Simplify cos(arctan( square root of 14))

Problem

cos(arctan(√(,14)))

Solution

1. Identify the inner expression as an angle θ such that θ=arctan(√(,14))

2. Rewrite the relationship using the definition of the inverse tangent function, which implies tan(θ)=√(,14)

3. Represent the angle θ in a right triangle where the side opposite to θ is √(,14) and the side adjacent to θ is 1 since tan(θ)=opposite/adjacent=√(,14)/1

4. Calculate the hypotenuse h using the Pythagorean theorem a2+b2=c2

h=√(,(√(,14))2+1)

h=√(,14+1)

h=√(,15)

5. Evaluate the cosine of the angle θ using the ratio cos(θ)=adjacent/hypotenuse

cos(θ)=1/√(,15)

6. Rationalize the denominator by multiplying the numerator and denominator by √(,15)

1/√(,15)⋅√(,15)/√(,15)=√(,15)/15

Final Answer

cos(arctan(√(,14)))=√(,15)/15

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