(a) If $3 \sin \theta + 4 \cos \theta = 5$, Find The Value Of $\tan \theta$.

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Introduction

In this problem, we are given an equation involving sine and cosine functions, and we need to find the value of the tangent function. The equation is $3 \sin \theta + 4 \cos \theta = 5$. To solve this problem, we will use the trigonometric identity $\sin^2 \theta + \cos^2 \theta = 1$ and the definition of the tangent function.

Step 1: Square both sides of the equation

We start by squaring both sides of the equation $3 \sin \theta + 4 \cos \theta = 5$. This gives us:

$$\left(3 \sin \theta + 4 \cos \theta\right)^2 = 5^2$$

Expanding the left-hand side, we get:

$$9 \sin^2 \theta + 24 \sin \theta \cos \theta + 16 \cos^2 \theta = 25$$

Step 2: Use the trigonometric identity $\sin^2 \theta + \cos^2 \theta = 1$

We can use the trigonometric identity $\sin^2 \theta + \cos^2 \theta = 1$ to simplify the equation. Rearranging the equation, we get:

$$9 \sin^2 \theta + 16 \cos^2 \theta = 25 - 24 \sin \theta \cos \theta$$

Step 3: Use the trigonometric identity $\sin^2 \theta + \cos^2 \theta = 1$ again

We can use the trigonometric identity $\sin^2 \theta + \cos^2 \theta = 1$ again to simplify the equation. Rearranging the equation, we get:

$$9 \sin^2 \theta + 16 \cos^2 \theta = 25 - 24 \sin \theta \cos \theta$$

$$9 \sin^2 \theta + 16 \left(1 - \sin^2 \theta\right) = 25 - 24 \sin \theta \cos \theta$$

Step 4: Simplify the equation

Simplifying the equation, we get:

$$9 \sin^2 \theta + 16 - 16 \sin^2 \theta = 25 - 24 \sin \theta \cos \theta$$

$$-7 \sin^2 \theta + 16 = 25 - 24 \sin \theta \cos \theta$$

Step 5: Rearrange the equation

Rearranging the equation, we get:

$$-7 \sin^2 \theta + 24 \sin \theta \cos \theta - 9 = 0$$

Step 6: Factor the equation

Factoring the equation, we get:

$$\left(-7 \sin \theta + 9\right)\left(\sin \theta - 1\right) = 0$$

Step 7: Solve for $\sin \theta$

Solving for $\sin \theta$, we get:

$$\sin \theta = \frac{9}{7} \text{ or } \sin \theta = 1$$

Step 8: Solve for $\cos \theta$

Solving for $\cos \theta$, we get:

$$\cos \theta = \frac{4}{7} \text{ or } \cos \theta = 0$$

Step 9: Find the value of $\tan \theta$

Using the definition of the tangent function, we can find the value of $\tan \theta$:

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

Substituting the values of $\sin \theta$ and $\cos \theta$, we get:

$$\tan \theta = \frac{\frac{9}{7}}{\frac{4}{7}} \text{ or } \tan \theta = \frac{1}{0}$$

Step 10: Simplify the expression

Simplifying the expression, we get:

$$\tan \theta = \frac{9}{4} \text{ or } \tan \theta = \infty$$

Conclusion

In this problem, we were given an equation involving sine and cosine functions, and we needed to find the value of the tangent function. We used the trigonometric identity $\sin^2 \theta + \cos^2 \theta = 1$ and the definition of the tangent function to solve the problem. The final answer is $\boxed{\frac{9}{4}}$.

Discussion

This problem is a classic example of how to use trigonometric identities to solve equations involving sine and cosine functions. The key to solving this problem is to use the trigonometric identity $\sin^2 \theta + \cos^2 \theta = 1$ to simplify the equation and then use the definition of the tangent function to find the value of $\tan \theta$. This problem requires a good understanding of trigonometric identities and the definition of the tangent function.

Applications

This problem has many applications in mathematics and physics. For example, it can be used to solve problems involving right triangles, circular functions, and trigonometric identities. It can also be used to model real-world problems involving waves, vibrations, and oscillations.

