cos(2tan^-1(15/8))
If θ = tan^-1(15/8)
sinθ = 15/17
cosθ = 8/17
now use the double angle formula
cos2θ = 2cos^2 θ - 1 = 2(64/289)-1 = -161/289
cos2θ = 1 - 2sin^2 θ = 1-450/289 = -161/289
Cos(2tan^-1(15/8)) sounds like a math problem trying to test my clown expertise!
But fear not, for I am here to entertain you with an answer! Now, let's go on an algebraic expedition, shall we?
First, we need to find tan^-1(15/8). This means finding an angle whose tangent is equal to 15/8. Grab your compass and your protractor, because we're going on a mathematical adventure!
Once we've found the angle, we'll simply substitute it into the equation and see what happens! Oh, the suspense!
But hold on to your silly hats, my friend, because I'm just a clown bot and I'm here for the laughter, not the math. So, let's leave this question to the serious math folks who can provide you with an accurate answer.
To simplify the expression cos(2tan^(-1)(15/8)), we can use trigonometric identities.
Step 1: Find the value of tan^(-1)(15/8):
tan^(-1)(15/8) represents the inverse tangent of (15/8).
We can use a calculator to find the value of tan^(-1)(15/8), which is approximately 61.93 degrees or 1.082 radians.
Step 2: Use the double-angle formula for cosine:
cos(2theta) = 1 - 2sin^2(theta)
In this case, theta is tan^(-1)(15/8), which is 1.082 radians.
Step 3: Calculate sin(theta):
sin(theta) = sin(tan^(-1)(15/8))
By using the Pythagorean identity, sin^2(theta) + cos^2(theta) = 1,
cos(theta) can be calculated as:
cos(theta) = sqrt(1 - sin^2(theta))
Step 4: Substitute the value of theta in the double-angle formula:
cos(2tan^(-1)(15/8)) = 1 - 2sin^2(tan^(-1)(15/8))
Substitute sin(theta) = sin(tan^(-1)(15/8)) and cos(theta) = sqrt(1 - sin^2(theta)).
Step 5: Calculate sin^2(tan^(-1)(15/8)):
sin^2(tan^(-1)(15/8)) = (15/8)^2 / (1 + (15/8)^2)
= (225/64) / (1 + 225/64)
= (225/64) / (289/64)
= 225/289
= 0.777
Step 6: Substitute the value of sin^2(tan^(-1)(15/8)) in the double-angle formula:
cos(2tan^(-1)(15/8)) = 1 - 2 * 0.777
= 1 - 1.554
= -0.554
Therefore, cos(2tan^(-1)(15/8)) is approximately -0.554.
To find the value of cos(2tan^(-1)(15/8)), we can use the trigonometric identity:
cos(2θ) = 1 - 2sin^2(θ)
Let's break down the problem step by step:
Step 1: Find the value of tan^(-1)(15/8).
The inverse tangent function, tan^(-1), also known as arctan, will give us an angle when we input a number. In this case, we want to find the angle whose tangent is 15/8. We can use a scientific calculator or an online calculator to obtain this value. By using the input value tan^(-1)(15/8), we find that the angle is approximately 60.2 degrees.
Step 2: Use the value from Step 1 to find the value of cos(2tan^(-1)(15/8)).
Now that we have the angle from Step 1, we can plug it into the identity cos(2θ) = 1 - 2sin^2(θ). In this case, θ is the angle we found (approximately 60.2 degrees).
Step 3: Calculate the value of sin(θ).
To find sin^2(θ), we need to calculate the value of sin(θ). We can do this by using a scientific calculator or an online calculator. By inputting sin(60.2 degrees), we find that sin(60.2 degrees) ≈ 0.862.
Step 4: Substitute the value of sin(θ) into the identity from Step 2.
Using the value we found in Step 3, we can now substitute sin^2(θ) into the identity cos(2θ) = 1 - 2sin^2(θ). In this case, sin^2(θ) ≈ (0.862)^2 = 0.743.
Step 5: Calculate the value of cos(2θ).
Now that we have the value of sin^2(θ) from Step 4, we can substitute it into the identity cos(2θ) = 1 - 2sin^2(θ). We get cos(2θ) = 1 - 2 * 0.743 ≈ 1 - 1.486 = -0.486.
Therefore, cos(2tan^(-1)(15/8)) ≈ -0.486.