For which value(s) of $$ x, x \in\left[0^{\circ} ; 360^{\circ}\right] $$ is the expression in 4.1 undefined

Question

UpStudy AI · Accepted Answer

We must first identify when a trigonometric expression is undefined. Typically, an expression like

$$
\tan x=\frac{\sin x}{\cos x}
$$

is undefined wherever its denominator is zero. That is, wherever

$$
\cos x = 0.
$$

Within the interval $$ x\in\left[0^{\circ},\,360^{\circ}\right] $$, the cosine function equals zero at

$$
x=90^{\circ} \quad \text{and} \quad x=270^{\circ}.
$$

Thus, if the expression in 4.1 is indeed $$ \tan x $$, then it is undefined when $$ x=90^{\circ} $$ and $$ x=270^{\circ} $$.

Step by step:

1. Write the expression (assuming it is the tangent function):

$$
   \tan x = \frac{\sin x}{\cos x}.
   $$

2. Set the condition for being undefined by requiring the denominator to be zero:

$$
   \cos x = 0.
   $$

3. Determine the values of $$ x $$ in $$\left[0^{\circ},\,360^{\circ}\right]$$ for which $$\cos x=0$$. We know that:

$$
   \cos x = 0 \quad \text{when} \quad x = 90^{\circ} \quad \text{or} \quad x = 270^{\circ}.
   $$

4. Conclude that the expression is undefined at these values.

Thus, the expression in 4.1 is undefined for

$$
x = 90^{\circ} \quad \text{and} \quad x = 270^{\circ}.
$$
The expression is undefined at $$ x = 90^\circ $$ and $$ x = 270^\circ $$.

For which value(s) of \( x, x \in\left[0^{\circ} ; 360^{\circ}\right] \) is the expression in 4.1 undefined