Let $\sin A = -\frac{3}{5}$ With $A$ In Quadrant III. Find $\cos(2A$\].

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Introduction

In trigonometry, the double angle formula is a fundamental concept that relates the cosine of a double angle to the cosine and sine of the original angle. The double angle formula for cosine is given by $\cos(2A) = \cos^2 A - \sin^2 A$. This formula is essential in solving trigonometric problems and is widely used in various mathematical applications.

Understanding the Given Information

We are given that $\sin A = -\frac{3}{5}$, and $A$ lies in Quadrant III. This means that the angle $A$ is in the third quadrant of the unit circle, where both sine and cosine values are negative.

Using the Pythagorean Identity

To find $\cos A$, we can use the Pythagorean identity, which states that $\sin^2 A + \cos^2 A = 1$. Since we are given the value of $\sin A$, we can substitute it into the Pythagorean identity to solve for $\cos A$.

Calculating $\cos A$

Using the Pythagorean identity, we have:

$$\left(-\frac{3}{5}\right)^2 + \cos^2 A = 1$$

Simplifying the equation, we get:

$$\frac{9}{25} + \cos^2 A = 1$$

Subtracting $\frac{9}{25}$ from both sides, we get:

$$\cos^2 A = \frac{16}{25}$$

Taking the square root of both sides, we get:

$$\cos A = \pm \frac{4}{5}$$

Since $A$ lies in Quadrant III, where cosine values are negative, we take the negative value:

$$\cos A = -\frac{4}{5}$$

Applying the Double Angle Formula

Now that we have the values of $\sin A$ and $\cos A$, we can apply the double angle formula to find $\cos(2A)$:

$$\cos(2A) = \cos^2 A - \sin^2 A$$

Substituting the values of $\cos A$ and $\sin A$, we get:

$$\cos(2A) = \left(-\frac{4}{5}\right)^2 - \left(-\frac{3}{5}\right)^2$$

Simplifying the equation, we get:

$$\cos(2A) = \frac{16}{25} - \frac{9}{25}$$

Subtracting $\frac{9}{25}$ from $\frac{16}{25}$, we get:

$$\cos(2A) = \frac{7}{25}$$

Conclusion

In this article, we used the given information that $\sin A = -\frac{3}{5}$ and $A$ lies in Quadrant III to find $\cos(2A)$. We first used the Pythagorean identity to find the value of $\cos A$, and then applied the double angle formula to find $\cos(2A)$. The final answer is $\cos(2A) = \frac{7}{25}$.

Additional Information

The double angle formula is a fundamental concept in trigonometry, and is widely used in various mathematical applications. It is essential to understand the Pythagorean identity and the double angle formula to solve trigonometric problems.

Example Problems

• Find $\cos(2A)$ if $\sin A = \frac{1}{2}$ and $A$ lies in Quadrant I.
• Find $\cos(2A)$ if $\sin A = -\frac{1}{3}$ and $A$ lies in Quadrant III.

Solved Problems

• Find $\cos(2A)$ if $\sin A = \frac{2}{3}$ and $A$ lies in Quadrant II.
• Find $\cos(2A)$ if $\sin A = -\frac{4}{5}$ and $A$ lies in Quadrant III.

Practice Problems

• Find $\cos(2A)$ if $\sin A = \frac{3}{4}$ and $A$ lies in Quadrant I.
• Find $\cos(2A)$ if $\sin A = -\frac{2}{3}$ and $A$ lies in Quadrant III.

Final Answer

The final answer is $\boxed{\frac{7}{25}}$.

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Introduction

In our previous article, we used the given information that $\sin A = -\frac{3}{5}$ and $A$ lies in Quadrant III to find $\cos(2A)$. We first used the Pythagorean identity to find the value of $\cos A$, and then applied the double angle formula to find $\cos(2A)$. In this article, we will answer some frequently asked questions related to the problem.

Q&A

Q: What is the Pythagorean identity?

A: The Pythagorean identity is a fundamental concept in trigonometry that states $\sin^2 A + \cos^2 A = 1$. This identity is used to find the value of $\cos A$ when the value of $\sin A$ is given.

Q: How do I find $\cos A$ when $\sin A$ is given?

A: To find $\cos A$ when $\sin A$ is given, you can use the Pythagorean identity. Substitute the value of $\sin A$ into the Pythagorean identity and solve for $\cos A$.

Q: What is the double angle formula for cosine?

A: The double angle formula for cosine is given by $\cos(2A) = \cos^2 A - \sin^2 A$. This formula is used to find the value of $\cos(2A)$ when the values of $\cos A$ and $\sin A$ are given.

Q: How do I apply the double angle formula to find $\cos(2A)$?

A: To apply the double angle formula, substitute the values of $\cos A$ and $\sin A$ into the formula and simplify the expression.

Q: What is the final answer to the problem?

A: The final answer to the problem is $\cos(2A) = \frac{7}{25}$.

Q: Can I use the double angle formula to find $\sin(2A)$?

A: Yes, you can use the double angle formula to find $\sin(2A)$. The double angle formula for sine is given by $\sin(2A) = 2\sin A \cos A$.

Q: How do I find $\sin(2A)$ when $\sin A$ and $\cos A$ are given?

A: To find $\sin(2A)$ when $\sin A$ and $\cos A$ are given, substitute the values of $\sin A$ and $\cos A$ into the double angle formula for sine and simplify the expression.

Additional Information

• The Pythagorean identity is a fundamental concept in trigonometry that is used to find the value of $\cos A$ when the value of $\sin A$ is given.
• The double angle formula for cosine is given by $\cos(2A) = \cos^2 A - \sin^2 A$.
• The double angle formula for sine is given by $\sin(2A) = 2\sin A \cos A$.

Example Problems

• Find $\cos(2A)$ if $\sin A = \frac{1}{2}$ and $A$ lies in Quadrant I.
• Find $\cos(2A)$ if $\sin A = -\frac{1}{3}$ and $A$ lies in Quadrant III.
• Find $\sin(2A)$ if $\sin A = \frac{2}{3}$ and $A$ lies in Quadrant II.
• Find $\sin(2A)$ if $\sin A = -\frac{4}{5}$ and $A$ lies in Quadrant III.

Solved Problems

• Find $\cos(2A)$ if $\sin A = \frac{3}{4}$ and $A$ lies in Quadrant I.
• Find $\cos(2A)$ if $\sin A = -\frac{2}{3}$ and $A$ lies in Quadrant III.
• Find $\sin(2A)$ if $\sin A = \frac{1}{4}$ and $A$ lies in Quadrant I.
• Find $\sin(2A)$ if $\sin A = -\frac{3}{5}$ and $A$ lies in Quadrant III.

Practice Problems

• Find $\cos(2A)$ if $\sin A = \frac{2}{5}$ and $A$ lies in Quadrant I.
• Find $\cos(2A)$ if $\sin A = -\frac{1}{2}$ and $A$ lies in Quadrant III.
• Find $\sin(2A)$ if $\sin A = \frac{3}{5}$ and $A$ lies in Quadrant I.
• Find $\sin(2A)$ if $\sin A = -\frac{2}{3}$ and $A$ lies in Quadrant III.

Final Answer

The final answer is $\boxed{\frac{7}{25}}$.