Which Equation Can Be Used To Solve For B B B ?A. Tan ⁡ ( 30 ∘ ) = 5 B \tan \left(30^{\circ}\right)=\frac{5}{b} Tan ( 3 0 ∘ ) = B 5 ​ B. Tan ⁡ ( 300 ∘ ) = B 5 \tan (300^{\circ})=\frac{b}{5} Tan ( 30 0 ∘ ) = 5 B ​

Solving for $b$: A Comprehensive Guide to Trigonometric Equations

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It is a fundamental subject that has numerous applications in various fields, including physics, engineering, and navigation. In this article, we will focus on solving for $b$ in two given trigonometric equations. We will analyze each equation, identify the appropriate trigonometric function, and provide step-by-step solutions to find the value of $b$.

We are given two trigonometric equations:

A. $\tan \left(30^{\circ}\right)=\frac{5}{b}$ B. $\tan (300^{\circ})=\frac{b}{5}$

To solve for $b$, we need to understand the properties of the tangent function and how it relates to the given angles.

The Tangent Function

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

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

The tangent function has a periodicity of $180^{\circ}$, which means that the value of $\tan \theta$ repeats every $180^{\circ}$. This property is essential in solving trigonometric equations.

Equation A: $\tan \left(30^{\circ}\right)=\frac{5}{b}$

To solve for $b$, we can start by evaluating the tangent of $30^{\circ}$:

$\tan \left(30^{\circ}\right) = \frac{1}{\sqrt{3}}$

Now, we can substitute this value into the equation:

$\frac{1}{\sqrt{3}} = \frac{5}{b}$

To solve for $b$, we can multiply both sides of the equation by $b$ and then divide by $\frac{1}{\sqrt{3}}$:

$b = 5\sqrt{3}$

Therefore, the value of $b$ in equation A is $5\sqrt{3}$.

Equation B: $\tan (300^{\circ})=\frac{b}{5}$

To solve for $b$, we can start by evaluating the tangent of $300^{\circ}$:

$\tan (300^{\circ}) = \tan (180^{\circ} + 120^{\circ})$

Using the periodicity of the tangent function, we can rewrite this as:

$\tan (300^{\circ}) = \tan (120^{\circ})$

Now, we can evaluate the tangent of $120^{\circ}$:

$\tan (120^{\circ}) = -\sqrt{3}$

Now, we can substitute this value into the equation:

$-\sqrt{3} = \frac{b}{5}$

To solve for $b$, we can multiply both sides of the equation by $5$ and then divide by $-\sqrt{3}$:

$b = -5\sqrt{3}$

Therefore, the value of $b$ in equation B is $-5\sqrt{3}$.

In this article, we have analyzed two trigonometric equations and solved for the value of $b$. We have used the properties of the tangent function and the periodicity of the tangent function to evaluate the equations. We have found that the value of $b$ in equation A is $5\sqrt{3}$, while the value of $b$ in equation B is $-5\sqrt{3}$. These results demonstrate the importance of understanding the properties of trigonometric functions and how they can be applied to solve real-world problems.

• The tangent function is defined as the ratio of the sine and cosine functions.
• The tangent function has a periodicity of $180^{\circ}$.
• To solve for $b$ in a trigonometric equation, we need to evaluate the tangent function and then use algebraic manipulations to isolate the variable $b$.
• The value of $b$ in equation A is $5\sqrt{3}$, while the value of $b$ in equation B is $-5\sqrt{3}$.

For further reading on trigonometry and its applications, we recommend the following resources:

• "Trigonometry" by Michael Corral
• "Trigonometry for Dummies" by Mary Jane Sterling
• "Trigonometry: A Unit Circle Approach" by Charles P. McKeague and Mark D. Turner

$$

In our previous article, we explored two trigonometric equations and solved for the value of $b$. In this article, we will continue to delve into the world of trigonometry and answer some of the most frequently asked questions related to solving for $b$.

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. It is denoted by the symbol $\tan \theta$ and is equal to $\frac{\sin \theta}{\cos \theta}$.

Q: What is the periodicity of the tangent function?

A: The tangent function has a periodicity of $180^{\circ}$. This means that the value of $\tan \theta$ repeats every $180^{\circ}$.

Q: How do I evaluate the tangent function?

A: To evaluate the tangent function, you need to know the values of the sine and cosine functions for a given angle. You can use a trigonometric table or a calculator to find these values.

Q: What is the difference between equation A and equation B?

A: Equation A is $\tan \left(30^{\circ}\right)=\frac{5}{b}$, while equation B is $\tan (300^{\circ})=\frac{b}{5}$. The main difference between the two equations is the angle and the position of the variable $b$.

Q: How do I solve for $b$ in equation A?

A: To solve for $b$ in equation A, you need to evaluate the tangent of $30^{\circ}$ and then use algebraic manipulations to isolate the variable $b$. The value of $b$ in equation A is $5\sqrt{3}$.

Q: How do I solve for $b$ in equation B?

A: To solve for $b$ in equation B, you need to evaluate the tangent of $300^{\circ}$ and then use algebraic manipulations to isolate the variable $b$. The value of $b$ in equation B is $-5\sqrt{3}$.

Q: What are some common mistakes to avoid when solving for $b$?

A: Some common mistakes to avoid when solving for $b$ include:

• Not evaluating the tangent function correctly
• Not using the correct algebraic manipulations to isolate the variable $b$
• Not considering the periodicity of the tangent function
• Not checking the units of the answer

Q: What are some real-world applications of trigonometry?

A: Trigonometry has numerous real-world applications, including:

• Navigation: Trigonometry is used in navigation to calculate distances and angles between locations.
• Physics: Trigonometry is used in physics to describe the motion of objects and the behavior of waves.
• Engineering: Trigonometry is used in engineering to design and build structures such as bridges and buildings.
• Computer Science: Trigonometry is used in computer science to create 3D graphics and animations.

In this article, we have answered some of the most frequently asked questions related to solving for $b$ in trigonometric equations. We have covered topics such as the tangent function, periodicity, and real-world applications of trigonometry. We hope that this article has been helpful in clarifying any doubts you may have had about solving for $b$.

• The tangent function is defined as the ratio of the sine and cosine functions.
• The tangent function has a periodicity of $180^{\circ}$.
• To solve for $b$ in a trigonometric equation, you need to evaluate the tangent function and then use algebraic manipulations to isolate the variable $b$.
• The value of $b$ in equation A is $5\sqrt{3}$, while the value of $b$ in equation B is $-5\sqrt{3}$.

For further reading on trigonometry and its applications, we recommend the following resources:

• "Trigonometry" by Michael Corral
• "Trigonometry for Dummies" by Mary Jane Sterling
• "Trigonometry: A Unit Circle Approach" by Charles P. McKeague and Mark D. Turner

These resources provide a comprehensive introduction to trigonometry and its applications, and are suitable for students and professionals alike.