Point B has rectangular coordinates (-5, 12) Find the angle θ that describes this point.
tan(θ) = y / x = 12 / -5
the angle is in Quadrant II ... negative x , positive y
To find the angle θ that describes a point in rectangular coordinates, we can use trigonometry. The angle θ is typically measured counterclockwise from the positive x-axis.
In this case, the coordinates of point B are (-5, 12). Consider the triangle formed by connecting this point with the origin (0, 0) and the projection of the point onto the x-axis, which we will call point P.
To find the length of the side adjacent to angle θ, we can use the x-coordinate of point B, which is -5. The length of this side is simply the absolute value of the x-coordinate, so the length is 5.
To find the length of the side opposite to angle θ, we can use the y-coordinate of point B, which is 12. The length of this side is the y-coordinate itself.
Now, we have the lengths of the two sides of the triangle adjacent and opposite to angle θ. We can use the tangent function (tan θ = opposite/adjacent) to find θ.
tan θ = opposite/adjacent
tan θ = 12/5
To solve for θ, we can take the inverse tangent (also known as the arctangent) of both sides:
θ = arctan(12/5)
Using a calculator or a tool like the arctan function in a programming language, you can find that θ is approximately 68.2 degrees.
Therefore, the angle θ that describes the point B (-5, 12) is approximately 68.2 degrees.