Use the fundamental trigonometric identities as...

Use the fundamental trigonometric identities as needed.\\ ? Given that $\cos x \approx 0.5062$ and that $x$ is between 0 and $\frac{\pi}{2}$, approximate the given function values.\\ Part 1 of 6 (a) $\sin x \approx$ (Round to four decimal places.) Part 2 of 6 (b) $\sin \left(\frac{\pi}{2} - x\right) \approx$ (Round to four decimal places.) Part 3 of 6 (c) $\tan x \approx$ (Round to four decimal places.) Part 4 of 6 (d) $\cos \left(\frac{\pi}{2} - x\right) \approx$ (Round to four decimal places.) Part 5 of 6 (e) $\sec x \approx$ (Round to four decimal places.) Part 6 of 6 (f) $\cot \left(\frac{\pi}{2} - x\right) \approx$ (Round to four decimal places.)

Transcript

00:01 In this problem we are given that tan of x equals to 5 over 12 and x belongs to closed interval pi up to 3 pi over 2.

00:13 We are asked to find out the answer for sine x and cos of x.

00:18 We know that tan of x is equal to the length of the opposite side divided by the length of the adjacent side.

00:27 So here we have the length of the opposite side to be 5 and the length of the adjacent side.

00:31 Adjacent side to be 12.

00:33 Making use of the pythogamous theorem we have hypertineuse squared to be equal to opposite squared plus adjacent squared which gives us 25 plus 144 which is 169.

00:48 So hypotenuse equals to 13.

00:52 So let us substitute this we have a sign of x to be equal to opposite over hypotenuse so we get 5 over 30.

01:02 And cost of x equals to adjacent over hypotenuse.

01:09 So we have 12 over 13.

01:12 But since we are given that x belongs to closed interval pi 3 pi over 2, that is x belongs to the third quadrant.

01:22 We know that both sign as well as cost would be negative in this quadrant.

01:30 So we include the negative sign.

01:32 Let us compare this with all of the options.

01:36 We see that option a matches with our answer.

01:39 So therefore option a is the correct answer.

01:45 In the second problem, we are given that f of x is of the form a times sine of 2 times pi over b times x minus c plus d.

01:59 And here we are given that y equals to 2 times cause of teta minus.

02:05 2 pi.

02:06 We are asked to find out the value of b.

02:09 So here we have y to be equal to 2 times cos, but we require sign...

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