Find the Exact Value csc(15)

Problem

the cosecant of 15 raised to the composed with power

Solution

1. Identify the angle as a difference of two special angles from the unit circle.

the cosecant of 15 raised to the composed with power equals the fraction with numerator 1 and denominator the sine of 15 raised to the composed with power

the sine of 15 raised to the composed with power equals the sine of open paren 45 raised to the composed with power minus 30 raised to the composed with power close paren

2. Apply the sine difference formula the sine of open paren A minus B close paren equals sine A cosine B minus cosine A sine B to find the denominator.

the sine of open paren 45 raised to the composed with power minus 30 raised to the composed with power close paren equals the sine of open paren 45 raised to the composed with power close paren the cosine of 30 raised to the composed with power minus the cosine of open paren 45 raised to the composed with power close paren the sine of 30 raised to the composed with power

3. Substitute the known values for the trigonometric functions of 45 raised to the composed with power and 30 raised to the composed with power

the sine of 15 raised to the composed with power equals the fraction with numerator the square root of 2 and denominator 2 times the fraction with numerator the square root of 3 and denominator 2 minus the fraction with numerator the square root of 2 and denominator 2 times one half

the sine of 15 raised to the composed with power equals the fraction with numerator the square root of 6 minus the square root of 2 and denominator 4

4. Invert the result to find the cosecant value.

the cosecant of 15 raised to the composed with power equals the fraction with numerator 4 and denominator the square root of 6 minus the square root of 2

5. Rationalize the denominator by multiplying the numerator and denominator by the conjugate the square root of 6 plus the square root of 2

the cosecant of 15 raised to the composed with power equals 4 times open paren the square root of 6 plus the square root of 2 close paren over open paren the square root of 6 minus the square root of 2 close paren times open paren the square root of 6 plus the square root of 2 close paren

the cosecant of 15 raised to the composed with power equals the fraction with numerator 4 times open paren the square root of 6 plus the square root of 2 close paren and denominator 6 minus 2

the cosecant of 15 raised to the composed with power equals 4 times open paren the square root of 6 plus the square root of 2 close paren over 4

6. Simplify the expression by canceling the common factor.

the cosecant of 15 raised to the composed with power equals the square root of 6 plus the square root of 2

Final Answer

the cosecant of 15 raised to the composed with power equals the square root of 6 plus the square root of 2

Want more problems? Check here!