Find all the angles between 0 and, 720 whose tangent is 1/ �ã 3

To find all the angles between 0 and 720 degrees whose tangent is 1/3, we can use the inverse tangent function (also known as arctangent).

Since the tangent of an angle is equal to the ratio of the opposite side over the adjacent side in a right triangle, we can write the equation:

tan(x) = 1/3

To find the angles, we need to find the inverse tangent of 1/3:

x = arctan(1/3)

Since tangent has a periodicity of 180 degrees, we can add multiples of 180 degrees to find all the solutions.

Since we are looking for solutions between 0 and 720 degrees, we can start with the principal value:

x = arctan(1/3) ≈ 18.43 degrees

Now, we can add multiples of 180 degrees to find other solutions:

x = 18.43 + 180 ≈ 198.43 degrees
x = 18.43 + 360 ≈ 378.43 degrees
x = 18.43 + 540 ≈ 558.43 degrees

Therefore, the angles between 0 and 720 degrees whose tangent is 1/3 are approximately:
18.43 degrees, 198.43 degrees, 378.43 degrees, and 558.43 degrees.

To find all the angles between 0 and 720 degrees whose tangent is 1/3, we can use the inverse tangent function (also known as the arctangent function or tan^(-1)).

The tangent of an angle is equal to the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle in a right triangle.

In this case, we are given that the tangent of the angle is 1/3. Let's call this angle "θ".

So, we have:

tan(θ) = 1/3

To find the values of θ, we need to take the inverse tangent (arctangent) of both sides of the equation:

θ = tan^(-1)(1/3)

Using a calculator or a math tool, we can calculate the arctangent of 1/3.
Note: tan^(-1) is sometimes written as atan or arctan.

θ ≈ 18.435 degrees

Now, if we recall that trigonometric functions are periodic, we know that tan(θ) repeats every 180 degrees. Therefore, to find all the angles between 0 and 720 whose tangent is 1/3, we can add or subtract multiples of 180 degrees to our initial angle of 18.435 degrees.

The angles that satisfy the condition are:

θ = 18.435 degrees + k * 180 degrees

where k is an integer.

To limit the angles between 0 and 720, we can set the range of k as:

0 ≤ k ≤ 4

Substituting the values of k, we get:

θ1 = 18.435 degrees + 0 * 180 degrees = 18.435 degrees
θ2 = 18.435 degrees + 1 * 180 degrees = 198.435 degrees
θ3 = 18.435 degrees + 2 * 180 degrees = 378.435 degrees
θ4 = 18.435 degrees + 3 * 180 degrees = 558.435 degrees

Therefore, the angles between 0 and 720 degrees whose tangent is 1/3 are: 18.435 degrees, 198.435 degrees, 378.435 degrees, and 558.435 degrees.

Do you mean 1/(sqrt3) ?

The first such angle is 30 degrees. (Think of a right triangle with sides of 1, 2 and sqrt 3, with 2 being the hypotenuse. Its smallest angle is 30 degrees.)

The angle in the third quadrant which has a reference angle of of 30 degrees (from the -x axis) is 210 degrees, and that angle has the same tangent. Add 360 degrees to those angles and you get two more: 390 and 570 degrees.