Find The Exact Value Of Cos ⁡ ( Α + Β \cos (\alpha+\beta Cos ( Α + Β ], Given Sin ⁡ Α = 2 5 \sin \alpha=\frac{2}{5} Sin Α = 5 2 ​ For Α \alpha Α In Quadrant II And Cos ⁡ Β = 1 3 \cos \beta=\frac{1}{3} Cos Β = 3 1 ​ For Β \beta Β In Quadrant IV.A. $\frac{8

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Introduction

In trigonometry, the sum and difference formulas are essential for finding the values of trigonometric functions. The sum formula for cosine is one of the most important formulas in trigonometry, which states that $\cos (A+B) = \cos A \cos B - \sin A \sin B$. In this article, we will use this formula to find the exact value of $\cos (\alpha+\beta)$, given $\sin \alpha=\frac{2}{5}$ for $\alpha$ in Quadrant II and $\cos \beta=\frac{1}{3}$ for $\beta$ in Quadrant IV.

Understanding the Problem

To find the exact value of $\cos (\alpha+\beta)$, we need to use the sum formula for cosine. However, we are given the values of $\sin \alpha$ and $\cos \beta$, but not the values of $\cos \alpha$ and $\sin \beta$. We also need to consider the quadrants in which $\alpha$ and $\beta$ lie.

Quadrant II and Quadrant IV

In Quadrant II, the sine function is positive, and the cosine function is negative. In Quadrant IV, the sine function is negative, and the cosine function is positive. Since $\alpha$ lies in Quadrant II, we know that $\cos \alpha$ is negative. Similarly, since $\beta$ lies in Quadrant IV, we know that $\sin \beta$ is negative.

Finding the Values of $\cos \alpha$ and $\sin \beta$

To find the values of $\cos \alpha$ and $\sin \beta$, we can use the Pythagorean identity, which states that $\sin^2 x + \cos^2 x = 1$. We can rearrange this identity to solve for $\cos \alpha$ and $\sin \beta$.

For $\alpha$ in Quadrant II, we have:

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

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

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

$$\cos^2 \alpha = 1 - \frac{4}{25}$$

$$\cos^2 \alpha = \frac{21}{25}$$

$$\cos \alpha = \pm \sqrt{\frac{21}{25}}$$

Since $\alpha$ lies in Quadrant II, we know that $\cos \alpha$ is negative, so we take the negative square root:

$$\cos \alpha = -\sqrt{\frac{21}{25}}$$

For $\beta$ in Quadrant IV, we have:

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

$$\sin^2 \beta + \left(\frac{1}{3}\right)^2 = 1$$

$$\sin^2 \beta + \frac{1}{9} = 1$$

$$\sin^2 \beta = 1 - \frac{1}{9}$$

$$\sin^2 \beta = \frac{8}{9}$$

$$\sin \beta = \pm \sqrt{\frac{8}{9}}$$

Since $\beta$ lies in Quadrant IV, we know that $\sin \beta$ is negative, so we take the negative square root:

$$\sin \beta = -\sqrt{\frac{8}{9}}$$

Using the Sum Formula for Cosine

Now that we have the values of $\cos \alpha$, $\sin \alpha$, $\cos \beta$, and $\sin \beta$, we can use the sum formula for cosine to find the exact value of $\cos (\alpha+\beta)$.

$$\cos (\alpha+\beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta$$

$$\cos (\alpha+\beta) = \left(-\sqrt{\frac{21}{25}}\right)\left(\frac{1}{3}\right) - \left(\frac{2}{5}\right)\left(-\sqrt{\frac{8}{9}}\right)$$

$$\cos (\alpha+\beta) = -\frac{\sqrt{21}}{15} + \frac{4\sqrt{2}}{15}$$

$$\cos (\alpha+\beta) = \frac{-\sqrt{21} + 4\sqrt{2}}{15}$$

Conclusion

In this article, we used the sum formula for cosine to find the exact value of $\cos (\alpha+\beta)$, given $\sin \alpha=\frac{2}{5}$ for $\alpha$ in Quadrant II and $\cos \beta=\frac{1}{3}$ for $\beta$ in Quadrant IV. We also used the Pythagorean identity to find the values of $\cos \alpha$ and $\sin \beta$. The final answer is $\frac{-\sqrt{21} + 4\sqrt{2}}{15}$.

Final Answer

The final answer is: $\boxed{\frac{-\sqrt{21} + 4\sqrt{2}}{15}}$

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Introduction

In our previous article, we used the sum formula for cosine to find the exact value of $\cos (\alpha+\beta)$, given $\sin \alpha=\frac{2}{5}$ for $\alpha$ in Quadrant II and $\cos \beta=\frac{1}{3}$ for $\beta$ in Quadrant IV. In this article, we will answer some frequently asked questions related to this topic.

Q1: What is the sum formula for cosine?

A1: The sum formula for cosine is $\cos (A+B) = \cos A \cos B - \sin A \sin B$. This formula is used to find the value of the cosine of the sum of two angles.

Q2: How do I find the values of $\cos \alpha$ and $\sin \beta$?

A2: To find the values of $\cos \alpha$ and $\sin \beta$, you can use the Pythagorean identity, which states that $\sin^2 x + \cos^2 x = 1$. You can rearrange this identity to solve for $\cos \alpha$ and $\sin \beta$.

Q3: What is the Pythagorean identity?

A3: The Pythagorean identity is $\sin^2 x + \cos^2 x = 1$. This identity is used to find the values of $\cos \alpha$ and $\sin \beta$.

Q4: How do I know which quadrant $\alpha$ and $\beta$ lie in?

A4: To determine which quadrant $\alpha$ and $\beta$ lie in, you need to consider the signs of the sine and cosine functions. In Quadrant II, the sine function is positive, and the cosine function is negative. In Quadrant IV, the sine function is negative, and the cosine function is positive.

Q5: What is the final answer to the problem?

A5: The final answer to the problem is $\frac{-\sqrt{21} + 4\sqrt{2}}{15}$.

Q6: Can I use the sum formula for cosine to find the value of $\sin (\alpha+\beta)$?

A6: Yes, you can use the sum formula for sine to find the value of $\sin (\alpha+\beta)$. The sum formula for sine is $\sin (A+B) = \sin A \cos B + \cos A \sin B$.

Q7: How do I find the value of $\tan (\alpha+\beta)$?

A7: To find the value of $\tan (\alpha+\beta)$, you can use the formula $\tan (A+B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}$.

Q8: Can I use the sum formula for cosine to find the value of $\cos (\alpha-\beta)$?

A8: Yes, you can use the sum formula for cosine to find the value of $\cos (\alpha-\beta)$. The sum formula for cosine is $\cos (A-B) = \cos A \cos B + \sin A \sin B$.

Q9: How do I find the value of $\sin (\alpha-\beta)$?

A9: To find the value of $\sin (\alpha-\beta)$, you can use the formula $\sin (A-B) = \sin A \cos B - \cos A \sin B$.

Q10: Can I use the sum formula for cosine to find the value of $\cos (2\alpha)$?

A10: Yes, you can use the sum formula for cosine to find the value of $\cos (2\alpha)$. The sum formula for cosine is $\cos (2A) = 2\cos^2 A - 1$.

Conclusion

In this article, we answered some frequently asked questions related to finding the exact value of $\cos (\alpha+\beta)$. We also provided some additional information on how to use the sum formula for cosine to find the values of $\sin (\alpha+\beta)$, $\tan (\alpha+\beta)$, $\cos (\alpha-\beta)$, $\sin (\alpha-\beta)$, and $\cos (2\alpha)$.