Given That \[$-4. \sin X=\frac{5}{13}\$\] And \[$\operatorname{tg} X\ \textless \ 0\$\], Find \[$\operatorname{ctg} 0.5 X\$\].

Introduction

Trigonometric equations are a fundamental concept in mathematics, and solving them requires a deep understanding of trigonometric functions and their relationships. In this article, we will explore how to solve a given trigonometric equation involving sine and tangent functions, and then find the value of cotangent of half the angle.

Given Equation

The given equation is:

$$-4 \sin x = \frac{5}{13}$$

We are also given that $\operatorname{tg} x < 0$. Our goal is to find the value of $\operatorname{ctg} 0.5 x$.

Step 1: Isolate the Sine Function

To start solving the equation, we need to isolate the sine function. We can do this by dividing both sides of the equation by $-4$:

$$\sin x = -\frac{5}{52}$$

Step 2: Find the Value of Cosine

Since we know the value of sine, we can use the Pythagorean identity to find the value of cosine:

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

Substituting the value of sine, we get:

$$\cos^2 x = 1 - \left(-\frac{5}{52}\right)^2$$

Simplifying, we get:

$$\cos^2 x = 1 - \frac{25}{2704}$$

$$\cos^2 x = \frac{2679}{2704}$$

Taking the square root of both sides, we get:

$$\cos x = \pm \sqrt{\frac{2679}{2704}}$$

Since $\operatorname{tg} x < 0$, we know that $x$ is in the third or fourth quadrant. In these quadrants, cosine is negative. Therefore, we take the negative square root:

$$\cos x = -\sqrt{\frac{2679}{2704}}$$

Step 3: Find the Value of Tangent

Now that we have the values of sine and cosine, we can find the value of tangent:

$$\operatorname{tg} x = \frac{\sin x}{\cos x}$$

Substituting the values, we get:

$$\operatorname{tg} x = \frac{-\frac{5}{52}}{-\sqrt{\frac{2679}{2704}}}$$

Simplifying, we get:

$$\operatorname{tg} x = \frac{5}{52} \cdot \sqrt{\frac{2704}{2679}}$$

$$\operatorname{tg} x = \frac{5}{52} \cdot \frac{52}{\sqrt{2679}}$$

$$\operatorname{tg} x = \frac{5}{\sqrt{2679}}$$

Step 4: Find the Value of Cotangent

Finally, we can find the value of cotangent:

$$\operatorname{ctg} x = \frac{1}{\operatorname{tg} x}$$

Substituting the value of tangent, we get:

$$\operatorname{ctg} x = \frac{1}{\frac{5}{\sqrt{2679}}}$$

Simplifying, we get:

$$\operatorname{ctg} x = \frac{\sqrt{2679}}{5}$$

Step 5: Find the Value of Cotangent of Half the Angle

Now that we have the value of cotangent, we can find the value of cotangent of half the angle:

$$\operatorname{ctg} 0.5 x = \frac{1}{\operatorname{tg} 0.5 x}$$

Using the half-angle formula for tangent, we get:

$$\operatorname{tg} 0.5 x = \frac{1 - \operatorname{tg}^2 x}{2 \operatorname{tg} x}$$

Substituting the value of tangent, we get:

$$\operatorname{tg} 0.5 x = \frac{1 - \left(\frac{5}{\sqrt{2679}}\right)^2}{2 \cdot \frac{5}{\sqrt{2679}}}$$

Simplifying, we get:

$$\operatorname{tg} 0.5 x = \frac{1 - \frac{25}{2679}}{\frac{10}{\sqrt{2679}}}$$

$$\operatorname{tg} 0.5 x = \frac{\frac{2654}{2679}}{\frac{10}{\sqrt{2679}}}$$

$$\operatorname{tg} 0.5 x = \frac{2654}{2679} \cdot \frac{\sqrt{2679}}{10}$$

$$\operatorname{tg} 0.5 x = \frac{2654}{50}$$

$$\operatorname{tg} 0.5 x = \frac{1327}{25}$$

Now that we have the value of tangent of half the angle, we can find the value of cotangent:

$$\operatorname{ctg} 0.5 x = \frac{1}{\operatorname{tg} 0.5 x}$$

Substituting the value of tangent, we get:

$$\operatorname{ctg} 0.5 x = \frac{1}{\frac{1327}{25}}$$

Simplifying, we get:

$$\operatorname{ctg} 0.5 x = \frac{25}{1327}$$

Conclusion

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Q: What is the main goal of the article?

A: The main goal of the article is to solve a given trigonometric equation involving sine and tangent functions, and then find the value of cotangent of half the angle.

Q: What are the given conditions of the problem?

A: The given conditions of the problem are:

• $-4 \sin x = \frac{5}{13}$
• $\operatorname{tg} x < 0$

Q: What is the first step in solving the equation?

A: The first step in solving the equation is to isolate the sine function by dividing both sides of the equation by $-4$.

Q: How do we find the value of cosine?

A: We find the value of cosine by using the Pythagorean identity, which states that $\cos^2 x = 1 - \sin^2 x$.

Q: What is the relationship between tangent and cotangent?

A: The relationship between tangent and cotangent is given by the formula $\operatorname{ctg} x = \frac{1}{\operatorname{tg} x}$.

Q: How do we find the value of cotangent of half the angle?

A: We find the value of cotangent of half the angle by using the half-angle formula for tangent, which states that $\operatorname{tg} 0.5 x = \frac{1 - \operatorname{tg}^2 x}{2 \operatorname{tg} x}$.

Q: What is the final answer to the problem?

A: The final answer to the problem is $\operatorname{ctg} 0.5 x = \frac{25}{1327}$.

Q: What are some common trigonometric identities that are used in solving trigonometric equations?

A: Some common trigonometric identities that are used in solving trigonometric equations include:

• $\sin^2 x + \cos^2 x = 1$
• $\tan^2 x + 1 = \sec^2 x$
• $\cot^2 x + 1 = \csc^2 x$

Q: How do we use trigonometric identities to simplify expressions?

A: We use trigonometric identities to simplify expressions by substituting the values of trigonometric functions into the identity and simplifying the resulting expression.

Q: What are some tips for solving trigonometric equations?

A: Some tips for solving trigonometric equations include:

• Start by isolating the trigonometric function that is being solved for.
• Use trigonometric identities to simplify the expression.
• Use the unit circle to visualize the trigonometric functions and their relationships.
• Check the solution by plugging it back into the original equation.

Q: What are some common mistakes to avoid when solving trigonometric equations?

A: Some common mistakes to avoid when solving trigonometric equations include:

• Not isolating the trigonometric function that is being solved for.
• Not using trigonometric identities to simplify the expression.
• Not checking the solution by plugging it back into the original equation.
• Not considering the domain and range of the trigonometric functions.