Problem 13 - A Graphical Approach to Precalculus with Limits

Question

To show that sec(- \(x\) ) = sec \(x\) for all \(x\) in the domain, we begin by writing $$ \sec (-x)=\frac{1}{\cos (-x)} $$ and then use the fact that \(\cos (-x)=\cos x\) for all \(x\) to complete the argument. Use this method to prove each of the following. $$\tan (-x)=-\tan x$$

Step-by-Step Solution

Answer

\( \tan(-x) = -\tan(x) \) by using trigonometric identities.

Step 1: Write Expression for Tan(-x)

The tangent function can be expressed in terms of the sine and cosine functions. Begin by expressing \( \tan(-x) \) as \( \frac{\sin(-x)}{\cos(-x)} \).

Step 2: Use Trigonometric Identities

Recall the identities: \( \sin(-x) = -\sin(x) \) and \( \cos(-x) = \cos(x) \). Use these to simplify the expression from Step 1: \( \tan(-x) = \frac{-\sin(x)}{\cos(x)} \).

Step 3: Simplify Expression

Now, simplify the expression from Step 2: \( \tan(-x) = -\left(\frac{\sin(x)}{\cos(x)}\right) = -\tan(x) \). This shows the negative sign placed outside the tangent function.

Key Concepts

Secant Function

The secant function is an important trigonometric function. It is the reciprocal of the cosine function. This means the secant of an angle, denoted by \( \sec(x) \), is equal to \( \frac{1}{\cos(x)} \). Understanding the secant function involves recognizing its relationship to cosine.
This is particularly important for realizing properties like symmetry. A useful symmetry property here is that \( \sec(-x) = \sec(x) \). This stems from the even property of the cosine function, meaning \( \cos(-x) = \cos(x) \). Therefore, since secant is the reciprocal of cosine, \( \sec(-x) \) simplifies to the same value as \( \sec(x) \).
Key takeaways about the secant function are:

• It's the reciprocal of cosine: \( \sec(x) = \frac{1}{\cos(x)} \)
• It shares the even property with cosine: \( \sec(-x) = \sec(x) \)

These properties are often used in solving trigonometric equations and proving identities. Understanding these basics can make working with secant much easier.

Tangent Function

The tangent function is another fundamental trigonometric function. It is defined as the ratio of the sine function to the cosine function. This is expressed as \( \tan(x) = \frac{\sin(x)}{\cos(x)} \). Knowing this helps break down more complex identities.
One of the important properties of the tangent function is its odd nature. This basically means \( \tan(-x) = -\tan(x) \). When we have \( \tan(-x) \), we look at it as \( \frac{\sin(-x)}{\cos(-x)} \). Using the trigonometric identities \( \sin(-x) = -\sin(x) \) and \( \cos(-x) = \cos(x) \), this becomes \( \frac{-\sin(x)}{\cos(x)} \), which simplifies to \(-\tan(x) \).
Key points for the tangent function include:

• It's the ratio of sine to cosine: \( \tan(x) = \frac{\sin(x)}{\cos(x)} \)
• It has an odd property: \( \tan(-x) = -\tan(x) \)

These elements are crucial for simplifying complex expressions involving tangent and proving various trigonometric identities.

Sine and Cosine Identities

Sine and cosine identities are foundational in trigonometry. They help in working with various trigonometric functions and solving equations. The most fundamental ones include \( \sin(-x) = -\sin(x) \) and \( \cos(-x) = \cos(x) \).
These identities help in understanding the symmetry and cyclical nature of trigonometric functions. The identity \( \sin(-x) = -\sin(x) \) indicates that sine is an odd function, meaning its graph is symmetric with respect to the origin. For cosine, \( \cos(-x) = \cos(x) \) shows that it's an even function with symmetry about the y-axis.
Important aspects to remember:

• Sine is an odd function: \( \sin(-x) = -\sin(x) \)
• Cosine is an even function: \( \cos(-x) = \cos(x) \)

Understanding these identities is crucial as they form the basis for more complex trigonometric identities and are widely used in calculus and physics. They also help students grasp the geometric interpretations of these functions, allowing for better insight into how angles and rotations work within trigonometry.