References

• [1] "Trigonometry" by Michael Corral
• [2] "Calculus" by Michael Spivak
• [3] "Differential Equations" by Morris Tenenbaum

Keywords

• Trigonometry
• Sine and cosine functions
• Tangent function
• Trigonometric identities
• Right triangles
• Circular functions
• Waves
• Vibrations
• Oscillations

Tags

• #trigonometry
• #sineandcosinefunctions
• #tangentfunction
• #trigonometricidentities
• #righttriangles
• #circularfunctions
• #waves
• #vibrations
• #oscillations
  

Introduction

In our previous article, we solved the problem of finding the value of the tangent function given the equation $3 \sin \theta + 4 \cos \theta = 5$. In this article, we will answer some common questions related to trigonometry and the tangent function.

Q: What is the tangent function?

A: The tangent function is a trigonometric function that is defined as the ratio of the sine function to the cosine function. It is denoted by the symbol $\tan \theta$ and is defined as:

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

Q: What are the key properties of the tangent function?

A: The key properties of the tangent function are:

• The tangent function is periodic with a period of $\pi$.
• The tangent function is an odd function, meaning that $\tan (-\theta) = -\tan \theta$.
• The tangent function has a vertical asymptote at $\theta = \frac{\pi}{2}$ and $\theta = \frac{3\pi}{2}$.

Q: How do I use the tangent function to solve problems?

A: To use the tangent function to solve problems, you need to follow these steps:

1. Identify the problem and determine the value of the tangent function.
2. Use the definition of the tangent function to express the problem in terms of the sine and cosine functions.
3. Use trigonometric identities to simplify the expression and solve for the unknown value.
4. Check your solution by plugging it back into the original equation.

Q: What are some common applications of the tangent function?

A: The tangent function has many applications in mathematics and physics, including:

• Solving problems involving right triangles and circular functions.
• Modeling real-world problems involving waves, vibrations, and oscillations.
• Solving problems involving trigonometric identities and equations.

Q: How do I graph the tangent function?

A: To graph the tangent function, you need to follow these steps:

1. Determine the period of the tangent function, which is $\pi$.
2. Determine the vertical asymptotes of the tangent function, which are at $\theta = \frac{\pi}{2}$ and $\theta = \frac{3\pi}{2}$.
3. Plot the points on the graph where the tangent function is defined.
4. Connect the points to form a smooth curve.

Q: What are some common mistakes to avoid when working with the tangent function?

A: Some common mistakes to avoid when working with the tangent function include:

• Not using the correct definition of the tangent function.
• Not simplifying the expression using trigonometric identities.
• Not checking the solution by plugging it back into the original equation.

Conclusion

In this article, we answered some common questions related to trigonometry and the tangent function. We discussed the definition and key properties of the tangent function, and provided tips on how to use it to solve problems. We also discussed some common applications of the tangent function and how to graph it. Finally, we provided some common mistakes to avoid when working with the tangent function.

Discussion

This article is a continuation of our previous article on trigonometry and the tangent function. In this article, we provided a more in-depth look at the tangent function and its applications. We hope that this article has been helpful in providing a better understanding of the tangent function and how to use it to solve problems.

Applications

This article has many applications in mathematics and physics, including:

• Solving problems involving right triangles and circular functions.
• Modeling real-world problems involving waves, vibrations, and oscillations.
• Solving problems involving trigonometric identities and equations.

References

• [1] "Trigonometry" by Michael Corral
• [2] "Calculus" by Michael Spivak
• [3] "Differential Equations" by Morris Tenenbaum

Keywords

• Trigonometry
• Sine and cosine functions
• Tangent function
• Trigonometric identities
• Right triangles
• Circular functions
• Waves
• Vibrations
• Oscillations

Tags

• #trigonometry
• #sineandcosinefunctions
• #tangentfunction
• #trigonometricidentities
• #righttriangles
• #circularfunctions
• #waves
• #vibrations
• #oscillations