Trigonometry Examples

$a = 1$ , $b = 2$ , $C = 20$

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.

$c = \sqrt{{(1)}^{2} + {(2)}^{2} - 2 \cdot 1 \cdot 2 \cos (20)}$

Step 4

Simplify the results.

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Step 4.1

One to any power is one.

$c = \sqrt{1 + {(2)}^{2} - 2 \cdot 1 \cdot (2 \cos (20))}$

Step 4.2

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

$c = \sqrt{1 + 4 - 2 \cdot 1 \cdot (2 \cos (20))}$

Step 4.3

Multiply $- 2 \cdot 1 \cdot 2$ .

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Step 4.3.1

Multiply $- 2$ by $1$ .

$c = \sqrt{1 + 4 - 2 \cdot (2 \cos (20))}$

Step 4.3.2

Multiply $- 2$ by $2$ .

$c = \sqrt{1 + 4 - 4 \cos (20)}$

Step 4.4

Evaluate $\cos (20)$ .

$c = \sqrt{1 + 4 - 4 \cdot 0.93969262}$

Step 4.5

Multiply $- 4$ by $0.93969262$ .

$c = \sqrt{1 + 4 - 3.75877048}$

Step 4.6

Add $1$ and $4$ .

$c = \sqrt{5 - 3.75877048}$

Step 4.7

Subtract $3.75877048$ from $5$ .

$c = \sqrt{1.24122951}$

Step 4.8

Evaluate the root.

$c = 1.1141048$

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 $A$ .

$\frac{\sin (A)}{1} = \frac{\sin (20)}{1.1141048}$

Step 7

Solve the equation for $A$ .

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Step 7.1

Divide $\sin (A)$ by $1$ .

$\sin (A) = \frac{\sin (20)}{1.1141048}$

Step 7.2

Simplify $\frac{\sin (20)}{1.1141048}$ .

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Step 7.2.1

Evaluate $\sin (20)$ .

$\sin (A) = \frac{0.34202014}{1.1141048}$

Step 7.2.2

Divide $0.34202014$ by $1.1141048$ .

$\sin (A) = 0.30699099$

Step 7.3

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

$A = \arcsin (0.30699099)$

Step 7.4

Simplify the right side.

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Step 7.4.1

Evaluate $\arcsin (0.30699099)$ .

$A = 17.87798714$

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.

$A = 180 - 17.87798714$

Step 7.6

Subtract $17.87798714$ from $180$ .

$A = 162.12201285$

Step 7.7

The solution to the equation $A = 17.87798714$ .

$A = 17.87798714, 162.12201285$

Step 7.8

Exclude the invalid angle.

$A = 17.87798714$

Step 8

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

$17.87798714 + 20 + B = 180$

Step 9

Solve the equation for $B$ .

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Step 9.1

Add $17.87798714$ and $20$ .

$37.87798714 + B = 180$

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 $37.87798714$ from both sides of the equation.

$B = 180 - 37.87798714$

Step 9.2.2

Subtract $37.87798714$ from $180$ .

$B = 142.12201285$

Step 10

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

$A = 17.87798714$

$B = 142.12201285$

$C = 20$

$a = 1$

$b = 2$

$c = 1.1141048$