Arrange these in order from least to greatest:
arctan(-sqrt3), arctan 0, arctan(1/2)
So far I got the first two values, arctan(-sqrt3), and that's 150 degrees. Arctan 0 would be zero degrees. I'll use just one answer for now, but I know there's more than one.
The last one I find a bit tricky, though, because it's not on the unit circle. I tried to use Pythagorean Theorem to get the third side (if triangles were involved) and got the square root of five. I can't use a calculator in this problem, but I checked anyway and found that arctan(1/2) is about 26.5 degrees.
Thus, I believe the order is arctan 0, arctan(1/2), and arctan(-sqrt3).
But is there any way I can figure out arctan(1/2) without the use of a calculator?
Thank you!
oops, i meant 120 degrees, not 150. my bad.
Your values are correct for a range of arctan from 0 to 180 degrees. Arctangents can take on values of arctan(x)±kπ due to its periodic nature. The period is 180 degrees.
If the range of the function is defined as from -90 to 90 degrees, the answer will be different.
To find the value of arctan(1/2) without a calculator, you can use the relationships between trigonometric functions and right triangles. Here's how you can do it:
1. Start with a right triangle where the opposite side is 1 and the adjacent side is 2. Since the ratio of opposite to adjacent is 1/2, we have found the triangle that corresponds to the value arctan(1/2).
|
|\
| \
| \
1 | \ 2
| \
------
This triangle has an angle θ such that tan(θ) = 1/2.
2. To find the angle θ, we need to determine the value of θ in degrees. Since the triangle is a right triangle, we can use the Pythagorean theorem.
Using the Pythagorean theorem: (opposite)^2 + (adjacent)^2 = (hypotenuse)^2
1^2 + 2^2 = (hypotenuse)^2
1 + 4 = (hypotenuse)^2
5 = (hypotenuse)^2
hypotenuse = √5
3. Now, we can find the sine and cosine of the angle θ using the sides of the triangle.
sine(θ) = opposite / hypotenuse = 1 / √5 = √5 / 5
cosine(θ) = adjacent / hypotenuse = 2 / √5 = 2√5 / 5
4. Finally, we can use the definitions of sine and cosine to find the angle in degrees.
sine(θ) = √5 / 5 = sin(θ)
Using the inverse sine function: θ = arcsin(√5 / 5)
Similarly, we can use the inverse cosine function to find the angle in degrees.
By evaluating arcsin(√5 / 5) or arccos(2√5 / 5) without a calculator, we can get the value of arctan(1/2) in degrees.
While this method is still using trigonometric functions, it does not rely on a calculator to find the value.