Find Exact Values Of X And R Easily - Step-by-Step Guide
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To find the exact values of x and r, one must delve into the realms of mathematics that revolve around equations, functions, and variables. This guide aims to simplify the process for you, providing a step-by-step method to tackle these problems effortlessly. By breaking down complex concepts into more manageable steps, you will gain confidence and clarity in solving equations. Let’s embark on this mathematical journey together! 🚀
Understanding Variables and Equations
Before we get into the specifics of finding exact values, it's important to grasp what variables and equations are.
Variables represent unknown values, commonly denoted by letters such as x, y, or r. Equations are mathematical statements asserting that two expressions are equal.
Example: An equation like ( 2x + 3 = 7 ) tells us that the expression ( 2x + 3 ) is equal to 7. Our goal is to solve for x.
Steps to Solve for x
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Isolate the Variable: Your first step is to isolate the variable (x in this case) on one side of the equation. For our example ( 2x + 3 = 7 ):
- Subtract 3 from both sides: [ 2x + 3 - 3 = 7 - 3 ] This simplifies to: [ 2x = 4 ]
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Divide by the Coefficient: Now, divide both sides by the coefficient of x: [ \frac{2x}{2} = \frac{4}{2} ] This gives us: [ x = 2 ]
Practical Example
Let’s illustrate this with a practical example:
Example Equation: ( 3x - 12 = 0 )
Step 1: Isolate the variable
- Add 12 to both sides: [ 3x - 12 + 12 = 0 + 12 ] Resulting in: [ 3x = 12 ]
Step 2: Divide by the coefficient
- Divide both sides by 3: [ \frac{3x}{3} = \frac{12}{3} ] Therefore: [ x = 4 ]
Finding Exact Values of r
Now that we’ve tackled x, let’s explore how to find exact values for r, which might represent a radius, a rate, or any other variable depending on the context of your problem.
Common Scenarios Involving r
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Using the Pythagorean Theorem: The most common application of r is in geometry, particularly in circles. The radius (r) can be determined from the diameter or from the coordinates of points on the circle.
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Exponential and Logarithmic Equations: In algebra, you may often find r involved in equations that require the use of logarithms to solve for exact values.
Example: Finding r in a Circle
Assume the equation of a circle is given as: [ x^2 + y^2 = r^2 ]
If you know the coordinates of a point on the circle, say (3, 4), you can find r.
Step 1: Substitute values into the equation
- Plug x = 3 and y = 4 into the equation: [ 3^2 + 4^2 = r^2 ]
Step 2: Solve for r
- Calculate: [ 9 + 16 = r^2 \Rightarrow 25 = r^2 ]
- Therefore, taking the square root gives: [ r = 5 ]
Working with Trigonometric Functions
Trigonometry often necessitates finding exact values for angles and sides in right-angled triangles. The primary functions to consider are sine (sin), cosine (cos), and tangent (tan).
Example: Find Exact Values for Angles
Suppose you are given a right triangle with a hypotenuse (h) of 10, and one side (a) of 6.
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Find the length of the other side (b): Using the Pythagorean theorem: [ h^2 = a^2 + b^2 ] Therefore: [ 10^2 = 6^2 + b^2 \Rightarrow 100 = 36 + b^2 \Rightarrow 64 = b^2 \Rightarrow b = 8 ]
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Calculate the angles: You can use the sine, cosine, and tangent to find angles A and B:
- For angle A: [ \sin A = \frac{a}{h} = \frac{6}{10} = 0.6 \Rightarrow A = \sin^{-1}(0.6) \approx 36.87^\circ ]
- For angle B, knowing that the angles in a triangle sum to 180°: [ B = 90^\circ - A = 90^\circ - 36.87^\circ \approx 53.13^\circ ]
Exact Values Table
Here’s a useful table summarizing some common trigonometric values for specific angles:
<table> <tr> <th>Angle (Degrees)</th> <th>Sin</th> <th>Cos</th> <th>Tan</th> </tr> <tr> <td>0°</td> <td>0</td> <td>1</td> <td>0</td> </tr> <tr> <td>30°</td> <td>0.5</td> <td>√3/2</td> <td>√3/3</td> </tr> <tr> <td>45°</td> <td>√2/2</td> <td>√2/2</td> <td>1</td> </tr> <tr> <td>60°</td> <td>√3/2</td> <td>0.5</td> <td>√3</td> </tr> <tr> <td>90°</td> <td>1</td> <td>0</td> <td>Undefined</td> </tr> </table>
Note on Precision
When dealing with exact values, ensure the use of a calculator that provides sufficient precision. For angles, always express your final answer in degrees or radians, as appropriate.
Conclusion
Finding exact values of variables such as x and r is a systematic process that involves isolating the variables, applying basic arithmetic operations, and understanding the properties of functions, whether algebraic or trigonometric. With practice, these methods will become second nature. Remember, the key is to break down each problem into simpler parts and tackle them one step at a time. Happy solving! 🎉