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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