Problem 153
Question
Determine whether each statement makes sense or does not make sense, and explain your reasoning. There are similarities and differences between solving \(4 x+1=3\) and \(4 \sin \theta+1=3: \ln\) the first equation, 1 need to isolate \(x\) to get the solution. In the trigonometric equation, I need to first isolate \(\sin \theta,\) but then 1 must continue to solve for \(\boldsymbol{\theta}\)
Step-by-Step Solution
Verified Answer
The statement makes sense. Even though the steps to isolate the variable are similar for algebraic and trigonometric equations, finding the solution requires different strategies. For the algebraic equation, once the variable is isolated, the solution is found. For the trigonometric equation, even after isolating \(\sin \theta\), we need to find the angle \(\theta\) whose sine value matches the isolated one.
1Step 1: Interpret Algebraic Equation
The algebraic equation is \(4x + 1 = 3\). To solve for \(x\), subtract 1 from both sides to isolate \(4x\), which gives \(4x = 3 - 1 = 2\). Then, divide both sides by 4 to find \(x\), leading to \(x = 2/4 = 0.5\).
2Step 2: Interpret Trigonometric Equation
Consider the trigonometric equation \(4 \sin \theta + 1 = 3\). As with the algebraic equation, subtract 1 from both sides to isolate \(4 \sin \theta\) and it becomes \(4 \sin \theta = 3 - 1 = 2\). Then, divide both sides by 4 to isolate \(\sin \theta\) which gives \(\sin \theta = 2/4 = 0.5\). We have not reached the solution yet as we need to find the angle \(\theta\) whose sine value is 0.5.
3Step 3: Finalize Trigonometric Equation Solution
Solving \(\sin \theta = 0.5\) is done by recalling or referring to the unit circle or sine table, or by using the arcsin function from a calculator. So, \(\theta = \arcsin (0.5) = 30^{\circ}\) or \(\pi/6\) in radian measure.
Key Concepts
Algebraic EquationsIsolate VariablesSine Function
Algebraic Equations
When we talk about algebraic equations, we are referring to mathematical statements where two expressions are set equal to each other and typically involve one or more variables. For example, in the equation, \(4x + 1 = 3\), our goal is to find the value of \(x\) that makes the statement true.
To do this, we perform operations that will 'isolate' the variable—meaning we'll end up with the variable on one side of the equation and a numerical answer on the other. The process is systematic: we first remove any additional terms from the side with the variable, then we address any coefficients interacting with the variable until it stands alone. This equation is considered linear because it forms a straight line when graphed and its highest degree (the exponent on the variable) is one.
Solving algebraic equations like this forms the foundation for more complex problem solving, including when equations involve trigonometric functions, like the sine function.
To do this, we perform operations that will 'isolate' the variable—meaning we'll end up with the variable on one side of the equation and a numerical answer on the other. The process is systematic: we first remove any additional terms from the side with the variable, then we address any coefficients interacting with the variable until it stands alone. This equation is considered linear because it forms a straight line when graphed and its highest degree (the exponent on the variable) is one.
Solving algebraic equations like this forms the foundation for more complex problem solving, including when equations involve trigonometric functions, like the sine function.
Isolate Variables
Isolating variables is a critical step in solving both algebraic and trigonometric equations. The concept is essential as it allows for understanding the value of the variable in question. The steps to isolate a variable are not only algebraic procedures but also the key to unlocking the answer to the equation. For example, in the simpler algebraic equation \(4x + 1 = 3\), isolation involves two main steps: subtracting 1 from both sides (to 'cancel' the addition of 1) and then dividing by 4 (to 'reverse' the multiplication by 4).
After these steps, we get \(x = 0.5\). This process shows how we systematically reverse operations to untangle the variable from other numbers. This skill is indispensable because it underpins dealing with any sort of equation, particularly when moving onto more advanced topics like calculus or differential equations.
After these steps, we get \(x = 0.5\). This process shows how we systematically reverse operations to untangle the variable from other numbers. This skill is indispensable because it underpins dealing with any sort of equation, particularly when moving onto more advanced topics like calculus or differential equations.
Sine Function
The sine function is one of the primary trigonometric functions. It relates to the concept of a right triangle and the unit circle. In the context of trigonometry, for a given angle \(\theta\), the sine function gives the ratio of the length of the side opposite the angle to the length of the hypotenuse in a right-angled triangle.
When we solve the trigonometric equation \(\text{4 sin \(\theta\) }+ 1 = 3\), we first isolate \(\text{sin \(\theta\)}\) similarly to isolating \(x\) in an algebraic equation. Once isolated, we seek the angle \(\theta\) that results in a sine of 0.5. This is where our understanding of trigonometry deepens as we need to consider the periodic nature of the sine function, which can have the same value at multiple angles within a 360-degree cycle. To find the specific solution, we may use the unit circle, a sine table, or the inverse sine function (arcsine) available on most calculators. In our case, \(\text{sin \(\theta\) }= 0.5\) corresponds to \(\theta = 30^{\text{\circ}}\) or \(\theta = \text{\(\frac{\pi}{6}\)}\), depending on whether we're working with degrees or radians.
When we solve the trigonometric equation \(\text{4 sin \(\theta\) }+ 1 = 3\), we first isolate \(\text{sin \(\theta\)}\) similarly to isolating \(x\) in an algebraic equation. Once isolated, we seek the angle \(\theta\) that results in a sine of 0.5. This is where our understanding of trigonometry deepens as we need to consider the periodic nature of the sine function, which can have the same value at multiple angles within a 360-degree cycle. To find the specific solution, we may use the unit circle, a sine table, or the inverse sine function (arcsine) available on most calculators. In our case, \(\text{sin \(\theta\) }= 0.5\) corresponds to \(\theta = 30^{\text{\circ}}\) or \(\theta = \text{\(\frac{\pi}{6}\)}\), depending on whether we're working with degrees or radians.
Other exercises in this chapter
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