If $\tan A = \frac{4}{5}$, Find $\sin A$.

$$$$

Introduction

In trigonometry, the tangent function is defined as the ratio of the opposite side to the adjacent side in a right-angled triangle. Given that $\tan A = \frac{4}{5}$, we are tasked with finding the value of $\sin A$. To do this, we need to use the relationship between the tangent and sine functions.

Understanding the Relationship Between Tangent and Sine

The tangent function is defined as $\tan A = \frac{\sin A}{\cos A}$. Since we are given that $\tan A = \frac{4}{5}$, we can write this as $\frac{\sin A}{\cos A} = \frac{4}{5}$. We also know that $\sin^2 A + \cos^2 A = 1$. Using these two equations, we can solve for $\sin A$.

Using the Pythagorean Identity

We can start by using the Pythagorean identity, $\sin^2 A + \cos^2 A = 1$. Since we are given that $\tan A = \frac{4}{5}$, we can write this as $\frac{\sin^2 A}{\cos^2 A} = \frac{4^2}{5^2}$. Simplifying this, we get $\frac{\sin^2 A}{\cos^2 A} = \frac{16}{25}$.

Solving for $\sin A$

We can now use the Pythagorean identity to solve for $\sin A$. We have $\sin^2 A + \cos^2 A = 1$, and we know that $\frac{\sin^2 A}{\cos^2 A} = \frac{16}{25}$. We can rewrite this as $\sin^2 A = \frac{16}{25} \cos^2 A$. Substituting this into the Pythagorean identity, we get $\frac{16}{25} \cos^2 A + \cos^2 A = 1$. Combining like terms, we get $\frac{41}{25} \cos^2 A = 1$.

Finding the Value of $\cos A$

We can now solve for $\cos A$. We have $\frac{41}{25} \cos^2 A = 1$, so $\cos^2 A = \frac{25}{41}$. Taking the square root of both sides, we get $\cos A = \pm \sqrt{\frac{25}{41}}$. Since $\cos A$ is positive in the first quadrant, we take the positive square root, so $\cos A = \frac{5}{\sqrt{41}}$.

Finding the Value of $\sin A$

We can now find the value of $\sin A$. We have $\sin^2 A = 1 - \cos^2 A$, so $\sin^2 A = 1 - \frac{25}{41}$. Simplifying this, we get $\sin^2 A = \frac{16}{41}$. Taking the square root of both sides, we get $\sin A = \pm \sqrt{\frac{16}{41}}$. Since $\sin A$ is positive in the first quadrant, we take the positive square root, so $\sin A = \frac{4}{\sqrt{41}}$.

Conclusion

In this article, we used the relationship between the tangent and sine functions to find the value of $\sin A$ given that $\tan A = \frac{4}{5}$. We used the Pythagorean identity to solve for $\sin A$, and found that $\sin A = \frac{4}{\sqrt{41}}$.

Final Answer

The final answer is $\boxed{\frac{4}{\sqrt{41}}}$.

References

• [1] "Trigonometry" by Michael Corral
• [2] "Calculus" by Michael Spivak

Additional Resources

• [1] Khan Academy: Trigonometry
• [2] MIT OpenCourseWare: Calculus

Related Articles

• [1] If $\sin A = \frac{3}{5}$, find $\cos A$.
• [2] If $\cos A = \frac{4}{5}$, find $\sin A$.

Tags

• Trigonometry
• Sine
• Cosine
• Tangent
• Pythagorean Identity
• Right-Angled Triangle
  

Introduction

In our previous article, we explored the relationship between the tangent and sine functions, and used this relationship to find the value of $\sin A$ given that $\tan A = \frac{4}{5}$. In this article, we will answer some common questions related to trigonometry and right-angled triangles.

Q: What is the difference between the sine, cosine, and tangent functions?

