<math><mstyle displaystyle="true"><mtable class="shapeTable" columnlines="solid" rowlines="solid"><mtr><mtd><mi>Side</mi></mtd><mtd><mi>Angle</mi></mtd></mtr><mtr><mtd><mtable><mtr><mtd><mrow><mi>b</mi><mo>=</mo></mrow></mtd><mtd><mn>15</mn></mtd></mtr><mtr><mtd><mrow><mi>c</mi><mo>=</mo></mrow></mtd><mtd></mtd></mtr><mtr><mtd><mrow><mi>a</mi><mo>=</mo></mrow></mtd><mtd><mn>9</mn></mtd></mtr></mtable></mtd><mtd><mtable><mtr><mtd><mrow><mi>A</mi><mo>=</mo></mrow></mtd><mtd></mtd></mtr><mtr><mtd><mrow><mi>B</mi><mo>=</mo></mrow></mtd><mtd></mtd></mtr><mtr><mtd><mrow><mi>C</mi><mo>=</mo></mrow></mtd><mtd><mn>106</mn></mtd></mtr></mtable></mtd></mtr></mtable></mstyle></math>

Use the law of cosines to find the unknown side of the triangle, given the other two sides and the included angle.

Solve the equation.

Substitute the known values into the equation.

Raise <math><mstyle displaystyle="true"><mn>9</mn></mstyle></math> to the power of <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Raise <math><mstyle displaystyle="true"><mn>15</mn></mstyle></math> to the power of <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mo>-</mo><mn>2</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>9</mn></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mo>-</mo><mn>18</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>15</mn></mstyle></math> .

Evaluate <math><mstyle displaystyle="true"><mi>cos</mi><mrow><mo>(</mo><mn>106</mn><mo>)</mo></mrow></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mo>-</mo><mn>270</mn></mstyle></math> by <math><mstyle displaystyle="true"><mo>-</mo><mn>0.27563735</mn></mstyle></math> .

Add <math><mstyle displaystyle="true"><mn>81</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>225</mn></mstyle></math> .

Add <math><mstyle displaystyle="true"><mn>306</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>74.42208607</mn></mstyle></math> .

Evaluate the root.

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.

Substitute the known values into the law of sines to find <math><mstyle displaystyle="true"><mi>A</mi></mstyle></math> .

Multiply both sides of the equation by <math><mstyle displaystyle="true"><mn>9</mn></mstyle></math> .

Simplify both sides of the equation.

Simplify the left side.

Cancel the common factor of <math><mstyle displaystyle="true"><mn>9</mn></mstyle></math> .

Cancel the common factor.

Rewrite the expression.

Simplify the right side.

Simplify <math><mstyle displaystyle="true"><mn>9</mn><mfrac><mrow><mi>sin</mi><mrow><mo>(</mo><mn>106</mn><mo>)</mo></mrow></mrow><mrow><mn>19.50441196</mn></mrow></mfrac></mstyle></math> .

Evaluate <math><mstyle displaystyle="true"><mi>sin</mi><mrow><mo>(</mo><mn>106</mn><mo>)</mo></mrow></mstyle></math> .

Divide <math><mstyle displaystyle="true"><mn>0.96126169</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>19.50441196</mn></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mn>9</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>0.04928432</mn></mstyle></math> .

Take the inverse sine of both sides of the equation to extract <math><mstyle displaystyle="true"><mi>A</mi></mstyle></math> from inside the sine.

Simplify the right side.

Evaluate <math><mstyle displaystyle="true"><mi>arcsin</mi><mrow><mo>(</mo><mn>0.44355888</mn><mo>)</mo></mrow></mstyle></math> .

The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from <math><mstyle displaystyle="true"><mn>180</mn></mstyle></math> to find the solution in the second quadrant.

Subtract <math><mstyle displaystyle="true"><mn>26.33117358</mn></mstyle></math> from <math><mstyle displaystyle="true"><mn>180</mn></mstyle></math> .

The solution to the equation <math><mstyle displaystyle="true"><mi>A</mi><mo>=</mo><mn>26.33117358</mn></mstyle></math> .

Exclude the invalid angle.

The sum of all the angles in a triangle is <math><mstyle displaystyle="true"><mn>180</mn></mstyle></math> degrees.

Add <math><mstyle displaystyle="true"><mn>26.33117358</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>106</mn></mstyle></math> .

Move all terms not containing <math><mstyle displaystyle="true"><mi>B</mi></mstyle></math> to the right side of the equation.

Subtract <math><mstyle displaystyle="true"><mn>132.33117358</mn></mstyle></math> from both sides of the equation.

Subtract <math><mstyle displaystyle="true"><mn>132.33117358</mn></mstyle></math> from <math><mstyle displaystyle="true"><mn>180</mn></mstyle></math> .

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

Do you know how to Solve the Triangle tri{15}{}{}{}{9}{106}? If not, you can write to our math experts in our application. The best solution for your task you can find above on this page.

Name | one billion five hundred seventy-four million six hundred sixty-nine thousand five hundred ninety-one |
---|

- 1574669591 has 8 divisors, whose sum is
**1612970064** - The reverse of 1574669591 is
**1959664751** - Previous prime number is
**53**

- Is Prime? no
- Number parity odd
- Number length 10
- Sum of Digits 53
- Digital Root 8

Name | one billion seven hundred million seven hundred sixty-six thousand seven hundred fifty-eight |
---|

- 1700766758 has 16 divisors, whose sum is
**2581804800** - The reverse of 1700766758 is
**8576670071** - Previous prime number is
**149**

- Is Prime? no
- Number parity even
- Number length 10
- Sum of Digits 47
- Digital Root 2

Name | eight hundred twenty-one million three hundred eighty thousand four hundred ninety |
---|

- 821380490 has 8 divisors, whose sum is
**1267272864** - The reverse of 821380490 is
**094083128** - Previous prime number is
**35**

- Is Prime? no
- Number parity even
- Number length 9
- Sum of Digits 35
- Digital Root 8