Find the Exact Value cos(63)

Problem

cos(63^∘)

Solution

1. Identify the expression as the cosine of an angle measured in degrees, cos(63^∘)

2. Recognize that 63^∘ is not a standard reference angle (like 30^∘ 45^∘ or 60^∘.

3. Determine if the angle can be expressed as a sum or difference of standard angles. Since 63 is not a simple combination of standard angles, the exact value is typically expressed using its trigonometric form or radical constants derived from the pentagon (18^∘ 36^∘ 54^∘ 72^∘.

4. Apply the sum formula cos(A+B)=cos(A)*cos(B)−sin(A)*sin(B) using 63^∘=45^∘+18^∘

5. Substitute the known values: cos(45^∘)=√(,2)/2 sin(45^∘)=√(,2)/2 cos(18^∘)=√(,10+2√(,5))/4 and sin(18^∘)=(√(,5)−1)/4

6. Calculate the product:

cos(63^∘)=√(,2)/2⋅√(,10+2√(,5))/4−√(,2)/2⋅(√(,5)−1)/4

7. Simplify the expression by combining the terms over a common denominator:

cos(63^∘)=(√(,20+4√(,5))−√(,10)+√(,2))/8

Final Answer

cos(63^∘)=(√(,20+4√(,5))−√(,10)+√(,2))/8

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