Find the Exact Value sin(arctan(7))

Problem

sin(arctan(7))

Solution

1. Identify the inner expression as an angle θ=arctan(7) which implies tan(θ)=7

2. Represent the relationship using a right triangle where the side opposite to θ is 7 and the side adjacent to θ is 1

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

h=√(,7+1)

h=√(,49+1)

h=√(,50)

4. Simplify the radical for the hypotenuse.

h=5√(,2)

5. Determine the value of sin(θ) using the ratio of the opposite side over the hypotenuse.

sin(θ)=7/(5√(,2))

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

sin(θ)=(7√(,2))/(5⋅2)

sin(θ)=(7√(,2))/10

Final Answer

sin(arctan(7))=(7√(,2))/10

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