Problem 123 - Algebra and Trigonometry

Question

Will help you prepare for the material covered in the next section. Solve: $$-\frac{\pi}{2}

Step-by-Step Solution

Answer

The solution to the given inequality is \(-\frac{3\pi}{4} < x < \frac{\pi}{4}\).

Step 1: Identify the Inequality

First, identify the original inequality given in the exercise. In this case, it's given by \(-\frac{\pi}{2}

Step 2: Isolate X

Subtract \(\frac{\pi}{4}\) from all sections of the inequality to isolate \(x\). When doing such operation, it's crucial to do it for all three parts of the inequality to keep the equation true. Therefore, the new inequality can be written as \(-\frac{\pi}{2} - \frac{\pi}{4} < x < \frac{\pi}{2} - \frac{\pi}{4}\).

Step 3: Simplify the Inequality

The next step is to simplify the inequality. Simplify \(-\frac{\pi}{2} - \frac{\pi}{4}\) and \(\frac{\pi}{2} - \frac{\pi}{4}\) to get simplified terms. After doing the calculations, the inequality becomes \(-\frac{3\pi}{4} < x < \frac{\pi}{4}\).

Key Concepts

Trigonometric Inequalities

Understanding trigonometric inequalities is crucial for solving problems that impose conditions on trigonometric functions. These inequalities often express limitations on angles within the context of the unit circle or trigonometric identities. For example, considering angles for which the sine or cosine functions produce values within a certain range. Simplifying and solving these inequalities require a good grasp of trigonometric identities, the properties of inequalities, and sometimes, the graphical representation of trigonometric functions.
To solve a trigonometric inequality, one typically needs to perform several steps, including isolating the trigonometric function, using identities to simplify the equation, and considering the periodic nature of trigonometric functions to find all the solutions within a given interval. It's essential to understand how angles are measured and expressed, as these inequalities frequently involve radians or degrees.

Isolate the Variable

When solving inequalities, it's necessary to isolate the variable on one side to better understand its values in relation to the rest of the equation. This process involves performing the same operations on all parts of the inequality to maintain balance. It’s important to remember that when multiplying or dividing an inequality by a negative number, the inequality sign must be flipped.

Steps to Isolate the Variable

• Identify the variable and the terms attached to it.
• Determine what operations will allow you to isolate the variable, usually through addition, subtraction, multiplication, or division.
• Apply these operations to all sections of the inequality.
• Simplify the resulting expression to clearly present the range of values the variable can take.

Following these steps correctly makes solving complex inequalities more manageable, and mastering this skill opens the door to tackling a wide array of mathematical problems.

Simplifying Expressions

Simplifying expressions is a fundamental skill in algebra that helps us to clearly and concisely express the relationships within an equation or inequality. This step is vital when working with inequalities, as it can help eliminate unnecessary complications and provide a clearer solution set.

Strategies for Simplifying Expressions

• Combine like terms by adding or subtracting them as needed.
• Use the distributive property to eliminate parentheses.
• Apply inverse operations to simplify complex fractions.
• Factor expressions where possible to reveal hidden patterns or simplify complicated terms.

Simplification not only makes it easier to visualize and interpret the solution but also reduces the chance of making errors during calculations. Clear and simplified expressions form the cornerstone of a solid mathematical understanding.

Inequality Notation

Inequality notation is how we express the relationship between two expressions in terms of their inequalities. The major inequality signs are '<' (less than), '>' (greater than), '\(\leq\)' (less than or equal to), and '\(\geq\)' (greater than or equal to).
In a compound inequality, which involves two inequalities combined into one statement like '\(-\frac{3\pi}{4} < x < \frac{\pi}{4}\)', you're looking at a range for the variable. This particular notation tells us that the value of 'x' is within a specific interval on the number line. Understanding how to interpret and write these inequalities is vital for effectively communicating the solutions to mathematical problems.

Important Points to Note in Inequality Notation

• The variable or expression on the left is either less than or greater than the one on the right, based on the inequality sign used.
• If an inequality involves a variable and a number, the variable's possible values are all the numbers that make the inequality true.
• In 'strict' inequalities using '<' or '>', the endpoint values are not included in the solution set.
• In 'non-strict' inequalities using '\(\leq\)' or '\(\geq\)', the endpoint values are included in the solution set.

