If sin2x=7/25, siny=-2/sq root of 13, cot y>0, and -pi/4<x<pi/4, find sin(2y-a)
If siny = -2/√13 , and coty ≥0, then y is in quad III
so cosy = -3/√13
then sin 2y = 2sinycosy
= 2(-2/√13)(-3/√13) = 12/13
cos 2y = cos^2 y - sin^2 y
= 9/13 - 4/13 = 5/13
so sin 2y = 12/13 and cos 2y = 5/13
if sin 2x = 7/25
remember cos^2 (2x) = 1 - sin^2 (2x)
= 1 - 49/625 = 576/625
then cos 2x = 24/25 , (or from right-angled triangle 7-24-25)
and cos (2x) = 1 - 2sin^2 (x)
2sin^2 x = 1 - cos2x
2sin^2 x = 1 - 576/625 = 49/625
sin^2 x = 49/1250
sin x = ±7/25√2 --->(x = 11.4° for -45°<x<45° )
sinx = 7/ 25√2
and cos^2 x = 1 - sin^2 x = 1201/1250
cos x = ± √1201/ 25√2
cosx = √1201/ 25√2
so sinx = 7/ 25√2 , cosx = √1201/ 25√2
now finally,
arhhh, you have a typo
I am sure you meant:
sin(2y - x)
= sin 2y cosx - cos 2y sinx
= (plug in my values from above)
= (12/13)(√1201/25√2) - (5/13)(7/25√2)
= (12√1201 - 35)/(325√2)
What a mess! I did not write this out on paper first, so you better check my arithmetic
I checked with my calculator and actually found the angles.
It works !!!
To find the value of sin(2y - a), we need to determine the values of y and a first. Let's break down the problem and solve it step by step.
We are given the following information:
1. sin(2x) = 7/25
2. siny = -2/√13
3. coty > 0
4. -π/4 < x < π/4
To find y, we can use the fact that sin y = opposite/hypotenuse, and since siny = -2/√13, we can write:
sin y = -2/√13
We know that sin y < 0 in the given interval (-π/4 < x < π/4), so y lies in the third or fourth quadrant where sine is negative.
We can construct a right-angled triangle in the third or fourth quadrant, where y is the angle opposite to the side -2 (opposite) and the hypotenuse is √13.
Using the Pythagorean theorem, we get:
(-2)^2 + adjacent^2 = (√13)^2
4 + adjacent^2 = 13
adjacent^2 = 13 - 4
adjacent^2 = 9
adjacent = 3
Now we can find cot y using the formula cot y = adjacent/opposite:
cot y = 3/(-2)
cot y = -3/2
Since cot y > 0, we know that y lies in the first or fourth quadrant. However, we determined earlier that y is in the third or fourth quadrant. Thus, we conclude that y is in the fourth quadrant, where cot y is positive.
Now let's find x. We know that sin(2x) = 7/25, so we can solve for x by taking the inverse sine (also known as arcsine) of (7/25) and dividing it by 2:
2x = arcsin(7/25)
x = (arcsin(7/25)) / 2
Finally, let's find a. We know that sin(2x) = 7/25, so we can rewrite it as sin(2x) = sin(a) using the identity sin(2x) = 2sin(x)cos(x):
2sin(x)cos(x) = 7/25
Since -π/4 < x < π/4, we know that x lies in the first quadrant where both sine and cosine are positive. Therefore, we can write:
2sin(x)cos(x) = 7/25
sin(x)cos(x) = (7/25)/2
sin(x)cos(x) = 7/50
Using the double-angle identity sin(2x) = 2sin(x)cos(x), we have:
sin(2x) = 7/50
Now, we can equate sin(2x) with sin(a), so we get:
sin(a) = 7/50
To find a, we take the inverse sine (arcsine) of (7/50):
a = arcsin(7/50)
Now we have y = ?, x = (arcsin(7/25)) / 2, and a = arcsin(7/50).
Finally, we can find sin(2y - a):
sin(2y - a) = sin(2(√13/13) - arcsin(7/50))
Make sure all the angles are in radians, as stated in the given range (-π/4 < x < π/4).