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mr Goodwill [35]
2 years ago
7

For what values of θ is sinθ < cosθ when 0 ≤ θ < π ?

Mathematics
1 answer:
zysi [14]2 years ago
4 0
<h3>Answer:  0 \le \theta < \frac{\pi}{4}</h3>

=========================================================

How to get this answer:

Use the unit circle to note that \sin\theta = \cos\theta = \frac{\sqrt{2}}{2} when \theta = \frac{\pi}{4} (aka 45 degrees)

Beyond this point, cosine is smaller than sine. This means that anything from 0 to pi/4 will have sine be smaller than cosine. It might help to graph y = sin(x) and y = cos(x) on the interval from x = 0 to x = pi.

The two curves y = sin(x) and y = cos(x) intersect at the point \left(\frac{\pi}{4}, \frac{\sqrt{2}}{2}\right)

-------------------------------

Here's a more detailed picture of whats going on.

\sin \theta < \cos \theta\\\\\sin \theta < \sqrt{1-\sin^2\theta}\\\\\sin^2 \theta < 1-\sin^2\theta \\\\2\sin^2 \theta < 1\\\\\sin^2 \theta < \frac{1}{2}\\\\\sin \theta < \sqrt{\frac{1}{2}}\\\\\sin \theta < \frac{1}{\sqrt{2}}\\\\\sin \theta < \frac{\sqrt{2}}{2}\\\\\theta < \arcsin\left(\frac{\sqrt{2}}{2}\right)\\\\\theta < \frac{\pi}{4}\\\\

Intersect the intervals 0 \le \theta < \pi and \theta < \frac{\pi}{4} and you'll end up with the final answer 0 \le \theta < \frac{\pi}{4}

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