If
sin(x) = 1/3 and sec(y) = 29/21
, where x and y lie between 0 and π/2, evaluate the expression using trigonometric identities. (Enter an exact answer.)
cos(2y)
Well, well, well! It seems like you've stumbled upon a trigonometric party. Let's put on our clown shoes and get cracking!
First things first, let's find the value of cos(y) using the identity cos^2(y) = 1 - sin^2(y). Since you already know sin(x) = 1/3, we can solve for cos(y):
cos^2(y) = 1 - (1/3)^2
cos^2(y) = 1 - 1/9
cos^2(y) = 8/9
Now, let's find the value of sin(y) using the identity sin^2(y) = 1 - cos^2(y):
sin^2(y) = 1 - 8/9
sin^2(y) = 1/9
Since sec(y) = 29/21, we know that sec^2(y) = 1 + tan^2(y). Substituting sec^2(y) with its value:
(29/21)^2 = 1 + tan^2(y)
841/441 = 1 + tan^2(y)
tan^2(y) = 841/441 - 1
tan^2(y) = 841/441 - 441/441
tan^2(y) = 400/441
Now, let's find the value of cos(2y) using the identity cos(2y) = cos^2(y) - sin^2(y):
cos(2y) = cos^2(y) - sin^2(y)
cos(2y) = 8/9 - 1/9
cos(2y) = 7/9
So, the exact value of cos(2y) is 7/9. Have a giggly day!
To evaluate the expression cos(2y), we need to use the double angle identity for cosine, which is:
cos(2y) = 1 - 2sin^2(y)
Given that sec(y) = 29/21, we can use the definition of secant to find the value of cosine:
sec(y) = 1/cos(y)
Rewriting this equation, we get:
cos(y) = 1/sec(y)
Plugging in the given value of sec(y) = 29/21:
cos(y) = 1/(29/21)
= 21/29
Now, we can substitute the value of sin(x) = 1/3 into the double angle identity:
cos(2y) = 1 - 2sin^2(y)
= 1 - 2(sin(y))^2
= 1 - 2(1 - cos^2(y)) [Using the Pythagorean identity: sin^2(y) + cos^2(y) = 1]
= 1 - 2(1 - (21/29)^2)
Now we can evaluate the expression cos(2y), using the given values:
cos(2y) = 1 - 2(1 - (21/29)^2)
= 1 - 2(1 - (441/841))
= 1 - 2(1 - 441/841)
= 1 - 2(400/841)
= 1 - 800/841
= (841 - 800)/841
= 41/841
Therefore, cos(2y) = 41/841.
To evaluate the expression cos(2y), we can use the double-angle identity for cosine, which states that:
cos(2θ) = cos²(θ) - sin²(θ)
Since we are given the value of sec(y), we can use the identity: sec²(y) = 1 + tan²(y) to find the value of tan(y) first.
Given that sec(y) = 29/21, we know that:
sec²(y) = (29/21)²
Now, by rearranging the equation, we can solve for tan²(y):
tan²(y) = sec²(y) - 1
Substituting the value of sec²(y), we have:
tan²(y) = (29/21)² - 1
Next, we can find the value of tan(y) by taking the square root:
tan(y) = √[(29/21)² - 1]
Now that we know the value of sin(x), we can apply the Pythagorean identity to find cos(x):
sin²(x) + cos²(x) = 1
Given that sin(x) = 1/3, we have:
(1/3)² + cos²(x) = 1
Simplifying the equation, we have:
1/9 + cos²(x) = 1
Subtracting 1/9 from both sides, we get:
cos²(x) = 8/9
Now, we can find the value of cos(x) by taking the square root:
cos(x) = ±√(8/9)
However, we know that x lies between 0 and π/2, which means x is positive. Hence,
cos(x) = √(8/9)
Finally, we can evaluate the expression cos(2y) using the double-angle identity:
cos(2y) = cos²(y) - sin²(y)
Substituting the known values, we have:
cos(2y) = (cos(y))² - (sin(y))²
cos(2y) = (√(8/9))² - (tan(y))²
cos(2y) = 8/9 - tan²(y)
Now, we substitute the value of tan²(y) we calculated earlier:
cos(2y) = 8/9 - [(29/21)² - 1]
cos(2y) = 8/9 - [841/441 - 1]
cos(2y) = 8/9 - (841/441 - 441/441)
cos(2y) = 8/9 - (400)*(-1/441)
cos(2y) = 8/9 - 400/441
To get the exact answer, we can get the common denominator of 9 for both fractions:
cos(2y) = (8*49 - 400)/441
cos(2y) = (392 - 400)/441
cos(2y) = -8/441
Therefore, the exact value of cos(2y) is -8/441.
cos(y) = 21/29
cos(2y) = 2cos^2(y) - 1 = 2(21/29)^2 - 1 = 41/841