Find the Exact Value sin(68)

Problem

sin(68)

Solution

1. Identify the goal as finding the exact value of sin(68) Since 68 is not a standard angle, it can be expressed as the sum of two angles, such as 60+8 or 30+38 but these do not lead to a standard exact form easily.

2. Apply the sum-to-product or double-angle identities if possible. However, 68 is typically expressed in terms of radicals using the values for 18 and 15 or by using the half-angle and sum formulas.

3. Recognize that 68=60+8 or 90−22 A common way to express this exact value involves nested radicals derived from the pentagon constants (sin(18) and standard trigonometric identities.

4. Use the identity sin(68)=cos(22) The exact value for cos(22) is derived from the half-angle formula cos(44/2) or by relating it to the roots of specific polynomials.

5. State the exact value in its standard radical form. The value is:

sin(68)=√(,2)/8√(,10+2√(,5))+√(,3)/4*(√(,5)−1)

6. Simplify the expression into a single combined form if required.

Final Answer

sin(68)=√(,2)/8√(,10+2√(,5))+(√(,15)−√(,3))/8

Want more problems? Check here!