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Please help!
maths functions

Please help! maths functions-example-1
User Camembert
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1. I'm not sure how you're expected to "read off" where they intersect based on an imprecise hand-drawn graph, but we can still find these intersections exactly.


2\sin(x) = 3 \cos(x)


(\sin(x))/(\cos(x)) = \frac32


\tan(x) = \frac32


x = \tan^(-1)\left(\frac32\right) + 180^\circ n

where
n is an integer.

In the interval [0°, 360°], we have solutions at


x \approx 56.31^\circ \text{ or } x \approx 236.31^\circ

From the sketch of the plot, we do see that the intersections are roughly where we expect them to be. (The first is somewhere between 45° and 90°, while the second is somewhere between 225° and 270°.)

2. According to the plot and the solutions from (1), we have


3 \cos(x) > 2 \sin(x)

whenever
0^\circ < x < 56.31^\circ or
236.31^\circ < x < 360^\circ.

3. Rewrite the inequality as


2 \sin(x) - 3 \cos(x) &nbsp;\le 0 \implies 3 \cos(x) \ge 2 \sin(x)

The answer to (1) tells us where the equality
3\cos(x) = 2\sin(x) holds.

The answer to (2) tells us where the strict inequality
3\cos(x)>2\sin(x) holds.

Putting these solutions together, we have
2\sin(x) - 3\cos(x) \le 0 whenever
0^\circ < x \le 56.31^\circ or
236.61^\circ \le x < 360^\circ.

User Qwermike
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