Trigonometry Examples

$C = 127.5$ , $a = 6$ , $b = 10.74$

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.

$c = \sqrt{{(6)}^{2} + {(10.74)}^{2} - 2 \cdot 6 \cdot 10.74 \cos (127.5)}$

Step 4

Simplify the results.

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Step 4.1

Raise $6$ to the power of $2$ .

$c = \sqrt{36 + {(10.74)}^{2} - 2 \cdot 6 \cdot (10.74 \cos (127.5))}$

Step 4.2

Raise $10.74$ to the power of $2$ .

$c = \sqrt{36 + 115.3476 - 2 \cdot 6 \cdot (10.74 \cos (127.5))}$

Step 4.3

Multiply $- 2 \cdot 6 \cdot 10.74$ .

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Step 4.3.1

Multiply $- 2$ by $6$ .

$c = \sqrt{36 + 115.3476 - 12 \cdot (10.74 \cos (127.5))}$

Step 4.3.2

Multiply $- 12$ by $10.74$ .

$c = \sqrt{36 + 115.3476 - 128.88 \cos (127.5)}$

Step 4.4

The exact value of $\cos (127.5)$ is $- \frac{\sqrt{(4 - \sqrt{6} + \sqrt{2}) \cdot 2}}{4}$ .

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Step 4.4.1

Rewrite $127.5$ as an angle where the values of the six trigonometric functions are known divided by $2$ .

$c = \sqrt{36 + 115.3476 - 128.88 \cos (\frac{255}{2})}$

Step 4.4.2

Apply the cosine half-angle identity $\cos (\frac{x}{2}) = \pm \sqrt{\frac{1 + \cos (x)}{2}}$ .

$c = \sqrt{36 + 115.3476 - 128.88 (\pm \sqrt{\frac{1 + \cos (255)}{2}})}$

Step 4.4.3

Change the $\pm$ to $-$ because cosine is negative in the second quadrant.

$c = \sqrt{36 + 115.3476 - 128.88 (- \sqrt{\frac{1 + \cos (255)}{2}})}$

Step 4.4.4

Simplify $- \sqrt{\frac{1 + \cos (255)}{2}}$ .

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Step 4.4.4.1

The exact value of $\cos (255)$ is $- \frac{\sqrt{6} - \sqrt{2}}{4}$ .

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Step 4.4.4.1.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the third quadrant.

$c = \sqrt{36 + 115.3476 - 128.88 (- \sqrt{\frac{1 - \cos (75)}{2}})}$

Step 4.4.4.1.2

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

$c = \sqrt{36 + 115.3476 - 128.88 (- \sqrt{\frac{1 - \cos (30 + 45)}{2}})}$

Step 4.4.4.1.3

Apply the sum of angles identity $\cos (x + y) = \cos (x) \cos (y) - \sin (x) \sin (y)$ .

$c = \sqrt{36 + 115.3476 - 128.88 (- \sqrt{\frac{1 - (\cos (30) \cos (45) - \sin (30) \sin (45))}{2}})}$

Step 4.4.4.1.4

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

$c = \sqrt{36 + 115.3476 - 128.88 (- \sqrt{\frac{1 - (\frac{\sqrt{3}}{2} \cdot \cos (45) - \sin (30) \sin (45))}{2}})}$

Step 4.4.4.1.5

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

$c = \sqrt{36 + 115.3476 - 128.88 (- \sqrt{\frac{1 - (\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} - \sin (30) \sin (45))}{2}})}$

Step 4.4.4.1.6

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

$c = \sqrt{36 + 115.3476 - 128.88 (- \sqrt{\frac{1 - (\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} - \frac{1}{2} \cdot \sin (45))}{2}})}$

Step 4.4.4.1.7

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

$c = \sqrt{36 + 115.3476 - 128.88 (- \sqrt{\frac{1 - (\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2})}{2}})}$

Step 4.4.4.1.8

Simplify $- (\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2})$ .

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Step 4.4.4.1.8.1

Simplify each term.

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Step 4.4.4.1.8.1.1

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

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Step 4.4.4.1.8.1.1.1

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

$c = \sqrt{36 + 115.3476 - 128.88 (- \sqrt{\frac{1 - (\frac{\sqrt{3} \sqrt{2}}{2 \cdot 2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2})}{2}})}$

Step 4.4.4.1.8.1.1.2

Combine using the product rule for radicals.