A: The sine, cosine, and tangent functions are all ratios of the sides of a right-angled triangle. The sine function is defined as the ratio of the opposite side to the hypotenuse, the cosine function is defined as the ratio of the adjacent side to the hypotenuse, and the tangent function is defined as the ratio of the opposite side to the adjacent side.

Q: How do I use the Pythagorean identity to solve for $\sin A$ or $\cos A$?

A: The Pythagorean identity states that $\sin^2 A + \cos^2 A = 1$. If you are given the value of $\sin A$ or $\cos A$, you can use this identity to solve for the other value. For example, if you are given that $\sin A = \frac{3}{5}$, you can use the Pythagorean identity to find that $\cos A = \pm \sqrt{1 - \sin^2 A} = \pm \sqrt{1 - \frac{9}{25}} = \pm \sqrt{\frac{16}{25}} = \pm \frac{4}{5}$.

Q: How do I use the tangent function to find the value of $\sin A$ or $\cos A$?

A: The tangent function is defined as $\tan A = \frac{\sin A}{\cos A}$. If you are given the value of $\tan A$, you can use this function to find the value of $\sin A$ or $\cos A$. For example, if you are given that $\tan A = \frac{4}{5}$, you can use the tangent function to find that $\sin A = \frac{4}{\sqrt{41}}$ and $\cos A = \frac{5}{\sqrt{41}}$.

Q: What is the relationship between the sine, cosine, and tangent functions?

A: The sine, cosine, and tangent functions are all related to each other through the Pythagorean identity. The sine and cosine functions are also related to each other through the tangent function, which is defined as $\tan A = \frac{\sin A}{\cos A}$.

Q: How do I use the sine and cosine functions to find the value of $\tan A$?

A: The tangent function is defined as $\tan A = \frac{\sin A}{\cos A}$. If you are given the values of $\sin A$ and $\cos A$, you can use these values to find the value of $\tan A$. For example, if you are given that $\sin A = \frac{3}{5}$ and $\cos A = \frac{4}{5}$, you can use the tangent function to find that $\tan A = \frac{\sin A}{\cos A} = \frac{\frac{3}{5}}{\frac{4}{5}} = \frac{3}{4}$.

Q: What is the significance of the Pythagorean identity?

A: The Pythagorean identity states that $\sin^2 A + \cos^2 A = 1$. This identity is significant because it allows us to solve for the value of $\sin A$ or $\cos A$ given the value of the other function. It also allows us to relate the sine and cosine functions to each other through the tangent function.

Q: How do I use the sine and cosine functions to find the value of $\sin A$ or $\cos A$ in a right-angled triangle?

A: To find the value of $\sin A$ or $\cos A$ in a right-angled triangle, you can use the definitions of the sine and cosine functions. The sine function is defined as the ratio of the opposite side to the hypotenuse, and the cosine function is defined as the ratio of the adjacent side to the hypotenuse. For example, if you are given a right-angled triangle with a hypotenuse of length 5 and an opposite side of length 3, you can use the sine function to find that $\sin A = \frac{3}{5}$.

Conclusion

In this article, we have answered some common questions related to trigonometry and right-angled triangles. We have discussed the relationship between the sine, cosine, and tangent functions, and how to use these functions to solve for the value of $\sin A$ or $\cos A$ given the value of the other function. We have also discussed the significance of the Pythagorean identity and how to use it to solve for the value of $\sin A$ or $\cos A$.

Final Answer

The final answer is $\boxed{\frac{4}{\sqrt{41}}}$.

References

• [1] "Trigonometry" by Michael Corral
• [2] "Calculus" by Michael Spivak

Additional Resources

• [1] Khan Academy: Trigonometry
• [2] MIT OpenCourseWare: Calculus

Related Articles

• [1] If $\sin A = \frac{3}{5}$, find $\cos A$.
• [2] If $\cos A = \frac{4}{5}$, find $\sin A$.

Tags

• Trigonometry
• Sine
• Cosine
• Tangent
• Pythagorean Identity
• Right-Angled Triangle