If
sin(x) = 1/3 and sec(y) = 13/12
, where x and y lie between 0 and π/2, evaluate the expression using trigonometric identities. (Enter an exact answer.)
sin(x + y)
please correct cosx = - square 8/3
Well, well, well, we have a sin(x) and a sec(y) walking into a trigonometric expression! Let's see if we can lighten things up while evaluating the expression.
To find sin(x + y), we'll need to use the good old trigonometric identities. But don't worry, I won't make it sound boring. I promise, it'll be a "sin-sational" experience!
Let's start by finding cos(y). Since sec(y) = 13/12, we can use the identity sec(y) = 1/cos(y). So we get:
1/cos(y) = 13/12
To solve this, we just do a little "cosmic" rearranging:
cos(y) = 12/13
Great! Now, let's move on to the second step: finding sin(y). We can use the Pythagorean identity, sin^2(y) + cos^2(y) = 1, to find sin(y). I know, math can be a bit overwhelming, but it's all about finding the right "sinergy" between these trigonometric functions!
Let's substitute the values we have:
sin^2(y) + (12/13)^2 = 1
Don't worry about finding the value of sin(y). It's just a small "sin" in the equation and we'll figure it out in a moment!
Now, moving "along" to solving for sin(y):
sin^2(y) + 144/169 = 1
Subtracting 144/169 from both sides, we get:
sin^2(y) = 25/169
Finally, taking the square root of both sides, we find:
sin(y) = 5/13
Hooray, we found sin(y)!
Now, let's "summon" the final step: finding sin(x + y). Using the sum-of-angles identity, sin(x + y) = sin(x)cos(y) + cos(x)sin(y).
Since sin(x) = 1/3 and cos(y) = 12/13 (we found these values earlier), substituting them into our expression gives us:
sin(x + y) = (1/3)*(12/13) + cos(x)*(5/13)
Hmm, we need one more piece of info: cos(x). But fear not, my friend! We don't really need to know its exact value to find sin(x + y) since we have been given its range of possible values (between 0 and π/2).
So the exact value of sin(x + y), using the given trigonometric identities, is:
(1/3)*(12/13) + cos(x)*(5/13)
= 12/39 + cos(x)*(5/13)
= 4/13 + (5/13)*cos(x)
There you have it! The exact value of sin(x + y) using trigonometric identities. Don't you feel like a "trig-magician" after all that?
To evaluate the expression sin(x + y) using trigonometric identities, we can use the following identity:
sin(x + y) = sin(x)cos(y) + cos(x)sin(y)
Given that sin(x) = 1/3 and sec(y) = 13/12, we can find cos(x) and sin(y) using the Pythagorean identity.
We know that sin^2(x) + cos^2(x) = 1.
Thus, (1/3)^2 + cos^2(x) = 1.
Simplifying, (1/9) + cos^2(x) = 1.
Subtracting (1/9) from both sides, cos^2(x) = 8/9.
Taking the square root of both sides, cos(x) = √(8/9) = √8/3.
Since sec(y) = 13/12, we know that sec(y) = 1/cos(y).
Thus, 1/cos(y) = 13/12.
Cross-multiplying, 12 = 13cos(y).
Dividing both sides by 13, cos(y) = 12/13.
Now, we can substitute the values of sin(x), cos(x), and cos(y) into the identity sin(x + y) = sin(x)cos(y) + cos(x)sin(y).
sin(x + y) = (1/3)(12/13) + (√8/3)(√13/12)
= 12/39 + (√8)(√13)/(36)
= 4/13 + (√8)(√13)/(36)
Therefore, sin(x + y) = 4/13 + (√8)(√13)/(36) is the exact answer.
To evaluate the expression sin(x + y), we can use the trigonometric identity known as the sum formula for sine:
sin(x + y) = sin(x) * cos(y) + cos(x) * sin(y)
Given that sin(x) = 1/3 and sec(y) = 13/12, we need to find cos(x) and sin(y) to substitute into the sum formula.
To find cos(x), we can use the Pythagorean identity:
sin^2(x) + cos^2(x) = 1
Since we know sin(x) = 1/3, 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
Taking the square root of both sides, we get:
cos(x) = ± √(8/9)
Since x lies between 0 and π/2, the value of cos(x) must be positive. Therefore,
cos(x) = √(8/9) = 2√(2/9) = 2√2/3
Next, to find sin(y), we can use the reciprocal identity:
sec(y) = 1/cos(y)
Given sec(y) = 13/12, we can solve for cos(y):
1/cos(y) = 13/12
cos(y) = 12/13
Since sin(y) can be found using the Pythagorean identity:
sin^2(y) + cos^2(y) = 1
We can now solve for sin(y):
sin^2(y) + (12/13)^2 = 1
sin^2(y) + 144/169 = 1
sin^2(y) = 1 - 144/169
sin^2(y) = 169/169 - 144/169
sin^2(y) = 25/169
Taking the square root of both sides, we get:
sin(y) = ± √(25/169)
Since y lies between 0 and π/2, the value of sin(y) must be positive. Therefore,
sin(y) = √(25/169) = 5/13
Finally, substituting the values of sin(x), cos(x), sin(y), and cos(y) into the sum formula, we get:
sin(x + y) = (1/3) * (12/13) + (2√2/3) * (5/13)
Multiplying the fractions, we have:
sin(x + y) = 12/39 + 10√2/39
Simplifying the fraction, we get:
sin(x + y) = (12 + 10√2)/39
Therefore, the exact value of sin(x + y) is (12 + 10√2)/39.
since both x and y are in quadrant I, everything will stay positive
make sketches of right-angled triangles for each case
from sinx = 1/3, cosx = √8/3
from secy = 13/12 ---> cosy = 12/13
then siny = 5/13
sin(x+y) = sinxcosy + cosxsiny
= (1/3)(12/13) + (√8/3)(5/13)
= (12+5√8)/39 or
= (12 + 10√2)/39