$c = \sqrt{36 + 115.3476 - 128.88 (- \sqrt{\frac{1 - (\frac{\sqrt{3 \cdot 2}}{2 \cdot 2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2})}{2}})}$

Step 4.4.4.1.8.1.1.3

Multiply $3$ by $2$ .

$c = \sqrt{36 + 115.3476 - 128.88 (- \sqrt{\frac{1 - (\frac{\sqrt{6}}{2 \cdot 2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2})}{2}})}$

Step 4.4.4.1.8.1.1.4

Multiply $2$ by $2$ .

$c = \sqrt{36 + 115.3476 - 128.88 (- \sqrt{\frac{1 - (\frac{\sqrt{6}}{4} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2})}{2}})}$

Step 4.4.4.1.8.1.2

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

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Step 4.4.4.1.8.1.2.1

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

$c = \sqrt{36 + 115.3476 - 128.88 (- \sqrt{\frac{1 - (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2 \cdot 2})}{2}})}$

Step 4.4.4.1.8.1.2.2

Multiply $2$ by $2$ .

$c = \sqrt{36 + 115.3476 - 128.88 (- \sqrt{\frac{1 - (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4})}{2}})}$

Step 4.4.4.1.8.2

Combine the numerators over the common denominator.

$c = \sqrt{36 + 115.3476 - 128.88 (- \sqrt{\frac{1 - \frac{\sqrt{6} - \sqrt{2}}{4}}{2}})}$

Step 4.4.4.2

Write $1$ as a fraction with a common denominator.

$c = \sqrt{36 + 115.3476 - 128.88 (- \sqrt{\frac{\frac{4}{4} - \frac{\sqrt{6} - \sqrt{2}}{4}}{2}})}$

Step 4.4.4.3

Combine the numerators over the common denominator.

$c = \sqrt{36 + 115.3476 - 128.88 (- \sqrt{\frac{\frac{4 - (\sqrt{6} - \sqrt{2})}{4}}{2}})}$

Step 4.4.4.4

Rewrite $\frac{4 - (\sqrt{6} - \sqrt{2})}{4}$ in a factored form.

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Step 4.4.4.4.1

Apply the distributive property.

$c = \sqrt{36 + 115.3476 - 128.88 (- \sqrt{\frac{\frac{4 - \sqrt{6} + \sqrt{2}}{4}}{2}})}$

Step 4.4.4.4.2

Multiply $- - \sqrt{2}$ .

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Step 4.4.4.4.2.1

Multiply $- 1$ by $- 1$ .

$c = \sqrt{36 + 115.3476 - 128.88 (- \sqrt{\frac{\frac{4 - \sqrt{6} + 1 \sqrt{2}}{4}}{2}})}$

Step 4.4.4.4.2.2

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

$c = \sqrt{36 + 115.3476 - 128.88 (- \sqrt{\frac{\frac{4 - \sqrt{6} + \sqrt{2}}{4}}{2}})}$

Step 4.4.4.5

Multiply the numerator by the reciprocal of the denominator.

$c = \sqrt{36 + 115.3476 - 128.88 (- \sqrt{\frac{4 - \sqrt{6} + \sqrt{2}}{4} \cdot \frac{1}{2}})}$

Step 4.4.4.6

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

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Step 4.4.4.6.1

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

$c = \sqrt{36 + 115.3476 - 128.88 (- \sqrt{\frac{4 - \sqrt{6} + \sqrt{2}}{4 \cdot 2}})}$

Step 4.4.4.6.2

Multiply $4$ by $2$ .

$c = \sqrt{36 + 115.3476 - 128.88 (- \sqrt{\frac{4 - \sqrt{6} + \sqrt{2}}{8}})}$

Step 4.4.4.7

Rewrite $\sqrt{\frac{4 - \sqrt{6} + \sqrt{2}}{8}}$ as $\frac{\sqrt{4 - \sqrt{6} + \sqrt{2}}}{\sqrt{8}}$ .

$c = \sqrt{36 + 115.3476 - 128.88 (- \frac{\sqrt{4 - \sqrt{6} + \sqrt{2}}}{\sqrt{8}})}$

Step 4.4.4.8

Simplify the denominator.

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Step 4.4.4.8.1

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 4.4.4.8.1.1

Factor $4$ out of $8$ .

