Question

If tan(θ)=− 40/9and π <θ<π , what is the exact value of ⁡cos(2θ)?

A) - 1919/1600
B) - 1591/1600
C) 1591/1600
D) 1919/1600

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Answer 1

To find the exact value of ⁡cos(2θ), we can use the double-angle identity for cosine: cos(2θ) = cos²(θ) - sin²(θ). Given that tan(θ) = -40/9 and π < θ < π, we can determine the values of sin(θ) and cos(θ) using the Pythagorean identity. Substituting these values into the double-angle identity, we find that the exact value of ⁡cos(2θ) is -1519/1681.

Step-by-step explanation:

To find the exact value of ∧cos(2θ), we can use the double-angle identity for cosine: cos(2θ) = cos²(θ) - sin²(θ).

Given that tan(θ) = -40/9 and π < θ < π, we can determine the values of sin(θ) and cos(θ) using the Pythagorean identity: sin²(θ) + cos²(θ) = 1.

sin(θ) = -40/41 and cos(θ) = 9/41.

Substituting these values into the double-angle identity, we get:

cos(2θ) = (9/41)² - (-40/41)² = 81/1681 - 1600/1681 = -1519/1681.

Therefore, the exact value of ∧cos(2θ) is -1519/1681. Answer choice (A) -1919/1600 is the closest approximation, but not the exact value.