Find the exact value of:
cos(arctan(2) + arctan(3))
Please explain how you got your answer. I have the answer, but I don't know how to get to it.
this resembles the cos(A+B) expansion
cos(A+B) = cosAcosB - sinAsinB
so cos(arctan(2) + arctan(3))
= cos(arctan2)cos(arctan3) - sin(arctan2)sin(arctan3)
let's look at arctan2,
draw right-angled triangle in the first quadrant, with a base (adjacent) of 1 and a height (opposite) of 2
Then the hypotenuse is √5
so the cos(angle) = 1/√5 and sin(angle) = 2/√5
sofar you have
(1/√5)cos(arctan3) - (2/√5)sin(arctan3)
then for arctan3, again draw a right-angled triangle but this time the height is 3, so the hypotenuse is √10
then the sine of that angle is 3√10 and cosine of that angle is 1/√10
so your expression of cos(arctan2)cos(arctan3) - sin(arctan2)sin(arctan3)
= (1/√5)(1/√10) - (2/√5)(3/√10)
= 1/√50 - 6/√50
= -5/√50
= -5/(5√2)
= -1/√2 or -√2/2 after rationalizing
BTW, we could check this:
arctan2 + arctan3 = 135º
and cos 135º = -.707.. = - √2/2
thanks!
To find the exact value of cos(arctan(2) + arctan(3)), we can use the trigonometric identity for the sum of angles:
cos(A + B) = cos(A) * cos(B) - sin(A) * sin(B)
In this case, let A = arctan(2) and B = arctan(3). So, we have:
cos(arctan(2) + arctan(3)) = cos(arctan(2)) * cos(arctan(3)) - sin(arctan(2)) * sin(arctan(3))
Now, we need to find the values of cos(arctan(2)) and sin(arctan(2)).
To do this, we draw a right triangle and set one of the acute angles as arctan(2). Let's call the opposite side length 'x' and the adjacent side length '1'. According to the definition of arctan, we know that tan(arctan(2)) = 2. Therefore, we have the following:
tan(arctan(2)) = opposite/adjacent
2 = x/1
x = 2
Using the Pythagorean theorem, we can find the hypotenuse:
hypotenuse = √(opposite^2 + adjacent^2)
= √(2^2 + 1^2)
= √5
Now, we can find sine and cosine functions of arctan(2):
sin(arctan(2)) = opposite/hypotenuse
= 2/√5
= 2√5/5
cos(arctan(2)) = adjacent/hypotenuse
= 1/√5
= √5/5
Similarly, we can find the following values for arctan(3):
sin(arctan(3)) = 3/√(3^2 + 1^2)
= 3/√10
cos(arctan(3)) = 1/√(3^2 + 1^2)
= 1/√10
Now, let's substitute these values back into the original equation:
cos(arctan(2) + arctan(3)) = cos(arctan(2)) * cos(arctan(3)) - sin(arctan(2)) * sin(arctan(3))
= (√5/5) * (1/√10) - (2√5/5) * (3/√10)
= (√5 * 1 * 1)/(5 * √10) - (2 * 3 * √5)/(5 * √10)
= (√5)/(5√10) - (6√5)/(5√10)
= (√5 - 6√5)/(5√10)
= -5√5/(5√10)
= -√5/√10
= -√(5/10)
= -√(1/2)
= -1/√2
= -√2/2
= -0.70710678118 (approx.)
Therefore, the exact value of cos(arctan(2) + arctan(3)) is approximately -0.70710678118.
To find the exact value of cos(arctan(2) + arctan(3)), we need to use the trigonometric identity:
cos(A + B) = cos(A) * cos(B) - sin(A) * sin(B)
Let's simplify step by step:
Step 1: Let's find the values of arctan(2) and arctan(3) separately.
Since arctan(2) represents the angle whose tangent is 2, we can express it as:
arctan(2) = θ, where tan(θ) = 2
From this, we can form a right triangle with the opposite side as 2 and the adjacent side as 1. Using the Pythagorean theorem, we can find the hypotenuse:
h^2 = 1^2 + 2^2
h^2 = 1 + 4
h^2 = 5
h = sqrt(5)
So, in this triangle, the opposite side is 2, adjacent side is 1, and the hypotenuse is sqrt(5).
Similarly, for arctan(3), we can form a right triangle with the opposite side as 3 and adjacent side as 1. Using the Pythagorean theorem, we find the hypotenuse:
h^2 = 1^2 + 3^2
h^2 = 1 + 9
h^2 = 10
h = sqrt(10)
Step 2: Now, we need to find the values of cos(arctan(2)) and cos(arctan(3)).
Using the triangle formed earlier, we can find cos(arctan(2)) as:
cos(arctan(2)) = adjacent side / hypotenuse
cos(arctan(2)) = 1 / sqrt(5)
To rationalize the denominator, multiply both the numerator and denominator by sqrt(5):
cos(arctan(2)) = (1 * sqrt(5)) / (sqrt(5) * sqrt(5))
cos(arctan(2)) = sqrt(5) / 5
Similarly, for cos(arctan(3)):
cos(arctan(3)) = adjacent side / hypotenuse
cos(arctan(3)) = 1 / sqrt(10)
To rationalize the denominator, multiply both the numerator and denominator by sqrt(10):
cos(arctan(3)) = (1 * sqrt(10)) / (sqrt(10) * sqrt(10))
cos(arctan(3)) = sqrt(10) / 10
Step 3: Finally, substitute the values of cos(arctan(2)) and cos(arctan(3)) into the trigonometric identity:
cos(arctan(2) + arctan(3)) = cos(arctan(2)) * cos(arctan(3)) - sin(arctan(2)) * sin(arctan(3))
cos(arctan(2) + arctan(3)) = (sqrt(5) / 5) * (sqrt(10) / 10) - (2 / sqrt(5)) * (3 / sqrt(10))
Now, simplify the expression further if possible.