Find the Exact Value sec(225)

Problem

sec(225^∘)

Solution

1. Identify the reference angle by determining the quadrant of 225^∘ Since 180^∘<225^∘<270^∘ the angle is in Quadrant III.

2. Calculate the reference angle (θ_R) using the formula for Quadrant III: (θ_R)=θ−180^∘

(θ_R)=225^∘−180^∘=45^∘

3. Determine the sign of the secant function in Quadrant III. Since sec(θ)=1/cos(θ) and cosine is negative in Quadrant III, the result will be negative.

sec(225^∘)=−sec(45^∘)

4. Recall the exact value of cos(45^∘) which is √(,2)/2

5. Apply the reciprocal identity sec(45^∘)=1/cos(45^∘)

sec(45^∘)=2/√(,2)

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

(2√(,2))/2=√(,2)

7. Combine the value with the negative sign determined in step 3.

sec(225^∘)=−√(,2)

Final Answer

sec(225^∘)=−√(,2)

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