Question

Given that $\sin\theta = \frac{a}{b}$, then $\cos\theta$ is equal to :

• A. $\frac{b}{\sqrt{b^2-a^2}}$
• B. $\frac{b}{a}$
• C. $\frac{\sqrt{b^2-a^2}}{b}$
• D. $\frac{a}{\sqrt{b^2-a^2}}$

Hint:

Using a quick mental right-triangle is often faster:
If Opposite $= a$ and Hypotenuse $= b$, then by Pythagoras, the Adjacent side is $\sqrt{b^2 - a^2}$.
Since cosine is Adjacent / Hypotenuse, we immediately get $\frac{\sqrt{b^2 - a^2}}{b}$.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
We are given the value of $\sin\theta$ as $\frac{a}{b}$ and need to find the expression for $\cos\theta$ in terms of $a$ and $b$.

Step 2: Key Formula or Approach:
We can solve this using either the fundamental trigonometric identity or by modeling the ratios using a right-angled triangle.
Method 1: Use the Pythagorean identity:
\[ \sin^2\theta + \cos^2\theta = 1 \implies \cos\theta = \sqrt{1 - \sin^2\theta} \]
Method 2: Use the right-angled triangle definition:
\[ \sin\theta = \frac{\text{Opposite}}{\text{Hypotenuse}} \]
By Pythagoras theorem, $\text{Adjacent} = \sqrt{\text{Hypotenuse}^2 - \text{Opposite}^2}$.
Then, $\cos\theta = \frac{\text{Adjacent}}{\text{Hypotenuse}}$.

Step 3: Detailed Explanation:

• Let us apply the algebraic identity method:
We are given:
\[ \sin\theta = \frac{a}{b} \]

• Substitute this value into the expression for $\cos\theta$:
\[ \cos\theta = \sqrt{1 - \left(\frac{a}{b}\right)^2} \]

• Simplify the term inside the square root by squaring the fraction:
\[ \cos\theta = \sqrt{1 - \frac{a^2}{b^2}} \]

• Take the common denominator inside the square root:
\[ \cos\theta = \sqrt{\frac{b^2 - a^2}{b^2}} \]

• Separate the square root of the numerator and the denominator:
\[ \cos\theta = \frac{\sqrt{b^2 - a^2}}{\sqrt{b^2}} \]
Since $b$ represents a side length ratio and is positive, $\sqrt{b^2} = b$:
\[ \cos\theta = \frac{\sqrt{b^2 - a^2}}{b} \]


Step 4: Final Answer:
The value of $\cos\theta$ is $\frac{\sqrt{b^2-a^2}}{b}$.
Hence, option (C) is correct.