Question

$$ \begin{array}{l} \text { C } \text { . Find the equation of the tangent line to } y=\cos x \text { at }\\\ x=1 \end{array} $$

Short answer

The tangent line at \( x = 1 \) is \( y = -\sin(1)x + \sin(1) + \cos(1) \).\)

Step-by-step solution

Find the Derivative

To find the tangent line to the curve at a specific point, we first need to find the derivative of the function. The given function is \( y = \cos x \). The derivative of \( \cos x \) is \( y' = -\sin x \).

Evaluate the Derivative at the Given Point

Next, evaluate the derivative at the point \( x = 1 \) to find the slope of the tangent line. Substitute \( x = 1 \) into the derivative: \( y'(1) = -\sin(1) \). Keep in mind that \( \sin(1) \) is a numeric value that can be found using a calculator.

Calculate the Value of the Function at the Given Point

We need to find the y-coordinate of the point on the curve where the tangent touches. Substitute \( x = 1 \) into the original function: \( y = \cos(1) \). This will give us the y-coordinate of the point \((1, \cos(1))\).

Apply Point-Slope Formula

Use the point-slope form of the equation of a line, \( y - y_1 = m(x - x_1) \), where \((x_1, y_1)\) is the point on the curve and \(m\) is the slope. Here, \( m = -\sin(1), x_1 = 1, y_1 = \cos(1) \). The equation becomes: \( y - \cos(1) = -\sin(1)(x - 1) \).

Simplify the Equation

Now simplify the equation from Step 4: \( y - \cos(1) = -\sin(1)x + \sin(1) \). Then rearrange it in the standard form: \( y = -\sin(1)x + \sin(1) + \cos(1) \).

Key concepts

Derivative

When dealing with calculus and functions, the derivative is a powerful tool. A derivative provides us with the slope of the function at any given point. It essentially tells us how the function changes at a specific moment. For the function given in the exercise, which is \( y = \cos x \), the derivative can be calculated using standard rules for differentiation. The derivative of \( \cos x \) is \( y' = -\sin x \). This result is based on the differentiation rules of trigonometric functions. Once you have the derivative, you can determine how the function is changing at a specific point, which is crucial for finding the equation of a tangent line.

Trigonometric Functions

Trigonometric functions like sine and cosine are fundamental in calculus. These functions arise from the geometric interpretation of a unit circle. In the function \( y = \cos x \), \( \cos x \) represents the cosine of an angle \( x \), which corresponds to the x-coordinate of a point on a unit circle. Trigonometric functions are periodic and oscillate between -1 and 1, making them unique and interesting to work with. When we differentiate \( \cos x \), we use the property that the derivative results in another trigonometric function, \(-\sin x\). Understanding these functions and their derivatives allows for deeper problem-solving strategies in calculus.

Calculus Problem-Solving

Tackling calculus problems often involves systematic steps. In this particular problem, we are tasked with finding the equation of a tangent line. The steps include:

• Finding the derivative of the function to determine the general slope at any point.
• Substituting a specific point into the derivative to find the slope at that point.
• Calculating the corresponding function value to get the y-coordinate of the tangent point.
• Applying the point-slope formula to derive the tangent line equation.
• Simplifying the equation for clarity and accuracy.

These steps guide us through understanding the function behavior and applying mathematical frameworks to real calculations.

Point-Slope Form

The point-slope form is a fundamental element in geometry and algebra for determining linear equations. It uses the format \( y - y_1 = m(x - x_1) \), where \((x_1, y_1)\) is a specific point on the line, and \(m\) is the slope. In this exercise, once we calculated the slope \(-\sin(1)\) at \(x = 1\) and found the function value \(y = \cos(1)\), we could apply this form. The equation from point-slope form directly facilitates writing the equation of a tangent line. Rearranging the terms or simplifying allows obtaining a standard line equation that's easier to understand and use.