Understanding how to write and solve for variables within inequalities is crucial for many mathematical and real-life applications where precise ranges are necessary.

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Problem 123

Other exercises in this chapter

Problem 122

Prove that if \(x>0, \tan ^{-1} x+\tan ^{-1} \frac{1}{x}=\frac{\pi}{2}\).

View solution

Problem 122

Use the identity for \(\cos ^{2} x\) to graph one period of \(y=\cos ^{2} x\)

View solution

Problem 123

Assuming Earth to be a sphere of radius 4000 miles, how many miles north of the Equator is Miami, Florida, if it is \(26^{\circ}\) north from the Equator? Round

View solution

Problem 124

Will help you prepare for the material covered in the next section. $$ \text { Simplify: } \frac{-\frac{3 \pi}{4}+\frac{\pi}{4}}{2} $$

View solution

Key Concepts

Trigonometric Inequalities

Understanding trigonometric inequalities is crucial for solving problems that impose conditions on trigonometric functions. These inequalities often express limitations on angles within the context of the unit circle or trigonometric identities. For example, considering angles for which the sine or cosine functions produce values within a certain range. Simplifying and solving these inequalities require a good grasp of trigonometric identities, the properties of inequalities, and sometimes, the graphical representation of trigonometric functions.
To solve a trigonometric inequality, one typically needs to perform several steps, including isolating the trigonometric function, using identities to simplify the equation, and considering the periodic nature of trigonometric functions to find all the solutions within a given interval. It's essential to understand how angles are measured and expressed, as these inequalities frequently involve radians or degrees.

Isolate the Variable

When solving inequalities, it's necessary to isolate the variable on one side to better understand its values in relation to the rest of the equation. This process involves performing the same operations on all parts of the inequality to maintain balance. It’s important to remember that when multiplying or dividing an inequality by a negative number, the inequality sign must be flipped.

Steps to Isolate the Variable

• Identify the variable and the terms attached to it.
• Determine what operations will allow you to isolate the variable, usually through addition, subtraction, multiplication, or division.
• Apply these operations to all sections of the inequality.
• Simplify the resulting expression to clearly present the range of values the variable can take.

Following these steps correctly makes solving complex inequalities more manageable, and mastering this skill opens the door to tackling a wide array of mathematical problems.

Simplifying Expressions

Simplifying expressions is a fundamental skill in algebra that helps us to clearly and concisely express the relationships within an equation or inequality. This step is vital when working with inequalities, as it can help eliminate unnecessary complications and provide a clearer solution set.

Strategies for Simplifying Expressions

• Combine like terms by adding or subtracting them as needed.
• Use the distributive property to eliminate parentheses.
• Apply inverse operations to simplify complex fractions.
• Factor expressions where possible to reveal hidden patterns or simplify complicated terms.

Simplification not only makes it easier to visualize and interpret the solution but also reduces the chance of making errors during calculations. Clear and simplified expressions form the cornerstone of a solid mathematical understanding.

Inequality Notation

Inequality notation is how we express the relationship between two expressions in terms of their inequalities. The major inequality signs are '<' (less than), '>' (greater than), '\(\leq\)' (less than or equal to), and '\(\geq\)' (greater than or equal to).
In a compound inequality, which involves two inequalities combined into one statement like '\(-\frac{3\pi}{4} < x < \frac{\pi}{4}\)', you're looking at a range for the variable. This particular notation tells us that the value of 'x' is within a specific interval on the number line. Understanding how to interpret and write these inequalities is vital for effectively communicating the solutions to mathematical problems.

Important Points to Note in Inequality Notation

• The variable or expression on the left is either less than or greater than the one on the right, based on the inequality sign used.
• If an inequality involves a variable and a number, the variable's possible values are all the numbers that make the inequality true.
• In 'strict' inequalities using '<' or '>', the endpoint values are not included in the solution set.
• In 'non-strict' inequalities using '\(\leq\)' or '\(\geq\)', the endpoint values are included in the solution set.

Understanding how to write and solve for variables within inequalities is crucial for many mathematical and real-life applications where precise ranges are necessary.