Trigonometric Half Angle Identities — Quiz

Question 10 • Precalculus

What is the exact value of tangent of open paren 5 pi over 8 close paren ?

Answer

square root of the fraction with numerator 2 minus square root of 2 and denominator 2 plus square root of 2 end root

square root of the fraction with numerator 2 plus square root of 2 and denominator 2 minus square root of 2 end root

negative square root of the fraction with numerator 2 minus square root of 2 and denominator 2 plus square root of 2 end root

negative square root of the fraction with numerator 2 plus square root of 2 and denominator 2 minus square root of 2 end root

Explanation

Since 5π/8 = π − 3π/8, the identity tan(π − θ) = −tan θ gives tan(5π/8) = −tan(3π/8). Also, tan(3π/8) = cot(π/8) = 1 + √2. Choice D equals −√((2 + √2)/(2 − √2)); rationalizing the fraction shows its square root is 1 + √2. Therefore, the exact value is −(1 + √2), which is choice D.

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