Simplify arctan 1/3 + arctan 2/3(round to the nearest degree).
tan(arctan 1/3 + arctan 2/3)
= (1/3 + 2/3)/(1-(1/3)(2/3)) = 9/7
arctan(9/7) = 52.13°
To simplify the expression arctan(1/3) + arctan(2/3), we can use the identity:
arctan(x) + arctan(y) = arctan((x + y) / (1 - xy))
Let's substitute x = 1/3 and y = 2/3 into the identity:
arctan(1/3) + arctan(2/3) = arctan((1/3 + 2/3) / (1 - (1/3)(2/3)))
= arctan(3/3 / (1 - 2/9))
= arctan(3/3 / (9/9 - 2/9))
= arctan(3/3 / 7/9)
= arctan(3/3 * 9/7)
= arctan(9/7)
To find the approximate value, we can use a calculator or a table of values for the arctan function.
Using a calculator, we find that arctan(9/7) is approximately 55.66 degrees.
Therefore, arctan(1/3) + arctan(2/3) is approximately 55.66 degrees (rounded to the nearest degree).
To simplify the expression arctan(1/3) + arctan(2/3) and round to the nearest degree, we can use the trigonometric identity for the sum of two angles.
The identity states that arctan(a) + arctan(b) = arctan((a + b) / (1 - ab))
In this case, a = 1/3 and b = 2/3. Plugging these values into the identity gives:
arctan(1/3) + arctan(2/3) = arctan((1/3 + 2/3) / (1 - (1/3)(2/3)))
Simplifying further:
arctan(1/3) + arctan(2/3) = arctan(1) = 45 degrees
Therefore, the simplified value of arctan(1/3) + arctan(2/3) rounded to the nearest degree is 45 degrees.