91Ó°ÊÓ

Trigonometric functions are a fundamental part of calculus and mathematics. Understanding their derivatives helps in finding slopes of tangent lines. The function in our exercise is the tangent function, denoted as \( y = \tan x \). The derivative of the tangent function, which is \( \sec^2 x \), plays a vital role in our solution. To find this derivative, consider how each trigonometric function has a unique derivative. For example:
\( \sin x \) differentiates to \( \cos x \)
\( \cos x \) differentiates to \( -\sin x \)

When you differentiate \( \tan x \), remember that it can be expressed as a quotient, \( \frac{\sin x}{\cos x} \). Applying the quotient rule, the derivative becomes \( \sec^2 x \). This derivative, \( \sec^2 x \), tells us the slope of the tangent line at any point \( x \). By substituting specific values of \( x \), we find the slope at desired points, which are crucial for forming the equation of the tangent line.

The equation of a line can be represented in different forms, the most common being the slope-intercept form, \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. In the context of tangent lines, the point-slope form is very useful:
\[ y - y_1 = m(x - x_1) \]
Where \( (x_1, y_1) \) is a point on the line and \( m \) is the slope.

To find the tangent line of a given function at a specified point, you perform a few straightforward steps:

• First, determine the slope \( m \) using the derivative of the function.
• Next, identify a specific point on the curve where the tangent touches. This values helps in calculating \((x_1, y_1)\).
• Finally, plug these values into the point-slope form and simplify to get the equation in slope-intercept form if needed.

These steps ensure that you accurately find the equation of the tangent line, as illustrated in the problem solution.

Trigonometric functions, like sine, cosine, and tangent, are essential in describing phenomena involving angles and periodicity. The tangent function, \( \tan x \), is defined as the ratio of sine and cosine, \( \frac{\sin x}{\cos x} \), and is important in trigonometry due to its unique properties and behavior. At various angles such as \( x = 0 \), \( \pi/4 \), or \( -\pi/4 \), the tangent provides distinct values that help in many applications, including determining tangent lines.

When working with these trigonometric functions, it's vital to remember:

• Tangent is undefined at angles where cosine is zero, like \( \frac{\pi}{2} \).
• The periodicity of tangent is \( \pi \), meaning it repeats every \( \pi \) radians.
• For certain key angles, we can easily determine tangent values without computation, such as \( \tan(0) = 0 \), \( \tan(\pi/4) = 1 \), and \( \tan(-\pi/4) = -1 \).

Understanding these principles aids in solving calculus problems related to trigonometric functions efficiently.