First, split the angle into two angles where the values of the six trigonometric functions are known. In this case, <math><mstyle displaystyle="true"><mfrac><mrow><mn>19</mn><mi>π</mi></mrow><mrow><mn>12</mn></mrow></mfrac></mstyle></math> can be split into <math><mstyle displaystyle="true"><mfrac><mrow><mn>5</mn><mi>π</mi></mrow><mrow><mn>4</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mi>π</mi></mrow><mrow><mn>3</mn></mrow></mfrac></mstyle></math> .

Use the sum formula for cosine to simplify the expression. The formula states that <math><mstyle displaystyle="true"><mi>cos</mi><mrow><mo>(</mo><mi>A</mi><mo>+</mo><mi>B</mi><mo>)</mo></mrow><mo>=</mo><mo>-</mo><mrow><mo>(</mo><mi>cos</mi><mrow><mo>(</mo><mi>A</mi><mo>)</mo></mrow><mi>cos</mi><mrow><mo>(</mo><mi>B</mi><mo>)</mo></mrow><mo>+</mo><mi>sin</mi><mrow><mo>(</mo><mi>A</mi><mo>)</mo></mrow><mi>sin</mi><mrow><mo>(</mo><mi>B</mi><mo>)</mo></mrow><mo>)</mo></mrow></mstyle></math> .

The exact value of <math><mstyle displaystyle="true"><mi>cos</mi><mrow><mo>(</mo><mfrac><mrow><mi>π</mi></mrow><mrow><mn>3</mn></mrow></mfrac><mo>)</mo></mrow></mstyle></math> is <math><mstyle displaystyle="true"><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> .

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.

The exact value of <math><mstyle displaystyle="true"><mi>cos</mi><mrow><mo>(</mo><mfrac><mrow><mi>π</mi></mrow><mrow><mn>4</mn></mrow></mfrac><mo>)</mo></mrow></mstyle></math> is <math><mstyle displaystyle="true"><mfrac><mrow><msqrt><mn>2</mn></msqrt></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> .

The exact value of <math><mstyle displaystyle="true"><mi>sin</mi><mrow><mo>(</mo><mfrac><mrow><mi>π</mi></mrow><mrow><mn>3</mn></mrow></mfrac><mo>)</mo></mrow></mstyle></math> is <math><mstyle displaystyle="true"><mfrac><mrow><msqrt><mn>3</mn></msqrt></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> .

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

The exact value of <math><mstyle displaystyle="true"><mi>sin</mi><mrow><mo>(</mo><mfrac><mrow><mi>π</mi></mrow><mrow><mn>4</mn></mrow></mfrac><mo>)</mo></mrow></mstyle></math> is <math><mstyle displaystyle="true"><mfrac><mrow><msqrt><mn>2</mn></msqrt></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mrow><mo>(</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>)</mo></mrow><mrow><mo>(</mo><mo>-</mo><mfrac><mrow><msqrt><mn>2</mn></msqrt></mrow><mrow><mn>2</mn></mrow></mfrac><mo>)</mo></mrow></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> and <math><mstyle displaystyle="true"><mfrac><mrow><msqrt><mn>2</mn></msqrt></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> .

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

Multiply <math><mstyle displaystyle="true"><mo>-</mo><mfrac><mrow><msqrt><mn>3</mn></msqrt></mrow><mrow><mn>2</mn></mrow></mfrac><mrow><mo>(</mo><mo>-</mo><mfrac><mrow><msqrt><mn>2</mn></msqrt></mrow><mrow><mn>2</mn></mrow></mfrac><mo>)</mo></mrow></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> by <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mfrac><mrow><msqrt><mn>3</mn></msqrt></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> by <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mfrac><mrow><msqrt><mn>3</mn></msqrt></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> and <math><mstyle displaystyle="true"><mfrac><mrow><msqrt><mn>2</mn></msqrt></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> .

Combine using the product rule for radicals.

Multiply <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

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

Combine the numerators over the common denominator.

Factor <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> out of <math><mstyle displaystyle="true"><mo>-</mo><msqrt><mn>2</mn></msqrt></mstyle></math> .

Factor <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> out of <math><mstyle displaystyle="true"><msqrt><mn>6</mn></msqrt></mstyle></math> .

Factor <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> out of <math><mstyle displaystyle="true"><mo>-</mo><mrow><mo>(</mo><msqrt><mn>2</mn></msqrt><mo>)</mo></mrow><mo>-</mo><mn>1</mn><mrow><mo>(</mo><mo>-</mo><msqrt><mn>6</mn></msqrt><mo>)</mo></mrow></mstyle></math> .

Simplify the expression.

Rewrite <math><mstyle displaystyle="true"><mo>-</mo><mrow><mo>(</mo><msqrt><mn>2</mn></msqrt><mo>-</mo><msqrt><mn>6</mn></msqrt><mo>)</mo></mrow></mstyle></math> as <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn><mrow><mo>(</mo><msqrt><mn>2</mn></msqrt><mo>-</mo><msqrt><mn>6</mn></msqrt><mo>)</mo></mrow></mstyle></math> .

Move the negative in front of the fraction.

The result can be shown in multiple forms.

Exact Form:

Decimal Form:

Do you know how to Expand Using Sum/Difference Formulas cos((19pi)/12)? 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 | four hundred ninety-four million nine hundred twelve thousand five hundred seven |
---|

- 494912507 has 4 divisors, whose sum is
**496159536** - The reverse of 494912507 is
**705219494** - Previous prime number is
**397**

- Is Prime? no
- Number parity odd
- Number length 9
- Sum of Digits 41
- Digital Root 5

Name | one billion four hundred sixteen million four hundred forty-seven thousand four hundred forty-six |
---|

- 1416447446 has 8 divisors, whose sum is
**2249651880** - The reverse of 1416447446 is
**6447446141** - Previous prime number is
**17**

- Is Prime? no
- Number parity even
- Number length 10
- Sum of Digits 41
- Digital Root 5

Name | two hundred forty-four million nine hundred ninety-four thousand one hundred two |
---|

- 244994102 has 8 divisors, whose sum is
**367756200** - The reverse of 244994102 is
**201499442** - Previous prime number is
**1409**

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