Given That { X $}$ And { Y $}$ Are Acute Angles, With${ \begin{align*} \sin X &= \frac{\sqrt{3}}{2}, \ \tan Y &= 1, \end{align*} }$and { W = 3x - 2y $}$.Calculate The Value Of { \cos W$}$. You Must

Introduction

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. In this article, we will explore how to solve trigonometric equations involving acute angles, specifically the calculation of the value of $\cos w$ given the values of $\sin x$, $\tan y$, and the expression $w = 3x - 2y$.

Understanding the Given Information

We are given that $x$ and $y$ are acute angles, with the following trigonometric values:

• $\sin x = \frac{\sqrt{3}}{2}$
• $\tan y = 1$

We are also given the expression $w = 3x - 2y$. Our goal is to calculate the value of $\cos w$.

Recalling Trigonometric Identities

Before we proceed, let's recall some essential trigonometric identities that will help us solve this problem:

• $\sin^2 x + \cos^2 x = 1$
• $\tan x = \frac{\sin x}{\cos x}$
• $\cos^2 x = 1 - \sin^2 x$

Finding the Value of $\cos x$

Using the first trigonometric identity, we can find the value of $\cos x$:

$$\cos^2 x = 1 - \sin^2 x = 1 - \left(\frac{\sqrt{3}}{2}\right)^2 = 1 - \frac{3}{4} = \frac{1}{4}$$

Taking the square root of both sides, we get:

$$\cos x = \pm \frac{1}{2}$$

Since $x$ is an acute angle, we know that $\cos x$ is positive. Therefore, we can conclude that:

$$\cos x = \frac{1}{2}$$

Finding the Value of $\sin y$

Using the second trigonometric identity, we can find the value of $\sin y$:

$$\tan y = \frac{\sin y}{\cos y} = 1$$

Since $\tan y = 1$, we know that $\sin y = \cos y$. Therefore, we can conclude that:

$$\sin y = \cos y = \frac{1}{\sqrt{2}}$$

Finding the Value of $\cos y$

Using the first trigonometric identity, we can find the value of $\cos y$:

$$\cos^2 y = 1 - \sin^2 y = 1 - \left(\frac{1}{\sqrt{2}}\right)^2 = 1 - \frac{1}{2} = \frac{1}{2}$$

Taking the square root of both sides, we get:

$$\cos y = \pm \frac{1}{\sqrt{2}}$$

Since $y$ is an acute angle, we know that $\cos y$ is positive. Therefore, we can conclude that:

$$\cos y = \frac{1}{\sqrt{2}}$$

Finding the Value of $\cos w$

Now that we have found the values of $\cos x$ and $\cos y$, we can use the expression $w = 3x - 2y$ to find the value of $\cos w$:

$$\cos w = \cos (3x - 2y)$$

Using the angle subtraction formula, we get:

$$\cos w = \cos 3x \cos 2y + \sin 3x \sin 2y$$

Substituting the values of $\cos x$, $\cos y$, $\sin x$, and $\sin y$, we get:

$$\cos w = \left(\frac{1}{2}\right)^3 \left(\frac{1}{2}\right) + \left(\frac{\sqrt{3}}{2}\right)^3 \left(\frac{1}{\sqrt{2}}\right)$$

Simplifying the expression, we get:

$$\cos w = \frac{1}{32} + \frac{3\sqrt{3}}{32}$$

Combining the terms, we get:

$$\cos w = \frac{1 + 3\sqrt{3}}{32}$$

Conclusion

In this article, we have shown how to solve a trigonometric equation involving acute angles. We have used the given values of $\sin x$, $\tan y$, and the expression $w = 3x - 2y$ to find the value of $\cos w$. Our final answer is:

$$\cos w = \frac{1 + 3\sqrt{3}}{32}$$

Q: What are trigonometric equations?

A: Trigonometric equations are mathematical equations that involve trigonometric functions, such as sine, cosine, and tangent. These equations can be used to solve problems involving right triangles, circular functions, and other mathematical concepts.

Q: What are acute angles?

A: Acute angles are angles that are less than 90 degrees. In the context of trigonometric equations, acute angles are often used to simplify calculations and provide a more straightforward solution.

Q: How do I solve a trigonometric equation involving acute angles?

A: To solve a trigonometric equation involving acute angles, you can use the following steps:

1. Identify the given values of the trigonometric functions, such as sine, cosine, and tangent.
2. Use the trigonometric identities to simplify the equation and isolate the unknown angle.
3. Use the inverse trigonometric functions to find the value of the unknown angle.
4. Substitute the value of the unknown angle into the original equation to find the solution.

Q: What are some common trigonometric identities?

A: Some common trigonometric identities include:

• $\sin^2 x + \cos^2 x = 1$
• $\tan x = \frac{\sin x}{\cos x}$
• $\cos^2 x = 1 - \sin^2 x$

Q: How do I use the angle subtraction formula?

A: The angle subtraction formula is used to find the cosine of the difference between two angles. The formula is:

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

Q: What is the significance of the value of $\cos w$?

A: The value of $\cos w$ represents the cosine of the angle $w$. In the context of the original problem, the value of $\cos w$ is used to find the solution to the trigonometric equation.

Q: How do I simplify the expression for $\cos w$?

A: To simplify the expression for $\cos w$, you can use the following steps:

1. Substitute the values of $\cos x$, $\cos y$, $\sin x$, and $\sin y$ into the expression.
2. Simplify the expression by combining like terms.
3. Use the trigonometric identities to further simplify the expression.

Q: What is the final answer for $\cos w$?

A: The final answer for $\cos w$ is:

$$\cos w = \frac{1 + 3\sqrt{3}}{32}$$

Conclusion

In this article, we have provided a comprehensive guide to solving trigonometric equations involving acute angles. We have covered the basics of trigonometric equations, acute angles, and trigonometric identities. We have also provided a step-by-step solution to the original problem and answered some frequently asked questions. We hope that this article has provided a clear and concise explanation of how to solve trigonometric equations involving acute angles.