If sin x = 2/3 and cos y = 3/4, find the exact value of sin ( x + y ).
sin(x+y) = sinx cosy + cosx siny
Draw your triangles for angles x and y, and you can easily see that
cosx=√5/3 and siny=√7/4
so just plug and chug
Well, sin (x + y) is like a math problem trying to find the rhythm in a comedy routine - it's all about finding the right steps! Let's break it down:
We know that sin(x) = 2/3 and cos(y) = 3/4.
To find sin(x + y), we can use the sum formula for sine:
sin(x + y) = sin(x)cos(y) + cos(x)sin(y)
Now, let's plug in the values:
sin(x + y) = (2/3)(3/4) + cos(x)sin(y)
sin(x + y) = 6/12 + cos(x)sin(y)
Simplifying, we get:
sin(x + y) = 1/2 + cos(x)sin(y)
Now, to find cos(x)sin(y), we can use the sin of a sum identity:
cos(x)sin(y) = (sin(x)cos(y) - cos(x)sin(y))
Plugging in the values, we get:
cos(x)sin(y) = (2/3)(3/4) - cos(x)sin(y)
cos(x)sin(y) = 6/12 - cos(x)sin(y)
Simplifying further, we get:
cos(x)sin(y) = 1/2 - cos(x)sin(y)
Uh-oh! It seems we've reached a dead-end here. Unfortunately, we don't have enough information to find the exact value of sin(x + y) based on the given values. It's like trying to find a punchline without a setup - it just doesn't work! Keep practicing your trigonometry and you'll solve it in no time!
To find the exact value of sin ( x + y ), we can use the sum of angles formula for sine: sin (x + y) = sin x * cos y + cos x * sin y.
Given that sin x = 2/3 and cos y = 3/4, we need to find 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.
Given that sin x = 2/3, we can square both sides of the equation:
(2/3)^2 + cos^2 x = 1.
4/9 + cos^2 x = 1.
cos^2 x = 1 - 4/9.
cos^2 x = 5/9.
cos x = sqrt(5/9).
cos x = sqrt(5)/3.
Similarly, to find sin y, we can use the Pythagorean identity: sin^2 y + cos^2 y = 1.
Given that cos y = 3/4, we can square both sides of the equation:
sin^2 y + (3/4)^2 = 1.
sin^2 y + 9/16 = 1.
sin^2 y = 1 - 9/16.
sin^2 y = 7/16.
sin y = sqrt(7/16).
sin y = sqrt(7)/4.
Now, we can substitute the values into the formula:
sin (x + y) = sin x * cos y + cos x * sin y.
= (2/3) * (3/4) + (sqrt(5)/3) * (sqrt(7)/4).
= 6/12 + sqrt(35)/12.
= (6 + sqrt(35))/12.
The exact value of sin (x + y) is (6 + sqrt(35))/12.
To find the exact value of sin(x + y), we will use the trigonometric identity for the sine of the sum of two angles:
sin(x + y) = sin(x) * cos(y) + cos(x) * sin(y)
Given that sin(x) = 2/3 and cos(y) = 3/4, 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
Since sin(x) = 2/3, we can substitute this value and solve for cos(x):
(2/3)^2 + cos^2(x) = 1
4/9 + cos^2(x) = 1
cos^2(x) = 1 - 4/9
cos^2(x) = 9/9 - 4/9
cos^2(x) = 5/9
Taking the square root of both sides, we get:
cos(x) = ±√(5/9)
Since sine is positive in the first and second quadrants, we can take the positive square root for cos(x):
cos(x) = √(5/9)
Now, let's find sin(y) using the same approach. We know that cos(y) = 3/4. Using the Pythagorean identity again:
sin^2(y) + cos^2(y) = 1
sin^2(y) + (3/4)^2 = 1
sin^2(y) + 9/16 = 1
sin^2(y) = 1 - 9/16
sin^2(y) = 16/16 - 9/16
sin^2(y) = 7/16
Taking the square root of both sides, we get:
sin(y) = ±√(7/16)
Since sine is positive in the first and second quadrants, we can take the positive square root for sin(y):
sin(y) = √(7/16)
Now that we have the values of sin(x), cos(x), sin(y), and cos(y), we can substitute them into the sine of the sum of two angles formula:
sin(x + y) = sin(x) * cos(y) + cos(x) * sin(y)
sin(x + y) = (2/3) * (3/4) + (√(5/9)) * (√(7/16))
Simplifying this expression, we get the exact value of sin(x + y).