The point (5,-6) is on the terminal arm of the standard position angleӨ. Determine the smallest positive measure of Ө in radians.
It's in the fourth quadrant so tan is negative?
tan(-6/5)
how can that be positive?
Thanks in advance.
The smallest positive measure of Ө in radians is -2.677945045 radians.
To determine the smallest positive measure of θ in radians, we can use the tangent function since the point (5, -6) lies on the terminal arm of the angle.
Let's consider the formula for the tangent of an angle in trigonometry:
tan(θ) = y / x
Here, the x-coordinate represents the horizontal position of the point (5, -6), which is 5, and the y-coordinate represents the vertical position, which is -6.
So, substituting the values into the formula, we have:
tan(θ) = -6 / 5
Now, let's solve for θ. To determine the angle specifically in the fourth quadrant, we need to use the arctangent or inverse tangent function.
θ = atan(-6 / 5)
Please note that atan represents arctan or the inverse tangent function.
Now, let's calculate:
θ ≈ -1.107
The value we obtained for θ is negative because the point (5, -6) lies in the fourth quadrant, where the angles are negative. However, we need to find the smallest positive measure of θ, so we need to add 2π or 360 degrees to the obtained value.
θ ≈ -1.107 + 2π
To simplify further, let's convert the value 2π into decimal form:
θ ≈ -1.107 + 6.283
θ ≈ 5.176
Therefore, the smallest positive measure of θ in radians is approximately 5.176 radians.
I hope this explanation helps! Let me know if you have any further questions.
To determine the smallest positive measure of θ in radians, we can use the inverse tangent function (arctan) to find the angle whose tangent is equal to -6/5.
However, it is important to note that in the fourth quadrant, both sine and tangent are negative.
So, tan(θ) = -6/5
To find the angle θ, we can use the arctan function:
θ = arctan(-6/5)
Since we are looking for the smallest positive measure of θ, we can add 2π radians to the angle obtained from the arctan function, as every full rotation of 2π radians brings us back to the starting position.
θ = arctan(-6/5) + 2π
Now we can calculate the value:
θ ≈ 2.214297 radians
Therefore, the smallest positive measure of θ in radians is approximately 2.214297 radians.