Find the Exact Value sin(57)

Problem

sin(57^∘)

Solution

1. Identify the expression as the sine of an angle measured in degrees, sin(57^∘)

2. Recognize that 57^∘ is not a standard reference angle (like 30^∘ 45^∘ or 60^∘ and cannot be easily expressed as a sum or difference of standard angles using integers.

3. Apply the sum formula sin(A+B)=sin(A)*cos(B)+cos(A)*sin(B) using 57^∘=30^∘+27^∘ or 57^∘=60^∘−3^∘ However, these do not lead to a standard radical form.

4. Determine if a closed-form radical expression exists. The value of sin(3^∘) can be expressed using nested radicals, which allows for an exact expression for any integer multiple of 3^∘

5. Use the known exact radical form for sin(57^∘) which is derived from the values of sin(18^∘) and sin(15^∘) using the identity sin(57^∘)=sin(60^∘−3^∘) or by combining components of the regular pentagon and triangle.

6. State the exact value in its radical form.

Final Answer

sin(57^∘)=(1+√(,5)+√(,30−6√(,5)))/8

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