Question

Which of the following statements is not true?

• A. \(\sin 0^\circ = \cos 0^\circ\)
• B. \(\tan 30^\circ = \cot 60^\circ\)
• C. \(\sin 30^\circ = \cos 60^\circ\)
• D. \(\sin 45^\circ = \frac{1}{\sec 45^\circ}\)

Hint:

Co-functions (sine/cosine, tangent/cotangent, secant/cosecant) are equal when their angles sum to \(90^\circ\).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This question tests knowledge of specific values for trigonometric ratios and complementary angle relationships (\(\sin \theta = \cos(90^\circ - \theta)\)).
Step 2: Key Formula or Approach:
Evaluate each statement using standard values: \(\sin 0^\circ = 0, \cos 0^\circ = 1, \tan 30^\circ = 1/\sqrt{3}, \cot 60^\circ = 1/\sqrt{3}, \sin 30^\circ = 1/2, \cos 60^\circ = 1/2, \sin 45^\circ = 1/\sqrt{2}, \sec 45^\circ = \sqrt{2}\).
Step 3: Detailed Explanation:
1. Check (a): \(\sin 0^\circ = 0\) and \(\cos 0^\circ = 1\). Since \(0 \neq 1\), this is NOT TRUE.
2. Check (b): \(\tan 30^\circ = 1/\sqrt{3}\) and \(\cot 60^\circ = \tan(90-60) = \tan 30^\circ = 1/\sqrt{3}\). This is true.
3. Check (c): \(\sin 30^\circ = 1/2\) and \(\cos 60^\circ = \sin(90-60) = \sin 30^\circ = 1/2\). This is true.
4. Check (d): \(\frac{1}{\sec 45^\circ}\) is the definition of \(\cos 45^\circ\). Since \(\sin 45^\circ = \cos 45^\circ = 1/\sqrt{2}\), this is true.
Step 4: Final Answer:
The statement that is not true is (a).