Find the Exact Value sec(arctan(-3/5))

Problem

sec(arctan(−3/5))

Solution

1. Identify the inner function as an angle θ=arctan(−3/5) This implies that tan(θ)=−3/5

2. Determine the quadrant for θ Since the range of arctan(x) is (−π/2,π/2) and the argument is negative, θ must be in Quadrant IV.

3. Use the definition of the tangent function in terms of x and y coordinates, where tan(θ)=y/x We can let y=−3 and x=5

4. Calculate the radius r using the Pythagorean theorem r=√(,x2+y2)

r=√(,5+(−3)2)

r=√(,25+9)

r=√(,34)

5. Apply the definition of the secant function, which is sec(θ)=r/x

sec(θ)=√(,34)/5

Final Answer

sec(arctan(−3/5))=√(,34)/5

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