Find the Exact Value sin(87)

Problem

sin(87^∘)

Solution

1. Identify the expression as the sine of an angle measured in degrees, sin(87^∘)

2. Recognize that 87^∘ is not a standard reference angle (like 30^∘ 45^∘ or 60^∘.

3. Apply the sum formula for sine, sin(A+B)=sin(A)*cos(B)+cos(A)*sin(B) by breaking 87^∘ into the sum of 45^∘ and 42^∘

4. Determine that 42^∘ is also not a standard angle, meaning the exact value must be expressed using radicals or the original trigonometric form if no further simplification into standard constants is possible.

5. Evaluate using the half-angle or sum/difference identities if specific radical forms are required, but for sin(87^∘) the most common exact form involves nested radicals derived from sin(3^∘) or cos(3^∘)

6. State the exact value using the identity sin(87^∘)=cos(3^∘)

7. Expand cos(3^∘) using the difference formula cos(18^∘−15^∘) or similar combinations of standard values (72^∘,75^∘,e*t*c. to reach the radical expression.

Final Answer

sin(87^∘)=√(,2)/8*(√(,3)+1)√(,5+√(,5))+(√(,6)−√(,2))/16*(√(,5)−1)

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