Find a simplified expression for cos(tan ^(-1)(x/5))
Ah, I see you're playing with trigonometric acrobatics! Well, the expression you're looking for is cos(tan^(-1)(x/5)). Let's dive into the circus of math and simplify it together.
To start, let's imagine a right triangle with an angle A, where the opposite side has length x, and the adjacent side has length 5. We can find the hypotenuse H using the Pythagorean theorem.
Now, the tangent of angle A is equal to the opposite side divided by the adjacent side, which gives us x/5. Taking the inverse tangent of both sides, we get tan^(-1)(x/5) = A.
Alright, time to bring in the clown-car of trig identities. Remember the phrase "SOHCAHTOA?" Well, it will come in handy for this. Looking at the triangle, we have sin(A) = x/H, and cos(A) = 5/H.
Let's substitute these values back into the original expression. We get cos(tan^(-1)(x/5)) = cos(A).
Since we know that cos(A) = 5/H, we can simplify further. Substituting H with √(x^2 + 25) (from the Pythagorean theorem), we have cos(tan^(-1)(x/5)) = 5/√(x^2 + 25).
And voila! We have reached our final, simplified expression: cos(tan^(-1)(x/5)) = 5/√(x^2 + 25). Enjoy the trapeze act of mathematics!
To find a simplified expression for cos(tan^(-1)(x/5)), we can use trigonometric identities.
Let's start by using the identity tan^(-1)(a) = sin^(-1)(a/sqrt(1+a^2)).
So, tan^(-1)(x/5) = sin^(-1)(x/5√(1+(x/5)^2)).
Now, let's use another identity, sin^(-1)(a) = cos^(-1)(sqrt(1-a^2)).
Therefore, sin^(-1)(x/5√(1+(x/5)^2)) = cos^(-1)(√(1-(x/5√(1+(x/5)^2))^2)).
Now, let's simplify further.
√(1-(x/5√(1+(x/5)^2))^2) = √(1-x^2/(25(1+(x/5)^2))).
Simplifying the denominator, 25(1+(x/5)^2) = 25 + x^2.
Therefore, √(1-x^2/(25(1+(x/5)^2))) = √(1-x^2/(25+x^2)).
Finally, we can write the simplified expression as cos(tan^(-1)(x/5)) = √(1-x^2/(25+x^2)).
To find a simplified expression for cos(tan^(-1)(x/5)), we can use the trigonometric identity involving the tangent and cosine functions.
The identity is: cos(tan^(-1)(y)) = 1 / √(1 + y^2)
In this case, y = x/5. So we substitute it into the identity:
cos(tan^(-1)(x/5)) = 1 / √(1 + (x/5)^2)
Now, let's simplify further:
To simplify the expression, we need to rationalize the denominator by multiplying both the numerator and denominator by the conjugate of the denominator.
1 / √(1 + (x/5)^2) = 1 / √(1 + x^2/25)
Multiplying numerator and denominator by the conjugate (√(1 + x^2/25)), we get:
1 / √(1 + x^2/25) = (√(1 + x^2/25)) / ((√(1 + x^2/25)) * (√(1 + x^2/25)))
Expanding the denominator:
(√(1 + x^2/25)) / ((√(1 + x^2/25)) * (√(1 + x^2/25))) = (√(1 + x^2/25)) / (√(1 + x^2/25))^2
Simplifying the denominator:
(√(1 + x^2/25)) / (√(1 + x^2/25))^2 = (√(1 + x^2/25)) / (1 + x^2/25)
Finally, simplifying the expression:
√(1 + x^2/25) / (1 + x^2/25)
This is the simplified expression for cos(tan^(-1)(x/5)).