Given sin (x-y)=1/4, sin x cos y=3/5,and tan y=3/2,without using mathematical table or scientific calculator,find the values for:
(a)cos x sin y
(b)cos (x+y)
a) sin(x-y) = sinxcosy - cosxsiny
1/4 = 3/5 - cosxsiny
cosxsiny = 3/5 - 1/4 = 7/20
So now we know sinxcosy= 3/5 and cosxsiny = 7/20
form (cosxsiny)/sinxcosy) = (7/20)/(3/5)
cotx tany = 7/12
cotx (3/5) = 7/12
cotx = 35/36
tanx = 36/35
so sinx = 36/√2521
cosx = 35/√2521
from tany = 3/5
siny = 3/√13
cosy = 2/√13
cos(x+y) = cosxcosy - sinxsiny
= (35/√2521)(2/√13) - (36/√2521)(3/√13)
I need somebody to check my arithmetic because if I use my calculator to find the actual angles, I get contradictory results.
I could have made an arithmetic error, (happens lol)
I have a feeling that for the first 2 pieces of information, the third part, namely that tany = 3/5, is not possible.
To find the values of cos x sin y and cos (x+y), we can use the given trigonometric equations and some trigonometric identities:
(a) Finding cos x sin y:
From the equation sin x cos y = 3/5, we can solve for sin y by dividing both sides of the equation by cos y:
sin y = (3/5) / cos y = 3/5cos y.
Now, we are given that sin (x-y) = 1/4. We can use the difference of angles formula for sine to expand the equation:
sin x cos y - cos x sin y = 1/4.
Substituting the values we found:
(3/5cos y) cos y - cos x (3/5cos y) = 1/4.
Expanding and rearranging the equation:
(3/5)cos^2(y) - (3/5)cos x cos y = 1/4.
Since cos^2(y) = 1 - sin^2(y), we can substitute this expression into the equation:
(3/5)(1 - sin^2(y)) - (3/5)cos x cos y = 1/4.
Expanding and rearranging again:
3/5 - (3/5)sin^2(y) - (3/5)cos x cos y = 1/4.
Now, we have an equation in terms of sin^2(y) and cos x cos y. However, we need to find the value of cos x sin y. Let's look at the given equation tan y = 3/2.
(b) Finding cos (x+y):
We can use the given equation tan y = 3/2 to find the value of sin y and cos y. Recall that tan y = sin y / cos y.
Substituting the value of tan y:
(3/2) = sin y / cos y.
Cross-multiplying and rearranging:
2sin y = 3cos y.
Dividing both sides by 2cos y:
sin y / cos y = 3/2.
Comparing this equation with our original equation sin x cos y = 3/5, we can see that sin y / cos y = sin x / cos x.
Since sin y / cos y = sin x / cos x, we can deduce that cos x sin y = cos y sin x.
Now, let's revisit equation (a):
3/5 - (3/5)sin^2(y) - (3/5)cos x cos y = 1/4.
Since cos x sin y = cos y sin x, we can substitute sin x for cos y in the equation:
3/5 - (3/5)sin^2(y) - (3/5)(cos x sin y) = 1/4.
Now, we can substitute the value of cos x sin y:
3/5 - (3/5)sin^2(y) - (3/5)(cos y sin x) = 1/4.
Lastly, we need to solve this equation to find the value of cos x sin y. However, without additional information or restrictions on the values of x and y, we cannot determine an exact value. We can only express it in terms of x and y.