{0.99619469}{0.08550503}$

Step 8.4.2.2

Simplify the left side.

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Step 8.4.2.2.1

Cancel the common factor of $0.08550503$ .

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Step 8.4.2.2.1.1

Cancel the common factor.

$\frac{0.08550503 c}{0.08550503} = \frac{0.99619469}{0.08550503}$

Step 8.4.2.2.1.2

Divide $c$ by $1$ .

$c = \frac{0.99619469}{0.08550503}$

Step 8.4.2.3

Simplify the right side.

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Step 8.4.2.3.1

Divide $0.99619469$ by $0.08550503$ .

$c = 11.65071376$