If $2 \sin 3 \theta=\sqrt{3}$, Then The Value Of $\theta$ Is:(a) $30^{\circ}$(b) $45^{\circ}$(c) $20^{\circ}$(d) $90^{\circ}$

Introduction

Trigonometric equations are a fundamental concept in mathematics, and solving them requires a deep understanding of trigonometric functions and their properties. In this article, we will focus on solving a specific type of trigonometric equation involving sine and cosine functions. We will use the given equation $2 \sin 3 \theta=\sqrt{3}$ to find the value of $\theta$.

Understanding the Given Equation

The given equation is $2 \sin 3 \theta=\sqrt{3}$. This equation involves the sine function and a coefficient of 2. The sine function is periodic, meaning it repeats its values at regular intervals. In this case, the sine function is evaluated at $3 \theta$, which means the angle is multiplied by 3.

Simplifying the Equation

To simplify the equation, we can start by isolating the sine function. We can do this by dividing both sides of the equation by 2, which gives us:

$$\sin 3 \theta = \frac{\sqrt{3}}{2}$$

Recalling Trigonometric Identities

The next step is to recall some trigonometric identities that can help us simplify the equation further. One such identity is the unit circle definition of sine, which states that $\sin \theta = \frac{y}{r}$, where $y$ is the $y$-coordinate of a point on the unit circle and $r$ is the radius of the circle.

Using the Unit Circle Definition

Using the unit circle definition, we can rewrite the equation as:

$$\sin 3 \theta = \frac{y}{r}$$

where $y$ is the $y$-coordinate of a point on the unit circle and $r$ is the radius of the circle.

Finding the Value of $\theta$

Now that we have simplified the equation, we can use the unit circle definition to find the value of $\theta$. We know that the sine function is equal to $\frac{\sqrt{3}}{2}$, which corresponds to a specific angle on the unit circle.

Recalling Angles on the Unit Circle

One of the angles on the unit circle that has a sine value of $\frac{\sqrt{3}}{2}$ is $60^{\circ}$. However, we need to find the value of $\theta$ that satisfies the equation $2 \sin 3 \theta=\sqrt{3}$.

Using the Periodicity of the Sine Function

The sine function is periodic, meaning it repeats its values at regular intervals. In this case, the sine function is evaluated at $3 \theta$, which means the angle is multiplied by 3. This means that the value of $\theta$ that satisfies the equation will be a multiple of the period of the sine function.

Finding the Value of $\theta$

Using the periodicity of the sine function, we can find the value of $\theta$ that satisfies the equation. We know that the sine function has a period of $360^{\circ}$, which means that the value of $\theta$ will be a multiple of $\frac{360^{\circ}}{3} = 120^{\circ}$.

Solving for $\theta$

Now that we have found the value of $\theta$, we can solve for it. We know that $\theta = 120^{\circ} \times n$, where $n$ is an integer. However, we also know that the value of $\theta$ must be between $0^{\circ}$ and $360^{\circ}$.

Finding the Correct Value of $\theta$

Using the fact that the value of $\theta$ must be between $0^{\circ}$ and $360^{\circ}$, we can find the correct value of $\theta$. We know that $\theta = 120^{\circ} \times n$, where $n$ is an integer. Trying different values of $n$, we find that $\theta = 120^{\circ} \times 1 = 120^{\circ}$ is not a solution, but $\theta = 120^{\circ} \times 2 = 240^{\circ}$ is not a solution either. However, $\theta = 120^{\circ} \times 3 = 360^{\circ}$ is a solution.

Conclusion

In conclusion, we have solved the trigonometric equation $2 \sin 3 \theta=\sqrt{3}$ to find the value of $\theta$. We used the unit circle definition of sine, the periodicity of the sine function, and the fact that the value of $\theta$ must be between $0^{\circ}$ and $360^{\circ}$ to find the correct value of $\theta$. The final answer is $\boxed{60^{\circ}}$.

Final Answer

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Q: What is the main concept of the article?

A: The main concept of the article is solving a trigonometric equation involving sine and cosine functions.

Q: What is the given equation in the article?

A: The given equation is $2 \sin 3 \theta=\sqrt{3}$.

Q: What is the unit circle definition of sine?

A: The unit circle definition of sine states that $\sin \theta = \frac{y}{r}$, where $y$ is the $y$-coordinate of a point on the unit circle and $r$ is the radius of the circle.

Q: What is the periodicity of the sine function?

A: The sine function has a period of $360^{\circ}$, meaning it repeats its values at regular intervals.

Q: How do you find the value of $\theta$ that satisfies the equation?

A: To find the value of $\theta$ that satisfies the equation, you need to use the unit circle definition of sine, the periodicity of the sine function, and the fact that the value of $\theta$ must be between $0^{\circ}$ and $360^{\circ}$.

Q: What is the final answer to the equation?

A: The final answer to the equation is $\boxed{60^{\circ}}$.

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

A: Some common trigonometric identities that can help in solving trigonometric equations include the unit circle definition of sine, the periodicity of the sine function, and the Pythagorean identity $\sin^2 \theta + \cos^2 \theta = 1$.

Q: How do you simplify a trigonometric equation?

A: To simplify a trigonometric equation, you can use various techniques such as factoring, combining like terms, and using trigonometric identities.

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

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

• Not using the correct trigonometric identity
• Not simplifying the equation properly
• Not considering the periodicity of the trigonometric function
• Not checking the solution in the original equation

Q: How do you check the solution of a trigonometric equation?

A: To check the solution of a trigonometric equation, you need to substitute the solution back into the original equation and verify that it is true.

Q: What are some real-world applications of trigonometric equations?

A: Some real-world applications of trigonometric equations include:

• Navigation: Trigonometric equations are used to calculate distances and angles in navigation.
• Physics: Trigonometric equations are used to describe the motion of objects in physics.
• Engineering: Trigonometric equations are used to design and analyze structures in engineering.
• Computer Science: Trigonometric equations are used in computer graphics and game development.