Problem 56 - Contemporary Precalculus

Question

$$\text { Prove the identity.}$$ $$\tan (\pi-x)=-\tan x$$

Step-by-Step Solution

Answer

Question: Prove the trigonometric identity $$\tan (\pi-x)=-\tan x$$ Answer: We proved the trigonometric identity $$\tan (\pi-x)=-\tan x$$ using the formula for the difference of angles for sine and cosine functions. We showed that $$\tan(\pi-x) = -\frac{\sin x}{\cos x} = -\tan x$$.

Step 1: Write the expressions for sine and cosine of the angle difference

We will start by expressing the sine and cosine of the angle difference using the following formulas: $$\sin(\pi-x) = \sin(\pi)\cos(x)-\cos(\pi) \sin(x)$$ $$\cos(\pi-x) = \cos(\pi)\cos(x) + \sin(\pi) \sin(x)$$

Step 2: Simplify the sine and cosine expressions

We know that \(\sin(\pi)=0\) and \(\cos(\pi)=-1\). Plug in these values to the expressions in Step 1: $$\sin(\pi-x) = 0\cos(x)-(-1) \sin(x) = \sin x$$ $$\cos(\pi-x) = (-1)\cos(x) + 0\sin(x) = -\cos x$$

Step 3: Define tangent function in terms of sine and cosine

Now, recall that the tangent function is defined as the ratio of sine and cosine: $$\tan(x) = \frac{\sin(x)}{\cos(x)}$$

Step 4: Calculate the tangent of the angle difference and simplify

Using the expressions for sine and cosine from Step 2, find the tangent of the angle difference: $$\tan(\pi-x) = \frac{\sin(\pi-x)}{\cos(\pi-x)} = \frac{\sin x}{-\cos x} = -\frac{\sin x}{\cos x}$$

Step 5: Final comparison

Compare the expression for \(\tan(\pi-x)\) derived in Step 4 with the definition of tangent function and show they are equal: $$\tan(\pi-x) = -\frac{\sin x}{\cos x} = -\tan x$$ Thus, we have proved the identity $$\tan(\pi-x)=-\tan x$$.

Key Concepts

Sine and Cosine Functions

Sine and cosine are fundamental trigonometric functions that describe the relationship between the angles and sides of a right-angled triangle. These functions are crucial in various identity proofs and calculations involving angles.

The sine function, denoted as \( \sin(\theta) \), calculates the ratio of the length of the side opposite the angle \( \theta \) to the hypotenuse. Similarly, the cosine function, \( \cos(\theta) \), calculates the ratio of the adjacent side's length to the hypotenuse.

• \( \sin(\theta) = \frac{\text{opposite side}}{\text{hypotenuse}} \)
• \( \cos(\theta) = \frac{\text{adjacent side}}{\text{hypotenuse}} \)

Beyond geometric interpretations, sine and cosine can also be understood graphically. They represent periodic wave forms that repeat every \(2\pi\) radians. This makes them vital in understanding oscillatory processes.

By knowing the exact values of \( \sin(\pi) \) and \( \cos(\pi) \), which are 0 and -1 respectively, transformations for other angle identities can be calculated. For example, when manipulating angles like \((\pi - x)\), these values greatly simplify the expressions, as seen in our exercise solution.

Tangent Function

The tangent function is another essential trigonometric function, especially when dealing with angle identities. It is often represented as \( \tan(\theta) \) and is defined as the ratio of the sine to the cosine of the same angle.

This can be written mathematically as:

• \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \)

The tangent function can be more intuitive when considering the sides of a triangle:

• In right-angled triangles, \( \tan(\theta) \) equals \( \frac{\text{opposite side}}{\text{adjacent side}} \).

Understanding this ratio allows for easier calculation when dealing with angle transformations or finding unknown angle measures.

In the given problem, the tangent function is utilized to express the relationship of tangents in terms of the difference of angles. By simplifying \( \tan(\pi - x) \) using previously calculated sine and cosine values, we see its effect visually as a flip along the x-axis, resulting in \(-\tan(x)\). By exploring these relationships, we leverage the interplay between sine, cosine, and tangent to demonstrate the identity.

Angle Difference Identities

Angle difference identities are formulas used to express the trigonometric functions of the difference of two angles in terms of the sine, cosine, or tangent of the individual angles. These identities are potent tools in simplifying and solving complex trigonometric expressions.

For the sine and cosine functions, the angle difference identities are given by:

• \( \sin(A - B) = \sin A \cos B - \cos A \sin B \)
• \( \cos(A - B) = \cos A \cos B + \sin A \sin B \)

In solving the given problem, these identities allowed us to transform the expression for \( \sin(\pi - x) \) and \( \cos(\pi - x) \). By substituting specific values (such as \( \sin(\pi) = 0 \) and \( \cos(\pi) = -1 \)) into these equations, simplifications naturally followed.

While the tangent function uses its unique identity to represent \( \tan(A - B) \), it can also be derived from the sine and cosine differences as shown in the steps of solving \( \tan(\pi-x) \). By taking this approach, angle difference identities help in forming, proving, and understanding complicated trigonometric relationships, which in this case, demonstrated that \( \tan(\pi-x) = -\tan(x) \).