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QUESTION 1 17 sin 2015 = 0, where 20 E (90°; 270°). Determine without the use of a calculator and with the aid of a diagram the value of: cos 20​

User Wiener
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Answer: cos 20​ = -√3/2

Explanation:

Since sin 2015 = 0 and 20° is in the second quadrant, we know that the reference angle for 20° is 20° - 180° = -160°. So we need to find the value of cos(-160°).

We can use the fact that cos(-θ) = cos(θ) to find the value of cos(-160°) as follows:

cos(-160°) = cos(160°)

To find the cosine of 160°, we can use the identity cos(180° - θ) = -cos(θ) as follows:

cos(160°) = cos(180° - 20°) = -cos(20°)

Now, we need to determine the sign of cos(20°) in the second quadrant. Since 30°, 45° and 60° are angles in the first quadrant with exact values of √3/2 and 90° - 60° = 30°, 90° - 45° = 45°, 90° - 30° = 60° are their respective corresponding angles in the second quadrant, we can use the fact that cosine is a decreasing function in the second quadrant to conclude that:

1 > cos(20°) > √3/2

Therefore, since sin 2015 = 0, we know that cos(-160°) = -cos(20°) = 0, which implies that cos(20°) = 0.

Therefore, cos 20​ = -√3/2.

User Kbaccouche
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