Simplify by referring to the appropriate triangle or trigonometric identity:
cos(arctan(x))
sketch a right-angled triangle with some angle Ø
, label the opposite side x and the adjacent side 1
(this way tanØ = x/1 , and Ø = arctan(x/1)
the hypotenuse would be √(x^2 + 1)
and
cosØ = 1/√(x^2+1)
Ah, the tangled web of trigonometry! Well, let's try to simplify this expression. We know that arctan(x) gives us the angle whose tangent is x. So, in a right triangle, let's call that angle theta. Now, cos(theta) is defined as the ratio of the adjacent side to the hypotenuse.
Now, let's label the sides of our triangle. The opposite side will be x, the adjacent side will be 1 (since tangent is x), and the hypotenuse will be...well, let's call it h, for hypotenuse.
Now, we can use the Pythagorean theorem to find h. So, we have x^2 + 1^2 = h^2, which gives us h = sqrt(x^2 + 1).
Therefore, cos(arctan(x)) is equal to the adjacent side divided by the hypotenuse, which simplifies to 1/sqrt(x^2 + 1).
Ta-da! We've simplified it using the appropriate triangle.
To simplify the expression cos(arctan(x)), we can refer to the appropriate triangle or trigonometric identity.
Let's start by considering a right triangle with an acute angle θ, where the opposite side is x and the adjacent side is 1. By definition, the tangent of θ is equal to the ratio of the opposite side to the adjacent side:
tan(θ) = x/1 = x
We can rewrite this equation as:
θ = arctan(x)
The cosine of θ can be found by using the Pythagorean identity:
cos^2(θ) = 1 - sin^2(θ)
Since we know the tangent of θ is x, we can find the sine of θ using the Pythagorean identity:
sin^2(θ) = 1 - cos^2(θ)
Substituting the value of cos^2(θ) from the first equation into the second equation:
sin^2(θ) = 1 - (1 - sin^2(θ))
Expanding the equation:
sin^2(θ) = 1 - 1 + sin^2(θ)
Simplifying the equation:
sin^2(θ) = sin^2(θ)
Taking the square root of both sides:
sin(θ) = ± sin(θ)
Since θ is an acute angle, the sine of θ is always positive, so we can take the positive root:
sin(θ) = sin(θ)
Now, we can use the definition of cosine:
cos(θ) = √(1 - sin^2(θ))
Substituting the value of sin(θ) from above:
cos(θ) = √(1 - sin^2(θ)) = √(1 - sin^2(θ)) = ±√cos^2(θ)
Since the cosine of an acute angle is always positive, we can take the positive square root:
cos(θ) = √cos^2(θ) = cos(θ)
Therefore, we can simplify cos(arctan(x)) to just cos(θ) or cos(arctan(x)) = cos(θ).
To simplify the expression cos(arctan(x)), we can use the trigonometric identity:
cos(arctan(x)) = 1 / sqrt(1 + x^2)
Explanation:
The arctan function, which stands for the inverse tangent, returns an angle whose tangent is equal to x. So when we take the arctan(x), we are finding an angle whose tangent is x.
Now, let's consider a right triangle with an angle θ and sides adjacent (A) and opposite (O) to the angle θ. The tangent of θ is defined as the ratio of the length of the opposite side to the length of the adjacent side, i.e., tan(θ) = O / A.
In our case, if we take arctan(x), it means that tan(arctan(x)) = x. This implies that we have a right triangle with an angle arctan(x), and x is the ratio of the length of the opposite side to the length of the adjacent side.
Now let's apply the Pythagorean theorem. According to the theorem, in a right triangle, the square of the hypotenuse (H) is equal to the sum of the squares of the other two sides:
H^2 = A^2 + O^2
Since the tangent of θ is equal to x, we can rewrite the ratio as O / A = x. This means that O = x and A = 1. Substituting these values into the Pythagorean theorem, we have:
H^2 = 1^2 + x^2
H^2 = 1 + x^2
Now, the cosine of the angle θ is defined as the ratio of the length of the adjacent side to the hypotenuse, i.e., cos(θ) = A / H. In our case, θ is arctan(x), so the cosine of the angle arctan(x) is:
cos(arctan(x)) = A / H = 1 / sqrt(1 + x^2)
Therefore, the simplified expression for cos(arctan(x)) is 1 / sqrt(1 + x^2).