QUESTION 3 PLC-CONTROL TEST 1.20If 3 Tan ⁡ Θ = − 2 3 \tan \theta = -2 3 Tan Θ = − 2 And Sin ⁡ Θ \textgreater 0 \sin \theta \ \textgreater \ 0 Sin Θ \textgreater 0 , With The Aid Of A Diagram, Calculate The Value Of Cos ⁡ Θ ⋅ Sin ⁡ Θ \cos \theta \cdot \sin \theta Cos Θ ⋅ Sin Θ .Given: Sin ⁡ 23 ∘ = P \sin 23^{\circ} = P Sin 2 3 ∘ = P , WITHOUT

Introduction

Trigonometric equations are a fundamental concept in mathematics, and solving them requires a deep understanding of the relationships between the different trigonometric functions. In this article, we will focus on solving a specific trigonometric equation involving the tangent function, and then use the result to calculate the value of the product of the cosine and sine functions.

The Given Equation

The given equation is $3 \tan \theta = -2$. We are also given that $\sin \theta > 0$. Our goal is to calculate the value of $\cos \theta \cdot \sin \theta$ using a diagram.

Step 1: Understanding the Tangent Function

The tangent function is defined as the ratio of the sine and cosine functions:

$$\tan \theta = \frac{\sin \theta}{\cos \theta}$$

Using this definition, we can rewrite the given equation as:

$$3 \frac{\sin \theta}{\cos \theta} = -2$$

Step 2: Solving for $\sin \theta$ and $\cos \theta$

To solve for $\sin \theta$ and $\cos \theta$, we can rearrange the equation to isolate the ratio of the sine and cosine functions:

$$\frac{\sin \theta}{\cos \theta} = -\frac{2}{3}$$

Since $\sin \theta > 0$, we know that $\sin \theta$ is positive. Therefore, we can conclude that $\cos \theta$ is negative.

Step 3: Using a Diagram to Visualize the Solution

To visualize the solution, we can use a diagram to represent the relationship between the sine and cosine functions. Let's consider a right triangle with an acute angle $\theta$. The sine of $\theta$ is the ratio of the length of the opposite side to the length of the hypotenuse, while the cosine of $\theta$ is the ratio of the length of the adjacent side to the length of the hypotenuse.

Using this diagram, we can see that the tangent of $\theta$ is the ratio of the opposite side to the adjacent side. Since we know that $\tan \theta = -\frac{2}{3}$, we can conclude that the opposite side is $-2$ and the adjacent side is $-3$.

Step 4: Calculating the Value of $\cos \theta \cdot \sin \theta$

Now that we have the values of the opposite and adjacent sides, we can calculate the value of $\cos \theta \cdot \sin \theta$. Using the diagram, we can see that the product of the sine and cosine functions is equal to the area of the triangle:

$$\cos \theta \cdot \sin \theta = \frac{1}{2} \cdot (-2) \cdot (-3) = 3$$

Conclusion

In this article, we solved a trigonometric equation involving the tangent function and used the result to calculate the value of the product of the cosine and sine functions. We also used a diagram to visualize the solution and understand the relationships between the different trigonometric functions.

Given: $\sin 23^{\circ} = p$

To solve the given equation, we can use the fact that $\sin 23^{\circ} = p$. We can then use the trigonometric identity $\sin^2 \theta + \cos^2 \theta = 1$ to solve for $\cos 23^{\circ}$.

Step 1: Using the Trigonometric Identity

Using the trigonometric identity $\sin^2 \theta + \cos^2 \theta = 1$, we can rewrite the equation as:

$$\sin^2 23^{\circ} + \cos^2 23^{\circ} = 1$$

Step 2: Solving for $\cos 23^{\circ}$

Since we know that $\sin 23^{\circ} = p$, we can substitute this value into the equation:

$$p^2 + \cos^2 23^{\circ} = 1$$

Solving for $\cos 23^{\circ}$, we get:

$$\cos^2 23^{\circ} = 1 - p^2$$

Taking the square root of both sides, we get:

$$\cos 23^{\circ} = \pm \sqrt{1 - p^2}$$

Since $\cos 23^{\circ}$ is positive, we can conclude that:

$$\cos 23^{\circ} = \sqrt{1 - p^2}$$

Step 3: Calculating the Value of $\cos 23^{\circ} \cdot \sin 23^{\circ}$

Now that we have the value of $\cos 23^{\circ}$, we can calculate the value of $\cos 23^{\circ} \cdot \sin 23^{\circ}$:

$$\cos 23^{\circ} \cdot \sin 23^{\circ} = \sqrt{1 - p^2} \cdot p$$

Using the fact that $\sin 23^{\circ} = p$, we can simplify the expression:

$$\cos 23^{\circ} \cdot \sin 23^{\circ} = \sqrt{1 - p^2} \cdot p = \sqrt{1 - p^2} \cdot \sin 23^{\circ}$$

Conclusion

In this article, we solved a trigonometric equation involving the sine function and used the result to calculate the value of the product of the cosine and sine functions. We also used a diagram to visualize the solution and understand the relationships between the different trigonometric functions.

Final Answer

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Q: What is the tangent function?

A: The tangent function is a trigonometric function that is defined as the ratio of the sine and cosine functions:

$$\tan \theta = \frac{\sin \theta}{\cos \theta}$$

Q: How do I solve a trigonometric equation involving the tangent function?

A: To solve a trigonometric equation involving the tangent function, you can use the definition of the tangent function and rearrange the equation to isolate the ratio of the sine and cosine functions. You can then use a diagram to visualize the solution and understand the relationships between the different trigonometric functions.

Q: What is the relationship between the sine and cosine functions?

A: The sine and cosine functions are related by the trigonometric identity:

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

This identity can be used to solve for the value of one function given the value of the other function.

Q: How do I calculate the value of the product of the cosine and sine functions?

A: To calculate the value of the product of the cosine and sine functions, you can use the diagram to visualize the solution and understand the relationships between the different trigonometric functions. You can then use the values of the opposite and adjacent sides to calculate the value of the product.

Q: What is the value of $\cos 23^{\circ} \cdot \sin 23^{\circ}$?

A: The value of $\cos 23^{\circ} \cdot \sin 23^{\circ}$ is equal to the area of the triangle:

$$\cos 23^{\circ} \cdot \sin 23^{\circ} = \frac{1}{2} \cdot (-2) \cdot (-3) = 3$$

Q: How do I use the given equation $\sin 23^{\circ} = p$ to solve for $\cos 23^{\circ}$?

A: To solve for $\cos 23^{\circ}$, you can use the trigonometric identity $\sin^2 \theta + \cos^2 \theta = 1$ and substitute the value of $\sin 23^{\circ}$ into the equation. You can then solve for $\cos 23^{\circ}$ and calculate the value of $\cos 23^{\circ} \cdot \sin 23^{\circ}$.

Q: What is the final answer to the problem?

A: The final answer to the problem is $\boxed{3}$.

Common Mistakes to Avoid

• Not using the definition of the tangent function: Make sure to use the definition of the tangent function to solve the equation.
• Not using a diagram to visualize the solution: A diagram can help you understand the relationships between the different trigonometric functions and make it easier to solve the equation.
• Not using the trigonometric identity: Make sure to use the trigonometric identity to solve for the value of one function given the value of the other function.
• Not calculating the value of the product of the cosine and sine functions: Make sure to calculate the value of the product of the cosine and sine functions using the values of the opposite and adjacent sides.

Conclusion

In this article, we answered some frequently asked questions about trigonometric equations and provided some tips and tricks to help you solve them. We also provided a final answer to the problem and some common mistakes to avoid.