If sin(x) = 1/3 and sec(y) = 5/4, where x and y lie between 0 and π/2, evaluate sin(x + y).
Draw your triangles.
cosx = √8/3
cosy = 4/5, so siny = 3/5
Now just plug in your sum-of- angles formula.
To evaluate sin(x + y), we need to find the values of x and y. We are given that sin(x) = 1/3 and sec(y) = 5/4. To find the values of x and y, we can use inverse trigonometric functions.
1. Finding x:
Since sin(x) = 1/3, we can use the arcsin (inverse sine) function to find the value of x. Taking the arcsin of both sides of the equation:
arcsin(sin(x)) = arcsin(1/3)
x = arcsin(1/3)
2. Finding y:
Since sec(y) = 5/4, we can use the arcsec (inverse secant) function to find the value of y. However, the arcsec function is not commonly used, so we will use the reciprocal identity for secant:
sec(y) = 5/4
cos(y) = 4/5
Now, taking the arccos (inverse cosine) of both sides of the equation:
arccos(cos(y)) = arccos(4/5)
y = arccos(4/5)
Now that we have the values of x and y, we can find sin(x + y):
sin(x + y) = sin(arcsin(1/3) + arccos(4/5))
To evaluate this expression further, we can use the sum formula for sine:
sin(x + y) = sin(x)cos(y) + cos(x)sin(y)
We already know that sin(x) = 1/3 and cos(y) = 4/5 from our earlier calculations. To find cos(x) and sin(y), we can use trigonometric identities.
3. Finding cos(x):
We know that sin(x) = 1/3. Using the Pythagorean identity, sin^2(x) + cos^2(x) = 1, we can solve for cos(x):
(1/3)^2 + cos^2(x) = 1
1/9 + cos^2(x) = 1
cos^2(x) = 1 - 1/9
cos^2(x) = 8/9
cos(x) = √(8/9)
cos(x) = √(8)/√(9)
cos(x) = (√(8))/3
4. Finding sin(y):
We know that cos(y) = 4/5. Using the Pythagorean identity, sin^2(y) + cos^2(y) = 1, we can solve for sin(y):
sin^2(y) + (4/5)^2 = 1
sin^2(y) + 16/25 = 1
sin^2(y) = 1 - 16/25
sin^2(y) = 9/25
sin(y) = √(9/25)
sin(y) = √(9)/√(25)
sin(y) = 3/5
Now, substituting the values we found into the sum formula for sine, we can calculate sin(x + y):
sin(x + y) = (1/3)(4/5) + (√(8)/3)(3/5)
sin(x + y) = 4/15 + √(8)/5
So, sin(x + y) = 4/15 + √(8)/5.
To evaluate sin(x + y), we'll use the following trigonometric identity:
sin(x + y) = sin(x)cos(y) + cos(x)sin(y)
Given that sin(x) = 1/3 and sec(y) = 5/4, we can determine the values of cos(x) and sin(y) using their reciprocal trigonometric identities.
Reciprocal Identity for Sin(x):
sin^2(x) + cos^2(x) = 1
Given sin(x) = 1/3, we can find cos(x) as follows:
(1/3)^2 + cos^2(x) = 1
1/9 + cos^2(x) = 1
cos^2(x) = 1 - 1/9
cos^2(x) = 8/9
cos(x) = ±√(8/9)
Since x lies between 0 and π/2, cos(x) must also be positive:
cos(x) = √(8/9)
Reciprocal Identity for Sec(y):
sec(y) = 1/cos(y)
Given sec(y) = 5/4, we can find cos(y) as follows:
1/cos(y) = 5/4
cos(y) = 4/5
Now we have all the required values to evaluate sin(x + y):
sin(x + y) = sin(x)cos(y) + cos(x)sin(y)
= (1/3)(4/5) + (√(8/9))(sin(y))
= 4/15 + (√(8/9))(sin(y))
To determine the value of sin(y), we'll use the identity:
sin^2(y) + cos^2(y) = 1
Since cos(y) = 4/5, we have:
sin^2(y) + (4/5)^2 = 1
sin^2(y) + 16/25 = 1
sin^2(y) = 1 - 16/25
sin^2(y) = 9/25
sin(y) = ±3/5
Since y lies between 0 and π/2, sin(y) must also be positive:
sin(y) = 3/5
Plugging in the values, we have:
sin(x + y) = 4/15 + (√(8/9))(3/5)
= 4/15 + 3√(8/9)/5
To simplify further, let's rationalize the denominator of the second term:
sin(x + y) = 4/15 + 3√(8/9)/5
= 4/15 + 3√(8/9) * (√9/√9) / 5
= 4/15 + 3√(72/81) / 5
= 4/15 + 3√(8/9 * 9/9) / 5
= 4/15 + 3√(8/9) / √(25/9)
= 4/15 + 3√(8/9) / (5/3)
= 4/15 + 3/5 * √(8/9)
= (4/15 + 3/5 * √(8/9)) * (3/3)
= (12/45 + 9/15 * √(8/9)) / 3
= (12/45 + 9/15 * √(8/9)) / 3
= (12 + 27/15 * √(8/9)) / 45
= (12 + 9/5 * √(8/9)) / 45
= (60 + 9√(8/9)) / 45
= (20 + 3√8) / 15
Therefore, sin(x + y) = (20 + 3√8) / 15.