Problem 53 - Contemporary Precalculus

Question

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

Step-by-Step Solution

Answer

Question: Prove the identity \(\sin(x - \pi) = -\sin x\). Answer: To prove this identity, we used the angle subtraction formula for sine function: \(\sin(A - B) = \sin A \cos B - \cos A \sin B\). We substituted x for A and π for B and evaluated the trigonometric functions. Finally, we simplified the equation to get the result, \(\sin(x - \pi) = -\sin x\).

Step 1: Identify the angle subtraction formula

We will use the angle subtraction formula for sine function: $$\sin(A - B) = \sin A \cos B - \cos A \sin B$$

Step 2: Substitute the given angle values

Substitute x for A and π for B in the angle subtraction formula: $$\sin(x - \pi) = \sin x \cos \pi - \cos x \sin \pi$$

Step 3: Evaluate the trigonometric functions

Remember that \(\cos \pi = -1\) and \(\sin \pi = 0\). Plug these values into the equation: $$\sin(x - \pi) = \sin x (-1) - \cos x (0)$$

Step 4: Simplify the equation

Simplify the equation to get the final result: $$\sin(x - \pi) = -\sin x$$ We have proved the identity \((\sin(x-\pi) = -\sin x)\).

Key Concepts

Understanding the Angle Subtraction Formula

The angle subtraction formula is a key concept in trigonometry, allowing us to simplify and understand trigonometric expressions involving subtracting angles. Specifically, for the sine function, the formula is given by:

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

This formula enables us to calculate the sine of a difference between two angles, by breaking it into simpler components involving the sines and cosines of the individual angles. This separation is particularly useful because it turns a complex trigonometric expression into a more manageable form. You replace the expression involving both angles with products of trigonometric functions of simpler, individual angles. This transformation is central to solving various trigonometric identities and equations.

Exploring the Sine Function

The sine function is one of the fundamental trigonometric functions, deeply rooted in the geometry of a right triangle. It is defined as the ratio of the length of the opposite side to the length of the hypotenuse. In a unit circle context:

• \( \sin(\theta) \) is the y-coordinate of the point on the unit circle at an angle \( \theta \) from the positive x-axis.

The sine function is periodic with a period of \( 2\pi \), and it is odd, meaning that \( \sin(-x) = -\sin(x) \). This property is crucial when working with angle subtraction and addition formulas, as demonstrated in proving trigonometric identities. Understanding the sine curve's behavior allows us to predict its values and transformations under various conditions, such as angle shifts by \( \pi \).

Diving into Trigonometric Functions

Trigonometric functions are essential in mathematics, providing a link between angles and ratios of triangle sides. The main trigonometric functions include sine, cosine, and tangent, each with its unique properties and applications. These functions periodically repeat their values:

• \( \sin \) and \( \cos \) repeat every \( 2\pi \).
• \( \tan \) repeats every \( \pi \).

The intrinsic properties of trigonometric functions such as evenness or oddness, and periodicity are leveraged in deriving and proving identities like the one given in the exercise. They help convert complex angular expressions into more accessible terms. Mastery of these properties equips anyone working with angles to address more challenging mathematical problems seamlessly.

Proof in Trigonometry

Proofs in trigonometry rely on foundational identities and algebraic manipulation to demonstrate true expressions. For the identity \( \sin(x - \pi) = -\sin x \), we used the angle subtraction formula as our main tool, followed by the evaluation of standard trigonometric values. Proofs involve:

• Understanding the fundamental identity involved, such as angle subtraction.
• Substituting known values and simplifying the expressions.

This process demands a combination of both algebraic manipulation and trigonometric knowledge. By integrating these skills, one can logically deduce and verify the relationships between different trigonometric expressions. Such proofs confirm the underlying harmony in trigonometry and its reliability in solving real-world problems.