If Cos ⁡ Θ = 4 5 \cos \theta = \frac{4}{5} Cos Θ = 5 4 ​ And Cos ⁡ Φ = 7 5 2 \cos \phi = \frac{7}{5 \sqrt{2}} Cos Φ = 5 2 ​ 7 ​ , Find:a. Cos ⁡ ( Θ − Φ \cos (\theta - \phi Cos ( Θ − Φ ]b. Sin ⁡ Θ \sin \theta Sin Θ

If $\cos \theta = \frac{4}{5}$ and $\cos \phi = \frac{7}{5 \sqrt{2}}$, find:

a. $\cos (\theta - \phi)$

Introduction

In trigonometry, the cosine of the difference of two angles is a fundamental concept that has numerous applications in various fields, including physics, engineering, and mathematics. Given the values of $\cos \theta$ and $\cos \phi$, we can use the cosine difference formula to find the value of $\cos (\theta - \phi)$. In this article, we will explore the steps involved in finding the value of $\cos (\theta - \phi)$ using the given values of $\cos \theta$ and $\cos \phi$.

The Cosine Difference Formula

The cosine difference formula states that:

$$\cos (A - B) = \cos A \cos B + \sin A \sin B$$

This formula allows us to find the value of $\cos (\theta - \phi)$ using the values of $\cos \theta$, $\cos \phi$, $\sin \theta$, and $\sin \phi$.

Finding $\sin \theta$

To find the value of $\sin \theta$, we can use the Pythagorean identity:

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

Given that $\cos \theta = \frac{4}{5}$, we can substitute this value into the Pythagorean identity to find the value of $\sin \theta$.

$$\sin^2 \theta + \left(\frac{4}{5}\right)^2 = 1$$

$$\sin^2 \theta + \frac{16}{25} = 1$$

$$\sin^2 \theta = 1 - \frac{16}{25}$$

$$\sin^2 \theta = \frac{9}{25}$$

Taking the square root of both sides, we get:

$$\sin \theta = \pm \sqrt{\frac{9}{25}}$$

Since the value of $\sin \theta$ is positive in the first quadrant, we take the positive square root:

$$\sin \theta = \frac{3}{5}$$

Finding $\sin \phi$

To find the value of $\sin \phi$, we can use the Pythagorean identity:

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

Given that $\cos \phi = \frac{7}{5 \sqrt{2}}$, we can substitute this value into the Pythagorean identity to find the value of $\sin \phi$.

$$\sin^2 \phi + \left(\frac{7}{5 \sqrt{2}}\right)^2 = 1$$

$$\sin^2 \phi + \frac{49}{50} = 1$$

$$\sin^2 \phi = 1 - \frac{49}{50}$$

$$\sin^2 \phi = \frac{1}{50}$$

Taking the square root of both sides, we get:

$$\sin \phi = \pm \sqrt{\frac{1}{50}}$$

Since the value of $\sin \phi$ is positive in the first quadrant, we take the positive square root:

$$\sin \phi = \frac{1}{5 \sqrt{2}}$$

Finding $\cos (\theta - \phi)$

Now that we have found the values of $\sin \theta$ and $\sin \phi$, we can use the cosine difference formula to find the value of $\cos (\theta - \phi)$.

$$\cos (\theta - \phi) = \cos \theta \cos \phi + \sin \theta \sin \phi$$

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

$$\cos (\theta - \phi) = \frac{4}{5} \cdot \frac{7}{5 \sqrt{2}} + \frac{3}{5} \cdot \frac{1}{5 \sqrt{2}}$$

$$\cos (\theta - \phi) = \frac{28}{25 \sqrt{2}} + \frac{3}{25 \sqrt{2}}$$

$$\cos (\theta - \phi) = \frac{31}{25 \sqrt{2}}$$

Rationalizing the denominator, we get:

$$\cos (\theta - \phi) = \frac{31}{25 \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$$

$$\cos (\theta - \phi) = \frac{31 \sqrt{2}}{50}$$

Therefore, the value of $\cos (\theta - \phi)$ is $\frac{31 \sqrt{2}}{50}$.

b. $\sin \theta$

We have already found the value of $\sin \theta$ in the previous section:

$$\sin \theta = \frac{3}{5}$$

Therefore, the value of $\sin \theta$ is $\frac{3}{5}$.

Conclusion

In this article, we have used the cosine difference formula to find the value of $\cos (\theta - \phi)$ given the values of $\cos \theta$ and $\cos \phi$. We have also found the value of $\sin \theta$ using the Pythagorean identity. The results show that the value of $\cos (\theta - \phi)$ is $\frac{31 \sqrt{2}}{50}$ and the value of $\sin \theta$ is $\frac{3}{5}$.
Q&A: If $\cos \theta = \frac{4}{5}$ and $\cos \phi = \frac{7}{5 \sqrt{2}}$, find:

a. $\cos (\theta - \phi)$

b. $\sin \theta$

Q&A Session

Q: What is the cosine difference formula?

A: The cosine difference formula is a fundamental concept in trigonometry that states:

$$\cos (A - B) = \cos A \cos B + \sin A \sin B$$

This formula allows us to find the value of $\cos (\theta - \phi)$ using the values of $\cos \theta$, $\cos \phi$, $\sin \theta$, and $\sin \phi$.

Q: How do I find the value of $\sin \theta$?

A: To find the value of $\sin \theta$, we can use the Pythagorean identity:

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

Given that $\cos \theta = \frac{4}{5}$, we can substitute this value into the Pythagorean identity to find the value of $\sin \theta$.

Q: What is the Pythagorean identity?

A: The Pythagorean identity is a fundamental concept in trigonometry that states:

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

This identity allows us to find the value of $\sin \theta$ or $\cos \theta$ using the other value.

Q: How do I find the value of $\sin \phi$?

A: To find the value of $\sin \phi$, we can use the Pythagorean identity:

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

Given that $\cos \phi = \frac{7}{5 \sqrt{2}}$, we can substitute this value into the Pythagorean identity to find the value of $\sin \phi$.

Q: What is the value of $\cos (\theta - \phi)$?

A: The value of $\cos (\theta - \phi)$ is $\frac{31 \sqrt{2}}{50}$.

Q: What is the value of $\sin \theta$?

A: The value of $\sin \theta$ is $\frac{3}{5}$.

Q: Can I use the cosine difference formula to find the value of $\cos (\theta + \phi)$?

A: Yes, you can use the cosine difference formula to find the value of $\cos (\theta + \phi)$. The formula is:

$$\cos (A + B) = \cos A \cos B - \sin A \sin B$$

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

A: Yes, you can use the Pythagorean identity to find the value of $\cos \theta$. The formula is:

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

Given that $\sin \theta = \frac{3}{5}$, we can substitute this value into the Pythagorean identity to find the value of $\cos \theta$.

Q: What is the value of $\cos \theta$?

A: The value of $\cos \theta$ is $\frac{4}{5}$.

Conclusion

In this Q&A session, we have answered some common questions related to the cosine difference formula and the Pythagorean identity. We have also provided examples of how to use these formulas to find the values of $\cos (\theta - \phi)$, $\sin \theta$, and $\cos \theta$.