Two angles between 0 degrees and 360 degrees have a tangent undefined. What are the angles? Why is the tangent undefined for each of those?
To find the angles that have undefined tangent values, we need to determine the angles at which the tangent function is undefined.
The tangent function is undefined when the angle is either 90 degrees or 270 degrees.
Let's go through the steps for each angle:
1. For an angle of 90 degrees:
- The tangent of 90 degrees is calculated as tan(90°).
- However, the tangent of 90 degrees is undefined because at this angle, the line is vertical, resulting in a vertical asymptote.
- Therefore, the angle at which the tangent is undefined is 90 degrees.
2. For an angle of 270 degrees:
- The tangent of 270 degrees is calculated as tan(270°).
- Similar to 90 degrees, the tangent of 270 degrees is undefined because at this angle, the line is vertical, resulting in another vertical asymptote.
- Therefore, the angle at which the tangent is undefined is 270 degrees.
So, the angles that have an undefined tangent are 90 degrees and 270 degrees. The tangent is not defined at these angles due to the vertical lines resulting in vertical asymptotes.
To find the angles between 0 degrees and 360 degrees where the tangent is undefined, we need to understand the behavior of the tangent function. The tangent of an angle is defined as the ratio of the sine of the angle to the cosine of the angle. When we say the tangent is undefined, it means that the cosine of the angle is equal to zero.
Therefore, to find the angles where the tangent is undefined, we need to find the angles where the cosine function is equal to zero between 0 degrees and 360 degrees. The cosine function equals zero at two specific angles: 90 degrees (or π/2 radians) and 270 degrees (or 3π/2 radians).
So, the two angles between 0 degrees and 360 degrees where the tangent is undefined are 90 degrees and 270 degrees.
The reason the tangent is undefined at these angles is because the cosine function, which is in the denominator of the tangent function, becomes zero at these angles. Division by zero is undefined in mathematics, hence the tangent becomes undefined at these points.