Explanations 
Textbooks 
Exams 
GCSE 
IGCSE 
AS 
A Level 
International A Level 
University Admissions Tests 
Flashcards 
Vaia AI 
Notes 
Study Plans 
Study Sets 
Find a job 
Find a degree 
Student Deals 
Magazine 
Mobile App Chapter 16: Problem 2
Calculate the equation of the tangent to \(y=\sin x\) where \(x=\frac{\pi}{4}\).
Short Answer
Expert verified
Answer: The equation of the tangent line is y - √2/2 = (√2/2)(x - π/4).
Step by step solution
01
Finding the Derivative of y=
To find the equation of the tangent line, we need to first find the derivative of \(y=\sin x\). The derivative is the rate at which the function is changing, which essentially gives us the slope of the tangent line. Differentiate the function by using the basic rules of differentiation:
\(\frac{dy}{dx} = \cos x\)
Now we have the derivative of \(y\) with respect to \(x\).
02
Finding the Coordinates of the Point of Tangency
Substitute the given value of \(x=\frac{\pi}{4}\) into the curve equation to get the corresponding value for \(y\):
\(y = \sin (\frac{\pi}{4})\)
Since \(\sin (\frac{\pi}{4}) = \frac{\sqrt{2}}{2}\), the point of tangency is \((\frac{\pi}{4}, \frac{\sqrt{2}}{2})\).
03
Finding the Slope of the Tangent Line
Now that we have the point of tangency, we can find the slope of the tangent line by substituting \(x=\frac{\pi}{4}\) into the derivative we found in step 1:
\(m = \cos(\frac{\pi}{4})\)
Since \(\cos (\frac{\pi}{4}) = \frac{\sqrt{2}}{2}\), the slope of the tangent line is \(m = \frac{\sqrt{2}}{2}\).
04
Writing the Equation of the Tangent Line
Finally, we'll write the tangent line's equation. We know the slope (\(m\)) and the point of tangency (\((x_1, y_1) = (\frac{\pi}{4}, \frac{\sqrt{2}}{2})\)), so we can use the point-slope form of the equation of a line:
\(y - y_1 = m(x - x_1)\)
Substitute the values of \(m\), \(x_1\), and \(y_1\) into the equation, then simplify to get the equation of the tangent line:
\(y - \frac{\sqrt{2}}{2} = \frac{\sqrt{2}}{2}(x - \frac{\pi}{4})\)
This is the equation of the tangent line to \(y = \sin x\) at the point \((\frac{\pi}{4}, \frac{\sqrt{2}}{2})\).
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with Vaia!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Derivative of the Sine Function
Understanding the derivative of the sine function is crucial when it comes to calculus, especially for finding the slope of a tangent line to a curve. The sine function, denoted as \(y = \sin x\), is one of the primary trigonometric functions and describes a smooth, periodic oscillation.
When you take the derivative of the sine function, denoted as \(\frac{dy}{dx}\) or \(y'\), you're determining how the value of the sine function changes as \(x\) changes. The rule to remember is that the derivative of \(\sin x\) is \(\cos x\). This relationship comes from the limit definition of the derivative and is a part of the fundamental tools of calculus.
The derivative \(\cos x\) tells us how steep or flat the curve \(y = \sin x\) is at any given point. For a specific value of \(x\), the derivative gives the exact slope of the line that just touches the curve at that point, which is the very definition of a tangent line.
When you take the derivative of the sine function, denoted as \(\frac{dy}{dx}\) or \(y'\), you're determining how the value of the sine function changes as \(x\) changes. The rule to remember is that the derivative of \(\sin x\) is \(\cos x\). This relationship comes from the limit definition of the derivative and is a part of the fundamental tools of calculus.
The derivative \(\cos x\) tells us how steep or flat the curve \(y = \sin x\) is at any given point. For a specific value of \(x\), the derivative gives the exact slope of the line that just touches the curve at that point, which is the very definition of a tangent line.
Point-Slope Form
When we have the slope of a line and a point through which it passes, we can use the point-slope form of the equation to find the equation of the line. The general form is \(y - y_1 = m(x - x_1)\), where \(m\) is the slope and \(x_1, y_1\) are the coordinates of the given point.
For instance, if we know that a line has a slope of 3 and passes through the point (2, 4), the equation would be \(y - 4 = 3(x - 2)\). This format is particularly helpful when you're working with tangent lines since you'll often have a point of tangency and the slope derived from the function's derivative. It translates the abstract concept of slope into a concrete formula that immediately relates to the graphical representation of the line.
After determining your slope and the coordinates of your point, substituting these values into the point-slope form equation sets the stage for writing down the precise equation for the line that you're interested in, making this an indispensable tool in calculus.
For instance, if we know that a line has a slope of 3 and passes through the point (2, 4), the equation would be \(y - 4 = 3(x - 2)\). This format is particularly helpful when you're working with tangent lines since you'll often have a point of tangency and the slope derived from the function's derivative. It translates the abstract concept of slope into a concrete formula that immediately relates to the graphical representation of the line.
After determining your slope and the coordinates of your point, substituting these values into the point-slope form equation sets the stage for writing down the precise equation for the line that you're interested in, making this an indispensable tool in calculus.
Tangent to a Curve
A tangent to a curve is a straight line that touches the curve at a single point without crossing through it. The tangent line has the same direction as the curve's slope at that point. This concept is pivotal in calculus as tangents are used to approximate the behavior of curves and to solve practical problems involving rates of change.
The slope of the tangent line at a particular point on a curve can be found using the derivative of the curve’s function. For example, in the exercise above, we're working with the function \(y=\sin x\), and the tangent line to this curve at the point \(x=\frac{\pi}{4}\) has a slope equal to the derivative of \(y\) evaluated at that point. The slope indicates the steepness and direction at the point where the tangent line just 'kisses' the curve.
Being able to find the tangent line at a given point is invaluable for many applications, including physics and engineering problems where understanding the change at a specific instance is essential. It empowers you to make precise predictions and understand the intricacies of motion and change in a graphical manner.
The slope of the tangent line at a particular point on a curve can be found using the derivative of the curve’s function. For example, in the exercise above, we're working with the function \(y=\sin x\), and the tangent line to this curve at the point \(x=\frac{\pi}{4}\) has a slope equal to the derivative of \(y\) evaluated at that point. The slope indicates the steepness and direction at the point where the tangent line just 'kisses' the curve.
Being able to find the tangent line at a given point is invaluable for many applications, including physics and engineering problems where understanding the change at a specific instance is essential. It empowers you to make precise predictions and understand the intricacies of motion and change in a graphical manner.
