Simplify cos(arctan( square root of 7))

Problem

cos(arctan(√(,7)))

Solution

1. Identify the inner expression as an angle θ where θ=arctan(√(,7))

2. Apply the definition of the arctangent function, which implies tan(θ)=√(,7)

3. Relate the tangent function to a right triangle where the opposite side is √(,7) and the adjacent side is 1

4. Calculate the hypotenuse h using the Pythagorean theorem a2+b2=h2

1+(√(,7))2=h2

1+7=h2

8=h2

h=√(,8)=2√(,2)

5. Evaluate the cosine of the angle θ using the ratio of the adjacent side to the hypotenuse.

cos(θ)=1/(2√(,2))

6. Rationalize the denominator by multiplying the numerator and denominator by √(,2)

1/(2√(,2))⋅√(,2)/√(,2)=√(,2)/4

Final Answer

cos(arctan(√(,7)))=√(,2)/4

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