Find the Exact Value sec(225 degrees )

Problem

sec(225^∘)

Solution

1. Identify the reference angle by finding the difference between the given angle and the nearest x-axis. Since 225^∘ is in the third quadrant, the reference angle is 225^∘−180^∘=45^∘

2. Determine the sign of the secant function in the third quadrant. In the third quadrant, only tangent and cotangent are positive, so sec(225^∘) must be negative.

3. Recall the reciprocal identity for secant, which states that sec(θ)=1/cos(θ)

4. Evaluate the cosine of the reference angle. We know that cos(45^∘)=√(,2)/2

5. Apply the quadrant sign to the cosine value. In the third quadrant, cosine is negative, so cos(225^∘)=−√(,2)/2

6. Calculate the reciprocal to find the secant value.

sec(225^∘)=1/(−√(,2)/2)

7. Simplify the expression by rationalizing the denominator.

sec(225^∘)=−2/√(,2)

sec(225^∘)=−√(,2)

Final Answer

sec(225^∘)=−√(,2)

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