Question

at what which of the following sets of values is the tangent function undefined? (1 point)

at 45 and 135 degrees

at 90 and 270 degrees

at 90 and 180 degrees

at 60 and 120 degrees

Explanation:

Step1: Recall tangent function definition

The tangent function is defined as $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$. A function is undefined when its denominator is zero. So, we need to find when $\cos(\theta) = 0$.

Step2: Find angles where cosine is zero

The cosine function, $\cos(\theta)$, is zero at $\theta = 90^\circ + 180^\circ n$ (where $n$ is an integer). For $n = 0$, $\theta = 90^\circ$; for $n = 1$, $\theta = 90^\circ + 180^\circ = 270^\circ$. Let's check the options:

• Option 1: At $45^\circ$ and $135^\circ$, $\cos(45^\circ)=\frac{\sqrt{2}}{2}

eq0$, $\cos(135^\circ)=-\frac{\sqrt{2}}{2}
eq0$, so tangent is defined here.

• Option 2: At $90^\circ$ and $270^\circ$, $\cos(90^\circ)=0$, $\cos(270^\circ)=0$, so tangent is undefined here.
• Option 3: At $90^\circ$ (cosine zero, tangent undefined) and $180^\circ$ (cosine is - 1, tangent is 0, defined), so this option is incorrect.
• Option 4: At $60^\circ$ and $120^\circ$, $\cos(60^\circ)=\frac{1}{2}

eq0$, $\cos(120^\circ)=-\frac{1}{2}
eq0$, so tangent is defined here.

Answer:

at 90 and 270 degrees