Find the Exact Value sec(210 degrees )

Problem

sec(210)

Solution

1. Identify the reference angle by finding the distance from the angle to the nearest x-axis. Since 210 is in the third quadrant, subtract 180 from the angle.

210−180=30

2. Determine the sign of the secant function in the third quadrant. In the third quadrant, only tangent and cotangent are positive, so secant is negative.

sec(210)=−sec(30)

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

sec(30)=1/cos(30)

4. Substitute the known exact value for cos(30) which is √(,3)/2

sec(30)=1/√(,3)/2

5. Simplify the fraction by multiplying by the reciprocal.

sec(30)=2/√(,3)

6. Rationalize the denominator by multiplying the numerator and denominator by √(,3)

2/√(,3)⋅√(,3)/√(,3)=(2√(,3))/3

7. Apply the negative sign determined in step 2 to find the final value.

sec(210)=−(2√(,3))/3

Final Answer

sec(210)=−(2√(,3))/3

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