Evaluate The Expression:${ \begin{array}{l} =\frac{8}{3} \div \frac{-7}{9} \ =\frac{8}{3} \times \frac{-9}{7} \ =-\frac{24}{7} \end{array} }$2. Solve The Equation ${ 3 \sin X \cos X + 1 = 0\$} For [$0^{\circ} \leqslant X

Introduction

Mathematics is a vast and fascinating field that encompasses various branches, including algebra, geometry, trigonometry, and calculus. In this article, we will delve into the world of mathematical expressions and equations, focusing on evaluating expressions and solving equations. We will explore the concepts of division and multiplication of fractions, as well as trigonometric equations, and provide step-by-step solutions to the given problems.

Evaluating Expressions

Division of Fractions

When dividing fractions, we need to invert the second fraction and multiply. This is a fundamental concept in mathematics, and it is essential to understand the rules of division and multiplication of fractions.

Example 1: Evaluate the expression $\frac{8}{3} \div \frac{-7}{9}$.

To evaluate this expression, we need to invert the second fraction and multiply:

$$\frac{8}{3} \div \frac{-7}{9} = \frac{8}{3} \times \frac{9}{-7}$$

Now, we can multiply the numerators and denominators:

$$\frac{8}{3} \times \frac{9}{-7} = \frac{8 \times 9}{3 \times -7}$$

Simplifying the expression, we get:

$$\frac{8 \times 9}{3 \times -7} = \frac{72}{-21}$$

We can further simplify the expression by dividing both the numerator and denominator by their greatest common divisor, which is 3:

$$\frac{72}{-21} = \frac{24}{-7}$$

Therefore, the value of the expression $\frac{8}{3} \div \frac{-7}{9}$ is $-\frac{24}{7}$.

Multiplication of Fractions

When multiplying fractions, we simply multiply the numerators and denominators.

Example 2: Evaluate the expression $\frac{8}{3} \times \frac{-9}{7}$.

To evaluate this expression, we need to multiply the numerators and denominators:

$$\frac{8}{3} \times \frac{-9}{7} = \frac{8 \times -9}{3 \times 7}$$

Simplifying the expression, we get:

$$\frac{8 \times -9}{3 \times 7} = \frac{-72}{21}$$

We can further simplify the expression by dividing both the numerator and denominator by their greatest common divisor, which is 3:

$$\frac{-72}{21} = \frac{-24}{7}$$

Therefore, the value of the expression $\frac{8}{3} \times \frac{-9}{7}$ is $-\frac{24}{7}$.

Solving Trigonometric Equations

Introduction to Trigonometric Equations

Trigonometric equations are equations that involve trigonometric functions, such as sine, cosine, and tangent. These equations can be solved using various techniques, including algebraic manipulation and trigonometric identities.

Example 3: Solve the equation $3 \sin x \cos x + 1 = 0$ for $0^{\circ} \leqslant x < 360^{\circ}$.

To solve this equation, we need to isolate the trigonometric function. We can start by subtracting 1 from both sides of the equation:

$$3 \sin x \cos x = -1$$

Next, we can divide both sides of the equation by 3:

$$\sin x \cos x = -\frac{1}{3}$$

Now, we can use the trigonometric identity $\sin 2x = 2 \sin x \cos x$ to rewrite the equation:

$$\sin 2x = -\frac{2}{3}$$

To solve this equation, we need to find the values of $x$ that satisfy the equation $\sin 2x = -\frac{2}{3}$. We can use a calculator or a trigonometric table to find the values of $x$.

Using a calculator, we find that the values of $x$ that satisfy the equation $\sin 2x = -\frac{2}{3}$ are:

$$2x = 147.5^{\circ}, 212.5^{\circ}, 317.5^{\circ}, 372.5^{\circ}$$

Dividing both sides of the equation by 2, we get:

$$x = 73.75^{\circ}, 106.25^{\circ}, 158.75^{\circ}, 186.25^{\circ}$$

Therefore, the solutions to the equation $3 \sin x \cos x + 1 = 0$ for $0^{\circ} \leqslant x < 360^{\circ}$ are $73.75^{\circ}, 106.25^{\circ}, 158.75^{\circ}, 186.25^{\circ}$.

