Question

If $\cos y = 0$, then what is the value of $\frac{1}{2} \cos \frac{y}{2}$ ?

• A. 0
• B. $\frac{1}{2}$
• C. $\frac{1}{\sqrt{2}}$
• D. $\frac{1}{2\sqrt{2}}$

Hint:

For trigonometric multiple-choice questions, substituting standard values of angles that satisfy the initial condition is the fastest and most reliable method to solve the problem.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
We are given that $\cos y = 0$. We need to find the value of the trigonometric expression:
\[ \frac{1}{2} \cos \frac{y}{2} \]

Step 2: Key Formula or Approach:
1. Find the value of angle $y$ using standard values of trigonometric ratios. For standard acute/obtuse angles, $\cos 90^\circ = 0$.
2. After finding $y$, calculate the half-angle $\frac{y}{2}$.
3. Substitute the half-angle into the expression $\frac{1}{2} \cos \frac{y}{2}$ and evaluate the final result using standard trigonometric values.

Step 3: Detailed Explanation:

• 1. We are given the equation:
\[ \cos y = 0 \]

• 2. Since we know that the cosine of $90^\circ$ (or $\frac{\pi}{2}$ radians) is 0:
\[ y = 90^\circ \]

• 3. Now, let us find the value of the half-angle $\frac{y}{2}$:
\[ \frac{y}{2} = \frac{90^\circ}{2} = 45^\circ \]

• 4. Find the cosine of this half-angle using standard values:
\[ \cos 45^\circ = \frac{1}{\sqrt{2}} \]

• 5. Substitute this value back into our target expression $\frac{1}{2} \cos \frac{y}{2}$:
\[ \frac{1}{2} \cos 45^\circ = \frac{1}{2} \times \frac{1}{\sqrt{2}} = \frac{1}{2\sqrt{2}} \]


Step 4: Final Answer:
The value of $\frac{1}{2} \cos \frac{y}{2}$ is $\frac{1}{2\sqrt{2}}$, which corresponds to option (D).