Solve The Equation For { \theta$} : : : { \begin{aligned} 3 \tan \theta - 4 &= 0 \\ \therefore \tan \theta &= \frac{4}{3} \end{aligned} \}

by ADMIN 139 views

Solving Trigonometric Equations: A Step-by-Step Guide to Finding the Value of {\theta$}$

Trigonometric equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving a specific type of trigonometric equation involving the tangent function. We will use the given equation to find the value of {\theta$}$, and provide a step-by-step guide on how to approach similar problems.

The Given Equation

The given equation is:

3tan⁡θ−4=03 \tan \theta - 4 = 0

We are asked to solve for {\theta$}$, which means we need to find the value of {\theta$}$ that satisfies the equation.

Step 1: Isolate the Tangent Function

To solve the equation, we first need to isolate the tangent function. We can do this by adding 4 to both sides of the equation:

3tan⁡θ=43 \tan \theta = 4

Step 2: Divide Both Sides by 3

Next, we need to divide both sides of the equation by 3 to get:

tan⁡θ=43\tan \theta = \frac{4}{3}

Step 3: Find the Value of {\theta$}$

Now that we have isolated the tangent function, we can use the inverse tangent function (also known as the arctangent function) to find the value of {\theta$}$. The inverse tangent function is denoted by {\arctan x$}$, and it returns the angle whose tangent is equal to x.

In this case, we have:

θ=arctan⁡(43)\theta = \arctan \left(\frac{4}{3}\right)

Using a Calculator or Trigonometric Table

To find the value of {\theta$}$, we can use a calculator or a trigonometric table. Using a calculator, we get:

θ≈0.9273 radians\theta \approx 0.9273 \text{ radians}

or

θ≈53.13∘\theta \approx 53.13^\circ

Understanding the Solution

So, what does this solution mean? In other words, what is the value of {\theta$}$ that satisfies the given equation? The solution tells us that the tangent of {\theta$}$ is equal to {\frac{4}{3}$}$. This means that the ratio of the opposite side to the adjacent side in a right triangle is equal to {\frac{4}{3}$}$.

Visualizing the Solution

To better understand the solution, let's visualize it. We can draw a right triangle with an angle {\theta$}$ and opposite side of length 4 and adjacent side of length 3. The tangent of {\theta$}$ is then equal to the ratio of the opposite side to the adjacent side, which is {\frac{4}{3}$}$.

In this article, we solved a trigonometric equation involving the tangent function. We used the given equation to find the value of {\theta$}$, and provided a step-by-step guide on how to approach similar problems. We also used a calculator or trigonometric table to find the value of {\theta$}$, and visualized the solution to better understand it.

Here are some common trigonometric equations that you may encounter:

  • Sine equation: sin⁡θ=12\sin \theta = \frac{1}{2}
  • Cosine equation: cos⁡θ=13\cos \theta = \frac{1}{3}
  • Tangent equation: tan⁡θ=23\tan \theta = \frac{2}{3}
  • Secant equation: sec⁡θ=12\sec \theta = \frac{1}{2}
  • Cosecant equation: csc⁡θ=13\csc \theta = \frac{1}{3}
  • Cotangent equation: cot⁡θ=23\cot \theta = \frac{2}{3}

Here are some tips and tricks to help you solve trigonometric equations:

  • Use the inverse trigonometric functions: Use the inverse trigonometric functions (such as arctangent, arcsine, and arccosine) to find the value of the angle.
  • Use a calculator or trigonometric table: Use a calculator or trigonometric table to find the value of the angle.
  • Visualize the solution: Visualize the solution to better understand it.
  • Check your work: Check your work to make sure that you have solved the equation correctly.

Trigonometric equations have many real-world applications, including:

  • Navigation: Trigonometric equations are used in navigation to find the position and direction of a ship or aircraft.
  • Surveying: Trigonometric equations are used in surveying to find the distance and angle between two points.
  • Physics: Trigonometric equations are used in physics to describe the motion of objects.
  • Engineering: Trigonometric equations are used in engineering to design and analyze systems.

In conclusion, solving trigonometric equations is an important skill that has many real-world applications. By following the steps outlined in this article, you can solve trigonometric equations involving the tangent function. Remember to use the inverse trigonometric functions, a calculator or trigonometric table, and to visualize the solution to better understand it. With practice and patience, you can become proficient in solving trigonometric equations and apply them to real-world problems.
Frequently Asked Questions (FAQs) About Solving Trigonometric Equations

Q: What is a trigonometric equation?

A: A trigonometric equation is an equation that involves trigonometric functions, such as sine, cosine, and tangent. These equations are used to describe the relationships between the angles and side lengths of triangles.

Q: What is the tangent function?

A: The tangent function is a trigonometric function that is defined as the ratio of the opposite side to the adjacent side in a right triangle. It is denoted by the symbol tan.

Q: How do I solve a trigonometric equation?

A: To solve a trigonometric equation, you need to isolate the trigonometric function and then use the inverse trigonometric function to find the value of the angle. You can use a calculator or trigonometric table to find the value of the angle.

Q: What is the inverse tangent function?

A: The inverse tangent function is a function that returns the angle whose tangent is equal to a given value. It is denoted by the symbol arctan.

Q: How do I use a calculator to solve a trigonometric equation?

A: To use a calculator to solve a trigonometric equation, you need to enter the equation into the calculator and then use the inverse trigonometric function to find the value of the angle.

Q: What are some common trigonometric equations?

A: Some common trigonometric equations include:

  • Sine equation: sin⁡θ=12\sin \theta = \frac{1}{2}
  • Cosine equation: cos⁡θ=13\cos \theta = \frac{1}{3}
  • Tangent equation: tan⁡θ=23\tan \theta = \frac{2}{3}
  • Secant equation: sec⁡θ=12\sec \theta = \frac{1}{2}
  • Cosecant equation: csc⁡θ=13\csc \theta = \frac{1}{3}
  • Cotangent equation: cot⁡θ=23\cot \theta = \frac{2}{3}

Q: How do I visualize a trigonometric equation?

A: To visualize a trigonometric equation, you can draw a right triangle with an angle and opposite and adjacent sides. You can then use the trigonometric function to find the ratio of the opposite side to the adjacent side.

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

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

  • Navigation: Trigonometric equations are used in navigation to find the position and direction of a ship or aircraft.
  • Surveying: Trigonometric equations are used in surveying to find the distance and angle between two points.
  • Physics: Trigonometric equations are used in physics to describe the motion of objects.
  • Engineering: Trigonometric equations are used in engineering to design and analyze systems.

Q: How do I check my work when solving a trigonometric equation?

A: To check your work when solving a trigonometric equation, you can plug the value of the angle back into the original equation and make sure that it is true.

Q: What are some tips and tricks for solving trigonometric equations?

A: Some tips and tricks for solving trigonometric equations include:

  • Use the inverse trigonometric functions: Use the inverse trigonometric functions (such as arctangent, arcsine, and arccosine) to find the value of the angle.
  • Use a calculator or trigonometric table: Use a calculator or trigonometric table to find the value of the angle.
  • Visualize the solution: Visualize the solution to better understand it.
  • Check your work: Check your work to make sure that you have solved the equation correctly.

In conclusion, solving trigonometric equations is an important skill that has many real-world applications. By following the steps outlined in this article, you can solve trigonometric equations involving the tangent function. Remember to use the inverse trigonometric functions, a calculator or trigonometric table, and to visualize the solution to better understand it. With practice and patience, you can become proficient in solving trigonometric equations and apply them to real-world problems.