$c = \sqrt{36 + 115.3476 - 128.88 (- \frac{\sqrt{4 - \sqrt{6} + \sqrt{2}}}{\sqrt{4 (2)}})}$

Step 4.4.4.8.1.2

Rewrite $4$ as $2^{2}$ .

$c = \sqrt{36 + 115.3476 - 128.88 (- \frac{\sqrt{4 - \sqrt{6} + \sqrt{2}}}{\sqrt{2^{2} \cdot 2}})}$

Step 4.4.4.8.2

Pull terms out from under the radical.

$c = \sqrt{36 + 115.3476 - 128.88 (- \frac{\sqrt{4 - \sqrt{6} + \sqrt{2}}}{2 \sqrt{2}})}$

Step 4.4.4.9

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

$c = \sqrt{36 + 115.3476 - 128.88 (- (\frac{\sqrt{4 - \sqrt{6} + \sqrt{2}}}{2 \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}))}$

Step 4.4.4.10

Combine and simplify the denominator.

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Step 4.4.4.10.1

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

$c = \sqrt{36 + 115.3476 - 128.88 (- \frac{\sqrt{4 - \sqrt{6} + \sqrt{2}} \sqrt{2}}{2 \sqrt{2} \sqrt{2}})}$

Step 4.4.4.10.2

Move $\sqrt{2}$ .

$c = \sqrt{36 + 115.3476 - 128.88 (- \frac{\sqrt{4 - \sqrt{6} + \sqrt{2}} \sqrt{2}}{2 (\sqrt{2} \sqrt{2})})}$

Step 4.4.4.10.3

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

$c = \sqrt{36 + 115.3476 - 128.88 (- \frac{\sqrt{4 - \sqrt{6} + \sqrt{2}} \sqrt{2}}{2 (\sqrt{2} \sqrt{2})})}$

Step 4.4.4.10.4

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

$c = \sqrt{36 + 115.3476 - 128.88 (- \frac{\sqrt{4 - \sqrt{6} + \sqrt{2}} \sqrt{2}}{2 (\sqrt{2} \sqrt{2})})}$

Step 4.4.4.10.5

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

$c = \sqrt{36 + 115.3476 - 128.88 (- \frac{\sqrt{4 - \sqrt{6} + \sqrt{2}} \sqrt{2}}{2 {\sqrt{2}}^{1 + 1}})}$

Step 4.4.4.10.6

Add $1$ and $1$ .

$c = \sqrt{36 + 115.3476 - 128.88 (- \frac{\sqrt{4 - \sqrt{6} + \sqrt{2}} \sqrt{2}}{2 {\sqrt{2}}^{2}})}$

Step 4.4.4.10.7

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

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Step 4.4.4.10.7.1

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

$c = \sqrt{36 + 115.3476 - 128.88 (- \frac{\sqrt{4 - \sqrt{6} + \sqrt{2}} \sqrt{2}}{2 {(2^{\frac{1}{2}})}^{2}})}$

Step 4.4.4.10.7.2

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

$c = \sqrt{36 + 115.3476 - 128.88 (- \frac{\sqrt{4 - \sqrt{6} + \sqrt{2}} \sqrt{2}}{2 \cdot 2^{\frac{1}{2} \cdot 2}})}$

Step 4.4.4.10.7.3

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

$c = \sqrt{36 + 115.3476 - 128.88 (- \frac{\sqrt{4 - \sqrt{6} + \sqrt{2}} \sqrt{2}}{2 \cdot 2^{\frac{2}{2}}})}$

Step 4.4.4.10.7.4

Cancel the common factor of $2$ .

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Step 4.4.4.10.7.4.1

Cancel the common factor.

$c = \sqrt{36 + 115.3476 - 128.88 (- \frac{\sqrt{4 - \sqrt{6} + \sqrt{2}} \sqrt{2}}{2 \cdot 2^{\frac{2}{2}}})}$

Step 4.4.4.10.7.4.2

Rewrite the expression.

$c = \sqrt{36 + 115.3476 - 128.88 (- \frac{\sqrt{4 - \sqrt{6} + \sqrt{2}} \sqrt{2}}{2 \cdot 2})}$

Step 4.4.4.10.7.5

Evaluate the exponent.

$c = \sqrt{36 + 115.3476 - 128.88 (- \frac{\sqrt{4 - \sqrt{6} + \sqrt{2}} \sqrt{2}}{2 \cdot 2})}$

Step 4.4.4.11

Combine using the product rule for radicals.

