Q.Which of the following is not correct?
(A) $\sin\theta = -\dfrac{1}{5}$
(B) $\cos\theta = 1$
(C) $\sec\theta = \dfrac{1}{2}$
(D) $\tan\theta = 20$

The key idea here is understanding the range of trigonometric functions. Each trigonometric function has a specific set of values it can take.

1. The range of $\sin\theta$ is $[-1, 1]$. This means $-1 \le \sin\theta \le 1$.
2. The range of $\cos\theta$ is $[-1, 1]$. This means $-1 \le \cos\theta \le 1$.
3. The range of $\sec\theta$ is $(-\infty, -1] \cup [1, \infty)$. This means $\sec\theta \le -1$ or $\sec\theta \ge 1$.
4. The range of $\tan\theta$ is $(-\infty, \infty)$. This means $\tan\theta$ can take any real value.

Let's check each option:

• (A) $\sin\theta = -\frac{1}{5}$. Since $-1 \le -\frac{1}{5} \le 1$, this is a possible value.
• (B) $\cos\theta = 1$. Since $-1 \le 1 \le 1$, this is a possible value. …

The question asks us to identify which of the given trigonometric ratios is not possible. To do this, we need to recall the fundamental ranges of the sine, cosine, tangent, and secant functions. These ranges are derived directly from their definitions, often visualized using the unit circle.

Concept and Intuition

Trigonometric functions relate an angle to the ratios of sides of a right-angled triangle, or more generally, to the coordinates of a point on the unit circle.

1. Sine and Cosine: For any angle $\theta$, if we consider a point $(x, y)$ on the unit circle corresponding to $\theta$, then $x = \cos\theta$ and $y = \sin\theta$. Since the unit circle has a radius of 1, the coordinates $(x, y)$ must satisfy $x^2 + y^2 = 1$. This means that both $x$ and $y$ must lie between $-1$ and $1$, inclusive.

   $-1 \le \sin\theta \le 1$

   $-1 \le \cos\theta \le 1$

2. Secant: The secant function is defined as the reciprocal of the cosine function: $\sec\theta = \frac{1}{\cos\theta}$. Since $\cos\theta$ can only take values between $-1$ and $1$ (excluding $0$ for $\sec\theta$ to be defined), let's consider the implications for $\sec\theta$:

   • If $0 < \cos\theta \le 1$, then $\frac{1}{\cos\theta} \ge 1$.
   • If $-1 \le \cos\theta < 0$, then $\frac{1}{\cos\theta} \le -1$. This means that $\sec\theta$ can never take a value strictly between $-1$ and $1$. In other words, $|\sec\theta| \ge 1$.

   $\sec\theta \in (-\infty, -1] \cup [1, \infty)$

3. Tangent: The tangent function is defined as $\tan\theta = \frac{\sin\theta}{\cos\theta}$. As $\theta$ varies, $\sin\theta$ and $\cos\theta$ change. When $\cos\theta$ approaches $0$ (i.e., $\theta$ approaches $\frac{\pi}{2} + n\pi$ for integer $n$), the value of $\tan\theta$ approaches positive or negative infinity. This means $\tan\theta$ can take any real value.

   $\tan\theta \in (-\infty, \infty)$

Now, let's examine each option based on these ranges.

Step-by-step Solution

1. Analyze Option (A): $\sin\theta = -\dfrac{1}{5}$

   The range for $\sin\theta$ is $[-1, 1]$.

   Since $-1 \le -\frac{1}{5} \le 1$, this value is perfectly valid for $\sin\theta$.

2. Analyze Option (B): $\cos\theta = 1$

   The range for $\cos\theta$ is $[-1, 1]$.

   Since $-1 \le 1 \le 1$, this value is valid for $\cos\theta$. For example, $\cos(0) = 1$. …