Finding the Exact Value of Cosine

Oct 13, 2025 by miodeya.com 66 views

Introduction

In trigonometry, the sine and cosine functions are used to describe the relationships between the angles and side lengths of triangles. Given the value of the sine of an angle, we can use the Pythagorean identity to find the exact value of the cosine of that angle. In this article, we will explore how to find the exact value of $\cos \theta$ when $\sin \theta = \frac{2}{5}$ .

The Pythagorean Identity

The Pythagorean identity is a fundamental relationship between the sine and cosine functions. It states that for any angle $\theta$ :

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

This identity can be used to find the value of $\cos \theta$ when $\sin \theta$ is known.

Finding the Exact Value of Cosine

Given that $\sin \theta = \frac{2}{5}$ , we can use the Pythagorean identity to find the exact value of $\cos \theta$ . We start by squaring both sides of the equation:

\sin^2 \theta = \left(\frac{2}{5}\right)^2 = \frac{4}{25}

Next, we can substitute this value into the Pythagorean identity:

\frac{4}{25} + \cos^2 \theta = 1

To solve for $\cos^2 \theta$ , we can subtract $\frac{4}{25}$ from both sides of the equation:

\cos^2 \theta = 1 - \frac{4}{25} = \frac{21}{25}

Now that we have the value of $\cos^2 \theta$ , we can take the square root of both sides to find the value of $\cos \theta$ :

\cos \theta = \pm \sqrt{\frac{21}{25}}

Since the cosine function is positive in the first and fourth quadrants, we can take the positive square root:

\cos \theta = \sqrt{\frac{21}{25}} = \frac{\sqrt{21}}{5}

Conclusion

In this article, we used the Pythagorean identity to find the exact value of $\cos \theta$ when $\sin \theta = \frac{2}{5}$ . We started by squaring both sides of the equation and then substituted the value into the Pythagorean identity. Finally, we took the square root of both sides to find the value of $\cos \theta$ . The exact value of $\cos \theta$ is $\frac{\sqrt{21}}{5}$ .

Example Use Cases

The value of $\cos \theta$ can be used in a variety of applications, including:

Navigation: In navigation, the cosine function is used to calculate the direction of a ship or aircraft.
Physics: In physics, the cosine function is used to describe the motion of objects in two-dimensional space.
Engineering: In engineering, the cosine function is used to design and analyze the performance of mechanical systems.

Step-by-Step Solution

Here is a step-by-step solution to the problem:

Squaring both sides: Square both sides of the equation $\sin \theta = \frac{2}{5}$ to get $\sin^2 \theta = \frac{4}{25}$ .
Substituting into the Pythagorean identity: Substitute the value of $\sin^2 \theta$ into the Pythagorean identity to get $\frac{4}{25} + \cos^2 \theta = 1$ .
Solving for $\cos^2 \theta$ : Subtract $\frac{4}{25}$ from both sides of the equation to get $\cos^2 \theta = \frac{21}{25}$ .
Taking the square root: Take the square root of both sides of the equation to get $\cos \theta = \pm \sqrt{\frac{21}{25}}$ .
Choosing the positive square root: Choose the positive square root to get $\cos \theta = \sqrt{\frac{21}{25}} = \frac{\sqrt{21}}{5}$ .

Common Mistakes

Here are some common mistakes to avoid when solving for $\cos \theta$ :

Not squaring both sides: Failing to square both sides of the equation can lead to incorrect results.
Not substituting into the Pythagorean identity: Failing to substitute the value of $\sin^2 \theta$ into the Pythagorean identity can lead to incorrect results.
Not choosing the positive square root: Choosing the negative square root can lead to incorrect results.

Conclusion

Q: What is the Pythagorean identity?

A: The Pythagorean identity is a fundamental relationship between the sine and cosine functions. It states that for any angle $\theta$ :

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

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

A: To use the Pythagorean identity to find the value of $\cos \theta$ , you need to square both sides of the equation $\sin \theta = \frac{2}{5}$ to get $\sin^2 \theta = \frac{4}{25}$ . Then, substitute the value of $\sin^2 \theta$ into the Pythagorean identity to get $\frac{4}{25} + \cos^2 \theta = 1$ . Finally, solve for $\cos^2 \theta$ and take the square root of both sides to find the value of $\cos \theta$ .

Q: What is the exact value of $\cos \theta$ when $\sin \theta = \frac{2}{5}$ ?

A: The exact value of $\cos \theta$ when $\sin \theta = \frac{2}{5}$ is $\frac{\sqrt{21}}{5}$ .

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

A: Yes, you can use the Pythagorean identity to find the value of $\sin \theta$ . If you know the value of $\cos \theta$ , you can square both sides of the equation to get $\cos^2 \theta = \frac{21}{25}$ . Then, substitute the value of $\cos^2 \theta$ into the Pythagorean identity to get $\sin^2 \theta = 1 - \frac{21}{25} = \frac{4}{25}$ . Finally, take the square root of both sides to find the value of $\sin \theta$ .

Q: What are some common mistakes to avoid when using the Pythagorean identity?

A: Some common mistakes to avoid when using the Pythagorean identity include:

Not squaring both sides of the equation
Not substituting the value of $\sin^2 \theta$ into the Pythagorean identity
Not choosing the positive square root

Q: Can I use the Pythagorean identity to find the value of $\tan \theta$ ?

A: Yes, you can use the Pythagorean identity to find the value of $\tan \theta$ . If you know the values of $\sin \theta$ and $\cos \theta$ , you can use the formula $\tan \theta = \frac{\sin \theta}{\cos \theta}$ to find the value of $\tan \theta$ .

Q: What are some real-world applications of the Pythagorean identity?

A: Some real-world applications of the Pythagorean identity include:

Navigation: The Pythagorean identity is used to calculate the direction of a ship or aircraft.
Physics: The Pythagorean identity is used to describe the motion of objects in two-dimensional space.
Engineering: The Pythagorean identity is used to design and analyze the performance of mechanical systems.

Q: Can I use the Pythagorean identity to find the value of $\sec \theta$ ?

A: Yes, you can use the Pythagorean identity to find the value of $\sec \theta$ . If you know the value of $\cos \theta$ , you can use the formula $\sec \theta = \frac{1}{\cos \theta}$ to find the value of $\sec \theta$ .

Q: What are some tips for memorizing the Pythagorean identity?

A: Some tips for memorizing the Pythagorean identity include:

Writing the Pythagorean identity on flashcards
Creating a mnemonic device to help you remember the Pythagorean identity
Practicing problems that involve the Pythagorean identity

Conclusion

In conclusion, the Pythagorean identity is a fundamental relationship between the sine and cosine functions. By using the Pythagorean identity, you can find the value of $\cos \theta$ when $\sin \theta$ is known. The exact value of $\cos \theta$ when $\sin \theta = \frac{2}{5}$ is $\frac{\sqrt{21}}{5}$ .