Given $\sin A = \frac{7}{25}$, Determine $\cos A$ And $\tan A$ Exactly.a. $\cos A = \frac{24}{25}$ And $\tan A = \frac{7}{24}$ B. $\cos A = \frac{7}{25}$ And $\tan A = \frac{24}{25}$ C.

Introduction

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It is a fundamental subject that has numerous applications in various fields, including physics, engineering, and navigation. In this article, we will focus on solving trigonometric identities, specifically finding the cosine and tangent of an angle given the sine.

The Pythagorean Identity

The Pythagorean identity is a fundamental concept in trigonometry that relates the sine, cosine, and tangent of an angle. It states that for any angle A, the following equation holds:

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

This identity can be used to find the cosine of an angle given the sine, and vice versa.

Finding Cosine Given Sine

Given that $\sin A = \frac{7}{25}$, we can use the Pythagorean identity to find the cosine of angle A. We start by squaring both sides of the equation:

$$\sin^2 A = \left(\frac{7}{25}\right)^2 = \frac{49}{625}$$

Next, we substitute this value into the Pythagorean identity:

$$\frac{49}{625} + \cos^2 A = 1$$

To solve for $\cos^2 A$, we subtract $\frac{49}{625}$ from both sides of the equation:

$$\cos^2 A = 1 - \frac{49}{625} = \frac{576}{625}$$

Taking the square root of both sides, we get:

$$\cos A = \pm \sqrt{\frac{576}{625}} = \pm \frac{24}{25}$$

Since the cosine of an angle is always non-negative, we take the positive value:

$$\cos A = \frac{24}{25}$$

Finding Tangent Given Sine

Now that we have found the cosine of angle A, we can use the definition of tangent to find the tangent of angle A:

$$\tan A = \frac{\sin A}{\cos A} = \frac{\frac{7}{25}}{\frac{24}{25}} = \frac{7}{24}$$

Therefore, the correct answer is:

• $\cos A = \frac{24}{25}$ and $\tan A = \frac{7}{24}$

Conclusion

In this article, we have shown how to use the Pythagorean identity to find the cosine and tangent of an angle given the sine. We have also demonstrated how to use the definition of tangent to find the tangent of an angle. By following these steps, we can solve trigonometric identities and find the values of cosine and tangent for any given angle.

Additional Examples

To further illustrate the concept, let's consider another example. Suppose we are given that $\sin A = \frac{3}{5}$. We can use the Pythagorean identity to find the cosine of angle A:

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

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

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

$$\cos^2 A = 1 - \frac{9}{25} = \frac{16}{25}$$

Taking the square root of both sides, we get:

$$\cos A = \pm \sqrt{\frac{16}{25}} = \pm \frac{4}{5}$$

Since the cosine of an angle is always non-negative, we take the positive value:

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

Now that we have found the cosine of angle A, we can use the definition of tangent to find the tangent of angle A:

$$\tan A = \frac{\sin A}{\cos A} = \frac{\frac{3}{5}}{\frac{4}{5}} = \frac{3}{4}$$

Therefore, the correct answer is:

• $\cos A = \frac{4}{5}$ and $\tan A = \frac{3}{4}$

Final Thoughts

Introduction

In our previous article, we explored how to use the Pythagorean identity to find the cosine and tangent of an angle given the sine. In this article, we will provide a Q&A guide to help you better understand and apply trigonometric identities.

Q: What is the Pythagorean identity?

A: The Pythagorean identity is a fundamental concept in trigonometry that relates the sine, cosine, and tangent of an angle. It states that for any angle A, the following equation holds:

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

Q: How do I use the Pythagorean identity to find the cosine of an angle given the sine?

A: To find the cosine of an angle given the sine, you can use the Pythagorean identity as follows:

1. Square both sides of the equation: $\sin^2 A = \left(\frac{7}{25}\right)^2 = \frac{49}{625}$
2. Substitute this value into the Pythagorean identity: $\frac{49}{625} + \cos^2 A = 1$
3. Solve for $\cos^2 A$: $\cos^2 A = 1 - \frac{49}{625} = \frac{576}{625}$
4. Take the square root of both sides: $\cos A = \pm \sqrt{\frac{576}{625}} = \pm \frac{24}{25}$
5. Take the positive value: $\cos A = \frac{24}{25}$

Q: How do I use the Pythagorean identity to find the tangent of an angle given the sine?

A: To find the tangent of an angle given the sine, you can use the definition of tangent as follows:

$$\tan A = \frac{\sin A}{\cos A}$$

Q: What is the definition of tangent?

A: The tangent of an angle is defined as the ratio of the sine of the angle to the cosine of the angle:

$$\tan A = \frac{\sin A}{\cos A}$$

Q: How do I use the Pythagorean identity to find the sine of an angle given the cosine?

A: To find the sine of an angle given the cosine, you can use the Pythagorean identity as follows:

1. Square both sides of the equation: $\cos^2 A = \left(\frac{24}{25}\right)^2 = \frac{576}{625}$
2. Substitute this value into the Pythagorean identity: $\sin^2 A + \frac{576}{625} = 1$
3. Solve for $\sin^2 A$: $\sin^2 A = 1 - \frac{576}{625} = \frac{49}{625}$
4. Take the square root of both sides: $\sin A = \pm \sqrt{\frac{49}{625}} = \pm \frac{7}{25}$
5. Take the positive value: $\sin A = \frac{7}{25}$

Q: What are some common trigonometric identities?

A: Some common trigonometric identities include:

• The Pythagorean identity: $\sin^2 A + \cos^2 A = 1$
• The definition of tangent: $\tan A = \frac{\sin A}{\cos A}$
• The definition of cotangent: $\cot A = \frac{\cos A}{\sin A}$
• The definition of secant: $\sec A = \frac{1}{\cos A}$
• The definition of cosecant: $\csc A = \frac{1}{\sin A}$

Q: How do I apply trigonometric identities in real-world problems?

A: Trigonometric identities are used in a wide range of real-world problems, including:

• Navigation: Trigonometric identities are used to calculate distances and angles in navigation.
• Physics: Trigonometric identities are used to describe the motion of objects in physics.
• Engineering: Trigonometric identities are used to design and analyze structures in engineering.
• Computer Science: Trigonometric identities are used in computer graphics and game development.

Conclusion

In this article, we have provided a Q&A guide to help you better understand and apply trigonometric identities. We have covered topics such as the Pythagorean identity, the definition of tangent, and common trigonometric identities. By following these steps and practicing with real-world problems, you can become proficient in using trigonometric identities to solve a wide range of problems.