Find the Exact Value csc(-330 degrees )

Problem

the cosecant of open paren negative 330 raised to the composed with power close paren

Solution

1. Use the property of odd functions to rewrite the expression, noting that cosecant negative theta equals negative cosecant theta

the cosecant of open paren negative 330 raised to the composed with power close paren equals negative the cosecant of 330 raised to the composed with power

2. Find the reference angle for 330 raised to the composed with power by subtracting it from 360 raised to the composed with power because it is in the fourth quadrant.

theta sub ref equals 360 raised to the composed with power minus 330 raised to the composed with power equals 30 raised to the composed with power

3. Determine the sign of the cosecant function in the fourth quadrant. Since sine theta is negative in the fourth quadrant, cosecant theta is also negative.

the cosecant of 330 raised to the composed with power equals negative the cosecant of 30 raised to the composed with power

4. Substitute this back into the expression from step 1.

negative the cosecant of 330 raised to the composed with power equals negative open paren negative the cosecant of 30 raised to the composed with power close paren equals the cosecant of 30 raised to the composed with power

5. Evaluate the cosecant of the reference angle using the reciprocal identity cosecant theta equals 1 over sine theta

the cosecant of 30 raised to the composed with power equals the fraction with numerator 1 and denominator the sine of 30 raised to the composed with power equals the fraction with numerator 1 and denominator 1 divided by 2 equals 2

Final Answer

the cosecant of open paren negative 330 raised to the composed with power close paren equals 2

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