Angles x and y are located in the first quadrant such that sinx= 4/5 and cosy= 7/25.
determine an exact value for xin(x+Y)
If your question mean:
Determine an exact value for sin ( x + y )
then:
sin ( x + y ) = sin x ∙ cos y + cos x ∙ sin y
sin x = 4 / 5
cos x = ± √ ( 1 - sin² x )
In the first quadrant, all trigonometric functions are positive, so:
cos x = √ ( 1 - sin² x )
cos x = √ [ 1 - ( 4 / 5 )² ]
cos x = √ ( 1 - 16 / 25 )
cos x = √ ( 25 / 25 - 16 / 25 )
cos x = √ ( 9 / 25 )
cos x = √ 9 / √ 25
cos x = 3 / 5
cos y = 7 / 25
sin y = ± √ ( 1 - cos² y )
In the first quadrant, all trigonometric functions are positive, so:
sin y = √ ( 1 - cos² y )
sin y = √ [ 1 - ( 7 / 25 )² y )
sin y = √ ( 1 - 49 / 625 )
sin y = √ ( 625 / 625 - 49 / 625 )
sin y = √ ( 576 / 625 )
sin y = √ 576 / √ 625
sin y = 24 / 25
sin ( x + y ) = sin x ∙ cos y + cos x ∙ sin y
sin ( x + y ) = 4 / 5 ∙ 7 / 25 + 3 / 5 ∙ 24 / 25
sin ( x + y ) = 28 / 125 + 72 / 125
sin ( x + y ) = 100 / 125
sin ( x + y ) = 25 ∙ 4 / 25 ∙ 5
sin ( x + y ) = 4 / 5
Well, well, well! Let's find the value of sin(x + y), shall we?
Now, since sin(x) = 4/5 and cos(y) = 7/25, we need to find cos(x) and sin(y). But don't worry, I got you covered!
We know that sin^2(x) + cos^2(x) = 1. So, if we substitute sin(x) = 4/5, we get: (4/5)^2 + cos^2(x) = 1. Solving for cos(x), we find cos(x) = 3/5.
Similarly, using the same trigonometric identity, we have sin^2(y) + cos^2(y) = 1. Substituting cos(y) = 7/25, we get: sin^2(y) + (7/25)^2 = 1. Solving for sin(y), we find sin(y) = 24/25.
Now, we can find sin(x + y) using the sum formula for sine: sin(x + y) = sin(x)cos(y) + cos(x)sin(y). Substituting the values we found, we get: sin(x + y) = (4/5)(7/25) + (3/5)(24/25).
Simplifying this expression, we have sin(x + y) = 28/125 + 72/125. Adding these fractions, we get sin(x + y) = 100/125.
But we're not done yet! We want an exact value for sin(x + y), so we need to simplify further. And what do you know, 100/125 can be reduced! Dividing both numbers by 25, we get sin(x + y) = 4/5.
So, the exact value of sin(x + y) is 4/5. I hope this answer puts a smile on your face!
To determine an exact value for sin(x + y), we can use the following trigonometric identity:
sin(x + y) = sin(x)cos(y) + cos(x)sin(y)
Given that sin(x) = 4/5 and cos(y) = 7/25, we need to find the value of cos(x) and sin(y) to substitute into the formula.
To find cos(x), we can use the Pythagorean identity:
sin^2(x) + cos^2(x) = 1
We know that sin(x) = 4/5, so we can substitute this value into the equation:
(4/5)^2 + cos^2(x) = 1
16/25 + cos^2(x) = 1
cos^2(x) = 1 - 16/25
cos^2(x) = 9/25
cos(x) = √(9/25)
cos(x) = 3/5
Similarly, to find sin(y), we can use the Pythagorean identity:
sin^2(y) + cos^2(y) = 1
We know that cos(y) = 7/25, so we can substitute this value into the equation:
sin^2(y) + (7/25)^2 = 1
sin^2(y) + 49/625 = 1
sin^2(y) = 1 - 49/625
sin^2(y) = 576/625
sin(y) = √(576/625)
sin(y) = 24/25
Now we can substitute these values into the formula for sin(x + y):
sin(x + y) = sin(x)cos(y) + cos(x)sin(y)
sin(x + y) = (4/5)(7/25) + (3/5)(24/25)
sin(x + y) = 28/125 + 72/125
sin(x + y) = 100/125
sin(x + y) = 4/5
Therefore, the exact value of sin(x + y) is 4/5.
To find an exact value for sin(x+Y), we can use the trigonometric identity for sin(x+Y):
sin(x+Y) = sinxcosy + cosxsiny
Given that sinx = 4/5 and cosy = 7/25, we can substitute these values into the formula:
sin(x+Y) = (4/5)(7/25) + (cosx)(siny)
To find cosx and siny, we can use the Pythagorean identity:
cos^2x + sin^2x = 1
Since we know that sinx = 4/5, we can solve for cosx:
cos^2x + (4/5)^2 = 1
cos^2x + 16/25 = 1
cos^2x = 1 - 16/25
cos^2x = 25/25 - 16/25
cos^2x = 9/25
cosx = ±√(9/25)
cosx = ±3/5
Now, to find siny, we can use the same identity:
cos^2y + sin^2y = 1
Since we know that cosy = 7/25, we can solve for siny:
(7/25)^2 + sin^2y = 1
49/625 + sin^2y = 1
sin^2y = 1 - 49/625
sin^2y = 625/625 - 49/625
sin^2y = 576/625
siny = ±√(576/625)
siny = ±24/25
Now substitute the values of cosx, siny, and cosy into the formula for sin(x+Y):
sin(x+Y) = (4/5)(7/25) + (±3/5)(±24/25)
We can simplify this expression as follows:
sin(x+Y) = 28/125 + 72/125
sin(x+Y) = 100/125
Finally, simplify the fraction to get the exact value:
sin(x+Y) = 4/5
Therefore, the exact value of sin(x+Y) is 4/5.