Fatima Wants To Find The Value Of Sin ⁡ Θ \sin \theta Sin Θ , Given Cot ⁡ Θ = 4 7 \cot \theta=\frac{4}{7} Cot Θ = 7 4 ​ . Which Identity Would Be Best For Fatima To Use?A. Cos ⁡ Θ = 1 Sec ⁡ Θ \cos \theta=\frac{1}{\sec \theta} Cos Θ = S E C Θ 1 ​ B. Sin ⁡ 2 Θ + Cos ⁡ 2 Θ = 1 \sin^2 \theta+\cos^2 \theta=1 Sin 2 Θ + Cos 2 Θ = 1 C. $\csc

Finding the Value of $\sin \theta$ Using Trigonometric Identities

In trigonometry, there are various identities that can be used to find the values of different trigonometric functions. When given the value of one trigonometric function, we can use these identities to find the values of other trigonometric functions. In this article, we will discuss how to find the value of $\sin \theta$ using the given value of $\cot \theta$.

Understanding the Given Information

Fatima is given the value of $\cot \theta$ as $\frac{4}{7}$. We know that $\cot \theta = \frac{\cos \theta}{\sin \theta}$. Therefore, we can write $\frac{\cos \theta}{\sin \theta} = \frac{4}{7}$.

Choosing the Right Identity

Now, we need to choose the right identity to find the value of $\sin \theta$. Let's analyze the given options:

A. $\cos \theta = \frac{1}{\sec \theta}$

This identity is not helpful in this case because we are given the value of $\cot \theta$, not $\sec \theta$.

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

This identity is also not helpful because we are given the value of $\cot \theta$, not $\sin \theta$ or $\cos \theta$.

C. $\csc \theta = \frac{1}{\sin \theta}$

This identity is not directly helpful because we are given the value of $\cot \theta$, not $\csc \theta$.

However, we can use the identity $\cot \theta = \frac{\cos \theta}{\sin \theta}$ to find the value of $\cos \theta$ in terms of $\sin \theta$. We can then use the identity $\sin^2 \theta + \cos^2 \theta = 1$ to find the value of $\sin \theta$.

Using the Pythagorean Identity

Let's use the Pythagorean identity $\sin^2 \theta + \cos^2 \theta = 1$ to find the value of $\sin \theta$. We can rewrite this identity as $\cos^2 \theta = 1 - \sin^2 \theta$.

We are given the value of $\cot \theta$ as $\frac{4}{7}$. We can use this value to find the value of $\cos \theta$ in terms of $\sin \theta$. Let's assume $\cos \theta = k \sin \theta$, where $k$ is a constant.

Substituting this value into the Pythagorean identity, we get:

$k^2 \sin^2 \theta = 1 - \sin^2 \theta$

Simplifying this equation, we get:

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

Combining like terms, we get:

$(k^2 + 1) \sin^2 \theta = 1$

Now, we can substitute the value of $\cot \theta$ as $\frac{4}{7}$ into this equation. We get:

$(\frac{4}{7})^2 + 1 = k^2 + 1$

Simplifying this equation, we get:

$k^2 = \frac{16}{49}$

Taking the square root of both sides, we get:

$k = \pm \frac{4}{7}$

Since $\cos \theta = k \sin \theta$, we can write:

$\cos \theta = \pm \frac{4}{7} \sin \theta$

Now, we can use the Pythagorean identity to find the value of $\sin \theta$. We get:

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

Substituting the value of $\cos \theta$, we get:

$\sin^2 \theta + (\pm \frac{4}{7} \sin \theta)^2 = 1$

Simplifying this equation, we get:

$\sin^2 \theta + \frac{16}{49} \sin^2 \theta = 1$

Combining like terms, we get:

$(\frac{65}{49}) \sin^2 \theta = 1$

Now, we can solve for $\sin \theta$. We get:

$\sin^2 \theta = \frac{49}{65}$

Taking the square root of both sides, we get:

$\sin \theta = \pm \sqrt{\frac{49}{65}}$

Simplifying this expression, we get:

$\sin \theta = \pm \frac{7}{\sqrt{65}}$

Therefore, the value of $\sin \theta$ is $\pm \frac{7}{\sqrt{65}}$.

In this article, we discussed how to find the value of $\sin \theta$ using the given value of $\cot \theta$. We used the Pythagorean identity to find the value of $\sin \theta$. We also analyzed the given options and chose the right identity to find the value of $\sin \theta$. The final answer is $\pm \frac{7}{\sqrt{65}}$.
Q&A: Finding the Value of $\sin \theta$ Using Trigonometric Identities

In our previous article, we discussed how to find the value of $\sin \theta$ using the given value of $\cot \theta$. We used the Pythagorean identity to find the value of $\sin \theta$. In this article, we will answer some frequently asked questions related to finding the value of $\sin \theta$ using trigonometric identities.

Q: What is the Pythagorean identity?

A: The Pythagorean identity is a fundamental identity in trigonometry that states $\sin^2 \theta + \cos^2 \theta = 1$. This identity can be used to find the value of $\sin \theta$ or $\cos \theta$ if one of them is known.

Q: How can I use the Pythagorean identity to find the value of $\sin \theta$?

A: To use the Pythagorean identity to find the value of $\sin \theta$, you need to know the value of $\cos \theta$. You can then substitute the value of $\cos \theta$ into the Pythagorean identity and solve for $\sin \theta$.

Q: What if I don't know the value of $\cos \theta$? Can I still use the Pythagorean identity?

A: Yes, you can still use the Pythagorean identity even if you don't know the value of $\cos \theta$. You can use the identity $\cot \theta = \frac{\cos \theta}{\sin \theta}$ to find the value of $\cos \theta$ in terms of $\sin \theta$. You can then substitute this value into the Pythagorean identity and solve for $\sin \theta$.

Q: How can I find the value of $\cos \theta$ in terms of $\sin \theta$?

A: To find the value of $\cos \theta$ in terms of $\sin \theta$, you can use the identity $\cot \theta = \frac{\cos \theta}{\sin \theta}$. You can then rewrite this identity as $\cos \theta = \cot \theta \sin \theta$. This gives you the value of $\cos \theta$ in terms of $\sin \theta$.

Q: What if I get a negative value for $\sin \theta$? Is that possible?

A: Yes, it is possible to get a negative value for $\sin \theta$. The value of $\sin \theta$ can be positive or negative, depending on the quadrant in which the angle $\theta$ lies.

Q: How can I determine the quadrant in which the angle $\theta$ lies?

A: To determine the quadrant in which the angle $\theta$ lies, you can use the following rules:

• If $\sin \theta > 0$, then $\theta$ lies in the first or second quadrant.
• If $\sin \theta < 0$, then $\theta$ lies in the third or fourth quadrant.

Q: What if I get a complex value for $\sin \theta$? Is that possible?

A: No, it is not possible to get a complex value for $\sin \theta$. The value of $\sin \theta$ is always a real number.

In this article, we answered some frequently asked questions related to finding the value of $\sin \theta$ using trigonometric identities. We discussed how to use the Pythagorean identity to find the value of $\sin \theta$, how to find the value of $\cos \theta$ in terms of $\sin \theta$, and how to determine the quadrant in which the angle $\theta$ lies. We also discussed the possibility of getting a negative or complex value for $\sin \theta$.