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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