$ ext{If } \sin heta = \frac{a}{b}, heta ext{ is in first quadrant, then } \cos heta ext{ is equal to :}$

Q: $\text{If } \sin \theta = \frac{a}{b}, \theta \text{ is in first quadrant, then } \cos \theta \text{ is equal to :}$

To determine the value of $\cos \theta$ given that $\sin \theta = \frac{a}{b}$ and $\theta$ is in the first quadrant, we can use the Pythagorean identity for sine and cosine. The identity states: \[ \sin^2 \theta + \cos^2 \theta = 1 \] Given $\sin \theta = \frac{a}{b}$, we can substitute this into the identity: \[ \left(\frac{a}{b}\right)^2 + \cos^2 \theta = 1 \] Simplifying, we have: \[ \frac{a^2}{b^2} + \cos^2 \theta = 1 \] Subtract $\frac{a^2}{b^2}$ from both sides: \[ \cos^2 \theta = 1 - \frac{a^2}{b^2} \] \[ \cos^2 \theta = \frac{b^2}{b^2} - \frac{a^2}{b^2} \] \[ \cos^2 \theta = \frac{b^2 - a^2}{b^2} \] Taking the square root of both sides, and since $\theta$ is in the first quadrant where cosine is positive, we have: \[ \cos \theta = \frac{\sqrt{b^2 - a^2}}{b} \] Thus, the correct answer is Option 4: $\frac{\sqrt{b^2 - a^2}}{b}$. **Explanation for why other options are incorrect:** 1. **Option 1: $\frac{a}{\sqrt{b^2 - a^2}}$** This option incorrectly places $a$ in the numerator and $\sqrt{b^2 - a^2}$ in the denominator. The expression does not satisfy the Pythagorean identity when $\sin \theta = \frac{a}{b}$. 2. **Option 2: $\frac{b}{\sqrt{b^2 - a^2}}$** This option incorrectly suggests that $\cos \theta$ is greater than 1, which is not possible for angles in the first quadrant. It does not align with the derived expression for $\cos \theta$. 3. **Option 3: $\frac{b}{a}$** This option suggests a ratio that does not relate to the Pythagorean identity for sine and cosine. It implies a value that is not consistent with the given $\sin \theta = \frac{a}{b}$. Therefore, the only option that correctly represents $\cos \theta$ in accordance with the Pythagorean identity and the given conditions is Option 4: $\frac{\sqrt{b^2 - a^2}}{b}$.

Question

$	ext{If } \sin 	heta = \frac{a}{b}, 	heta 	ext{ is in first quadrant, then } \cos 	heta 	ext{ is equal to :}$

Accepted Answer

To determine the value of $\cos \theta$ given that $\sin \theta = \frac{a}{b}$ and $\theta$ is in the first quadrant, we can use the Pythagorean identity for sine and cosine. The identity states:

$$
\sin^2 \theta + \cos^2 \theta = 1
$$

Given $\sin \theta = \frac{a}{b}$, we can substitute this into the identity:

$$
\left(\frac{a}{b}\right)^2 + \cos^2 \theta = 1
$$

Simplifying, we have:

$$
\frac{a^2}{b^2} + \cos^2 \theta = 1
$$

Subtract $\frac{a^2}{b^2}$ from both sides:

$$
\cos^2 \theta = 1 - \frac{a^2}{b^2}
$$

$$
\cos^2 \theta = \frac{b^2}{b^2} - \frac{a^2}{b^2}
$$

$$
\cos^2 \theta = \frac{b^2 - a^2}{b^2}
$$

Taking the square root of both sides, and since $\theta$ is in the first quadrant where cosine is positive, we have:

$$
\cos \theta = \frac{\sqrt{b^2 - a^2}}{b}
$$

Thus, the correct answer is Option 4: $\frac{\sqrt{b^2 - a^2}}{b}$.

**Explanation for why other options are incorrect:**

1. **Option 1: $\frac{a}{\sqrt{b^2 - a^2}}$**

This option incorrectly places $a$ in the numerator and $\sqrt{b^2 - a^2}$ in the denominator. The expression does not satisfy the Pythagorean identity when $\sin \theta = \frac{a}{b}$.

2. **Option 2: $\frac{b}{\sqrt{b^2 - a^2}}$**

This option incorrectly suggests that $\cos \theta$ is greater than 1, which is not possible for angles in the first quadrant. It does not align with the derived expression for $\cos \theta$.

3. **Option 3: $\frac{b}{a}$**

This option suggests a ratio that does not relate to the Pythagorean identity for sine and cosine. It implies a value that is not consistent with the given $\sin \theta = \frac{a}{b}$.

Therefore, the only option that correctly represents $\cos \theta$ in accordance with the Pythagorean identity and the given conditions is Option 4: $\frac{\sqrt{b^2 - a^2}}{b}$.