Find the Exact Value csc((3pi)/14)

Problem

csc((3*π)/14)

Solution

1. Identify the expression as the cosecant of an angle, which is the reciprocal of the sine function.

csc((3*π)/14)=1/sin((3*π)/14)

2. Relate the angle to a more common denominator by using the complementary angle identity sin(θ)=cos(π/2−θ)

π/2−(3*π)/14=(7*π)/14−(3*π)/14

(7*π)/14−(3*π)/14=(4*π)/14

(4*π)/14=(2*π)/7

3. Substitute the identity into the expression to see if it simplifies to a standard radical form.

sin((3*π)/14)=cos((2*π)/7)

4. Determine if the value can be expressed in simple radicals. The value cos((2*π)/7) is a root of a cubic equation 8*x3+4*x2−4*x−1=0 and does not have a standard "exact value" involving only square roots.

5. Conclude that for most trigonometric contexts, the exact value is kept in its trigonometric form or expressed via the roots of its minimal polynomial.

Final Answer

csc((3*π)/14)=csc((3*π)/14)

Want more problems? Check here!