Question

Express cos A and tan A in terms of sin A.

Hint:

Using identities is often faster and more general than building triangles for variable-based proofs.

Answer 1

Step 1: Understanding the Concept:
Trigonometric identities allow us to express one ratio in terms of another.
Step 2: Key Formula or Approach:
1. \( \sin^2 A + \cos^2 A = 1 \)
2. \( \tan A = \frac{\sin A}{\cos A} \)
Step 3: Detailed Explanation:
To express \( \cos A \) in terms of \( \sin A \):
From the identity \( \sin^2 A + \cos^2 A = 1 \):
\[ \cos^2 A = 1 - \sin^2 A \]
Taking the square root (assuming \( A \) is an acute angle):
\[ \cos A = \sqrt{1 - \sin^2 A} \]
Now, to express \( \tan A \) in terms of \( \sin A \):
We know \( \tan A = \frac{\sin A}{\cos A} \).
Substitute the expression for \( \cos A \) derived above:
\[ \tan A = \frac{\sin A}{\sqrt{1 - \sin^2 A}} \]
Step 4: Final Answer:
\( \cos A = \sqrt{1 - \sin^2 A} \) and \( \tan A = \frac{\sin A}{\sqrt{1 - \sin^2 A}} \).