Given: { \sin (A)=\frac{4}{5}, \frac{\pi}{2}\ \textless \ A\ \textless \ \pi$}$ And { \sin (B)=\frac{-2 \sqrt{5}}{5}, \pi\ \textless \ B\ \textless \ \frac{3 \pi}{2}$}$.What Is The Value Of { \cos (A-B)$}$?A.

Solving Trigonometric Problems: Finding the Value of cos(A-B)

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 a trigonometric problem involving the value of cos(A-B), given the values of sin(A) and sin(B).

We are given two equations:

• $\sin (A)=\frac{4}{5}, \frac{\pi}{2} < A < \pi$
• $\sin (B)=\frac{-2 \sqrt{5}}{5}, \pi < B < \frac{3 \pi}{2}$

Our objective is to find the value of $\cos (A-B)$.

To solve this problem, we need to recall some trigonometric identities. The most relevant identity in this case is the cosine difference formula:

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

We also need to recall the Pythagorean identity:

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

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

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

Since we are given that $\sin A = \frac{4}{5}$, we can substitute this value into the equation:

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

Simplifying the equation, we get:

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

Subtracting $\frac{16}{25}$ from both sides, we get:

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

Taking the square root of both sides, we get:

$\cos A = \pm \frac{3}{5}$

Since $A$ is in the second quadrant, where cosine is negative, we take the negative value:

$\cos A = -\frac{3}{5}$

To find the value of $\cos B$, we can use the Pythagorean identity:

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

Since we are given that $\sin B = \frac{-2 \sqrt{5}}{5}$, we can substitute this value into the equation:

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

Simplifying the equation, we get:

$\frac{20}{25} + \cos^2 B = 1$

Subtracting $\frac{20}{25}$ from both sides, we get:

$\cos^2 B = \frac{5}{25}$

Taking the square root of both sides, we get:

$\cos B = \pm \frac{1}{5}$

Since $B$ is in the third quadrant, where cosine is negative, we take the negative value:

$\cos B = -\frac{1}{5}$

Now that we have found the values of $\cos A$ and $\cos B$, we can substitute them into the cosine difference formula:

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

Substituting the values, we get:

$\cos (A-B) = \left(-\frac{3}{5}\right)\left(-\frac{1}{5}\right) + \left(\frac{4}{5}\right)\left(\frac{-2 \sqrt{5}}{5}\right)$

Simplifying the equation, we get:

$\cos (A-B) = \frac{3}{25} - \frac{8 \sqrt{5}}{25}$

In this article, we have solved a trigonometric problem involving the value of cos(A-B), given the values of sin(A) and sin(B). We have used the cosine difference formula and the Pythagorean identity to find the values of cos(A) and cos(B), and then substituted them into the cosine difference formula to find the value of cos(A-B). The final answer is:

$\boxed{\frac{3}{25} - \frac{8 \sqrt{5}}{25}}$

$$

In our previous article, we solved a trigonometric problem involving the value of cos(A-B), given the values of sin(A) and sin(B). In this article, we will provide a Q&A section to help readers understand the problem and its solution better.

A: The given values of sin(A) and sin(B) are used to find the values of cos(A) and cos(B) using the Pythagorean identity. The Pythagorean identity states that the sum of the squares of the sine and cosine of an angle is equal to 1.

A: To find the value of cos(A), we use the Pythagorean identity:

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

We substitute the given value of sin(A) into the equation and solve for cos(A).

A: The quadrant in which the angle A lies is important because it determines the sign of the cosine of the angle. In the second quadrant, the cosine is negative.

A: To find the value of cos(B), we use the Pythagorean identity:

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

We substitute the given value of sin(B) into the equation and solve for cos(B).

A: The quadrant in which the angle B lies is important because it determines the sign of the cosine of the angle. In the third quadrant, the cosine is negative.

A: To find the value of cos(A-B), we use the cosine difference formula:

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

We substitute the values of cos(A), cos(B), sin(A), and sin(B) into the formula and simplify the equation.

A: The final answer to the problem is:

$\boxed{\frac{3}{25} - \frac{8 \sqrt{5}}{25}}$

• Not using the Pythagorean identity to find the values of cos(A) and cos(B).
• Not considering the quadrant in which the angles A and B lie.
• Not substituting the values of cos(A), cos(B), sin(A), and sin(B) into the cosine difference formula.
• Not simplifying the equation correctly.

In this article, we have provided a Q&A section to help readers understand the problem and its solution better. We have also highlighted some common mistakes to avoid when solving trigonometric problems. By following the steps outlined in this article, readers should be able to solve similar problems with ease.

• Always use the Pythagorean identity to find the values of cos(A) and cos(B).
• Consider the quadrant in which the angles A and B lie.
• Substitute the values of cos(A), cos(B), sin(A), and sin(B) into the cosine difference formula.
• Simplify the equation correctly.

By following these tips, readers should be able to solve trigonometric problems with confidence.