Given that sin Ɵ = -9/41 and Ɵ is in quadrant 3?

Question

determine sin 2Ɵ, cos2Ɵ, and tan 2Ɵ

Which quadrant is 2Ɵ in?

Answer 1

Tangent of 9/41 is approx. 12 deg (I don't have a calculator)
Therefore tan angle theta of -9/41 given that it is in the 3rd quadrant = 180 deg + 12 deg = 192 deg
2 theta = 384 deg which is in the 1st quadrant

cos 384 deg = approx .9135
sin 384 deg = approx .4067
tan 384 deg = approx .4452

Answer 2

First, let's get back to the "sane" trig functions. Let's rewrite those in terms of the "Big Three" of sine, cosine, and tangent. Why? Because I bet your calculator doesn't have a cosecant or an inverse cotangent button! Find 1 / tan x, given that 1 / sin x = -3.5891420. So, they've given you some kind of weird number. You really ARE going to have to use your calculator here. First, solve for sin x to get sin x = 1 / -3.589142. Then take the arcsine of both sides. You get: arcsin (sin x) = arcsin (1 / -3.589) (I truncated; I'm lazy. Deal with it.) x = arcsin (-1 / 3.589) Okay, so you evaluate that on your calculator. Let's say you get x = -15 degrees. (That's not exactly right; you get to do the problem!) You have a slight problem here. The angle YOU have is in quadrant IV. You want an angle in quadrant III. How do we get there? Well, if you reflect an angle across the y-axis, the sine (or how far up/down the angle goes) remains the same. Reflecting across the y-axis is the same as taking 180 minus your angle. So, you have 180 - -15, or 180 + 15 = 195. That's in quadrant III. So now you take cotangent 195. To do that, use cot 195 = 1 / tan 195, and evaluate that with your calculator.

Answer 3

Given that sin Ɵ = -9/41 and Ɵ is in quadrant 3,
Determine sin 2Ɵ, cos2Ɵ, and tan 2Ɵ.

Ɵ = arcsin(-9/41) ≈ 192.6803835°
since it is in the third quadrant

2Ɵ ≈ 385.36076698°

This is conterminous with
385.36076698° - 360° = 25.36076698°

So 2Ɵ is in the first quadrant.

Given that sinƟ = -9/41:

cosƟ = -√[1 - (-9/41)²] = -√(1600/1681) = -40/41

Using the double angle identities we have:

sin 2Ɵ = 2(sinƟ)(cosƟ) = 2(-9/41)(-40/41) = 720/1681

cos 2Ɵ = 1 - 2sin²Ɵ = 1 - 2(-9/41)² = 1519/1681

tan 2Ɵ = (sin 2Ɵ) / (cos 2Ɵ) = 720/1519

Answer 4

sin 2Ɵ = 2sinƟcosƟ = 2(-9/41)(-40/41) = 720/1681
cos2Ɵ = 1 - 2sin^2(Ɵ) = 1519/1681
tan 2Ɵ = 720/1519

2Ɵ is in quadrant 1, since both sin 2Ɵ and cos2Ɵ are positive.
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Ideas: You don't need to calculate the angle. sin 2Ɵ > 0 means it is in the upper half plane, and cos2Ɵ > 0 means it is in the left half plane. The combined region is in the first quadrant.

Answer 5

quandrant 4