Conclusion

In this article, we have explored the concepts of evaluating expressions and solving equations. We have discussed the rules of division and multiplication of fractions, as well as trigonometric equations. We have provided step-by-step solutions to the given problems, including evaluating expressions and solving trigonometric equations. We hope that this article has provided a comprehensive guide to evaluating expressions and solving equations, and that it has been helpful in understanding these important mathematical concepts.

References

• [1] "Algebra and Trigonometry" by Michael Sullivan
• [2] "Trigonometry" by Charles P. McKeague
• [3] "Mathematics for the Nonmathematician" by Morris Kline

Further Reading

• [1] "Calculus" by Michael Spivak
• [2] "Linear Algebra and Its Applications" by Gilbert Strang
• [3] "Differential Equations and Dynamical Systems" by Lawrence Perko
  

Introduction

In our previous article, we explored the concepts of evaluating expressions and solving equations. We discussed the rules of division and multiplication of fractions, as well as trigonometric equations. In this article, we will provide a Q&A guide to help you better understand these concepts.

Q&A

Q: What is the difference between division and multiplication of fractions?

A: Division of fractions involves inverting the second fraction and multiplying, while multiplication of fractions involves multiplying the numerators and denominators.

Q: How do I evaluate an expression with multiple fractions?

A: To evaluate an expression with multiple fractions, you need to follow the order of operations (PEMDAS). First, evaluate any expressions inside parentheses, then evaluate any exponential expressions, then multiply and divide from left to right, and finally add and subtract from left to right.

Q: How do I solve a trigonometric equation?

A: To solve a trigonometric equation, you need to isolate the trigonometric function. You can use algebraic manipulation and trigonometric identities to solve the equation.

Q: What is the difference between a trigonometric equation and a trigonometric identity?

A: A trigonometric equation is an equation that involves a trigonometric function, while a trigonometric identity is a statement that is true for all values of the trigonometric function.

Q: How do I find the values of x that satisfy a trigonometric equation?

A: To find the values of x that satisfy a trigonometric equation, you need to use a calculator or a trigonometric table to find the values of the trigonometric function that satisfy the equation.

Q: What is the difference between a sine and a cosine function?

A: The sine function represents the ratio of the length of the side opposite a given angle to the length of the hypotenuse, while the cosine function represents the ratio of the length of the side adjacent to a given angle to the length of the hypotenuse.

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

A: To use trigonometric identities to solve a trigonometric equation, you need to rewrite the equation using the trigonometric identity. Then, you can solve the equation using algebraic manipulation.

Q: What is the difference between a linear equation and a quadratic equation?

A: A linear equation is an equation that involves a linear function, while a quadratic equation is an equation that involves a quadratic function.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, you need to use the quadratic formula or factor the equation.

Q: What is the quadratic formula?

A: The quadratic formula is a formula that is used to solve quadratic equations. It is given by:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

Q: How do I use the quadratic formula to solve a quadratic equation?

A: To use the quadratic formula to solve a quadratic equation, you need to plug in the values of a, b, and c into the formula. Then, you can simplify the expression and find the values of x that satisfy the equation.

Conclusion

In this article, we have provided a Q&A guide to help you better understand the concepts of evaluating expressions and solving equations. We have discussed the rules of division and multiplication of fractions, as well as trigonometric equations. We hope that this article has been helpful in understanding these important mathematical concepts.

References

• [1] "Algebra and Trigonometry" by Michael Sullivan
• [2] "Trigonometry" by Charles P. McKeague
• [3] "Mathematics for the Nonmathematician" by Morris Kline

Further Reading

• [1] "Calculus" by Michael Spivak
• [2] "Linear Algebra and Its Applications" by Gilbert Strang
• [3] "Differential Equations and Dynamical Systems" by Lawrence Perko