Evaluate sec(210)

Problem

sec(210)

Solution

1. Identify the reference angle by finding the difference between the given angle and the nearest horizontal axis (180.

(θ_ref)=210−180=30

2. Determine the quadrant of the angle 210 Since 180<210<270 the angle lies in Quadrant III.

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

sec(210)=−sec(30)

4. Evaluate the trigonometric value using the reciprocal of the cosine of the reference angle.

cos(30)=√(,3)/2

sec(30)=2/√(,3)

5. Rationalize the denominator to simplify the expression.

2/√(,3)⋅√(,3)/√(,3)=(2√(,3))/3

6. Apply the sign determined in step 3 to the final value.

sec(210)=−(2√(,3))/3

Final Answer

sec(210)=−(2√(,3))/3

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