$c = \sqrt{36 + 115.3476 - 128.88 (- \frac{\sqrt{(4 - \sqrt{6} + \sqrt{2}) \cdot 2}}{2 \cdot 2})}$

Step 4.4.4.12

Multiply $2$ by $2$ .

$c = \sqrt{36 + 115.3476 - 128.88 (- \frac{\sqrt{(4 - \sqrt{6} + \sqrt{2}) \cdot 2}}{4})}$

Step 4.5

Multiply $- 128.88 (- \frac{\sqrt{(4 - \sqrt{6} + \sqrt{2}) \cdot 2}}{4})$ .

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Step 4.5.1

Multiply $- 1$ by $- 128.88$ .

$c = \sqrt{36 + 115.3476 + 128.88 (\frac{\sqrt{(4 - \sqrt{6} + \sqrt{2}) \cdot 2}}{4})}$

Step 4.5.2

Combine $128.88$ and $\frac{\sqrt{(4 - \sqrt{6} + \sqrt{2}) \cdot 2}}{4}$ .

$c = \sqrt{36 + 115.3476 + \frac{128.88 \sqrt{(4 - \sqrt{6} + \sqrt{2}) \cdot 2}}{4}}$

Step 4.5.3

Multiply $128.88$ by $\sqrt{(4 - \sqrt{6} + \sqrt{2}) \cdot 2}$ .

$c = \sqrt{36 + 115.3476 + \frac{313.82869188}{4}}$

Step 4.6

Divide $313.82869188$ by $4$ .

$c = \sqrt{36 + 115.3476 + 78.45717297}$

Step 4.7

Add $36$ and $115.3476$ .

$c = \sqrt{151.3476 + 78.45717297}$

Step 4.8

Add $151.3476$ and $78.45717297$ .

$c = \sqrt{229.80477297}$

Step 4.9

Evaluate the root.

$c = 15.15931307$

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 $A$ .

$\frac{\sin (A)}{6} = \frac{\sin (127.5)}{15.15931307}$

Step 7

Solve the equation for $A$ .

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Step 7.1

Multiply both sides of the equation by $6$ .

$6 \frac{\sin (A)}{6} = 6 \frac{\sin (127.5)}{15.15931307}$

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 $6$ .

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Step 7.2.1.1.1

Cancel the common factor.

$6 \frac{\sin (A)}{6} = 6 \frac{\sin (127.5)}{15.15931307}$

Step 7.2.1.1.2

Rewrite the expression.

$\sin (A) = 6 \frac{\sin (127.5)}{15.15931307}$

Step 7.2.2

Simplify the right side.

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Step 7.2.2.1

Simplify $6 \frac{\sin (127.5)}{15.15931307}$ .

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Step 7.2.2.1.1

Evaluate $\sin (127.5)$ .

$\sin (A) = 6 (\frac{0.79335334}{15.15931307})$

Step 7.2.2.1.2

Divide $0.79335334$ by $15.15931307$ .

$\sin (A) = 6 \cdot 0.05233438$

Step 7.2.2.1.3

Multiply $6$ by $0.05233438$ .

$\sin (A) = 0.31400631$

Step 7.3

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

$A = \arcsin (0.31400631)$

Step 7.4

Simplify the right side.

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Step 7.4.1

Evaluate $\arcsin (0.31400631)$ .

$A = 18.30083638$

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.

$A = 180 - 18.30083638$

Step 7.6

Subtract $18.30083638$ from $180$ .

$A = 161.69916361$

Step 7.7

The solution to the equation $A = 18.30083638$ .

$A = 18.30083638, 161.69916361$

Step 7.8

Exclude the invalid angle.

$A = 18.30083638$

Step 8

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

$18.30083638 + 127.5 + B = 180$

Step 9

Solve the equation for $B$ .

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Step 9.1

Add $18.30083638$ and $127.5$ .

$145.80083638 + B = 180$

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 $145.80083638$ from both sides of the equation.

$B = 180 - 145.80083638$

Step 9.2.2

Subtract $145.80083638$ from $180$ .

$B = 34.19916361$

Step 10

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

$A = 18.30083638$

$B = 34.19916361$

$C = 127.5$

$a = 6$

$b = 10.74$

$c = 15.15931307$