Question

Expression for \(\sin C-\sin D\) is (a) \(2 \cos \frac{C-D}{2} \sin \frac{C+D}{2}\) (b) \(2 \cos \frac{C+D}{2} \sin \frac{C-D}{2}\) (c) \(2 \sin \frac{C+D}{2} \sin \frac{C-D}{2}\) (d) \(2 \cos \frac{C+D}{2} \cos \frac{C-D}{2}\)

Short answer

The correct option is (b): \(2 \cos \frac{C+D}{2} \sin \frac{C-D}{2}\)

Step-by-step solution

Recall the Formula for sin(x) - sin(y)

The trigonometric identity for the difference of two sine functions, \(\sin x - \sin y = 2 \cos \frac{x+y}{2} \sin \frac{x-y}{2}\)

Apply the Formula to the Expression

Replace \(x\) with \(C\) and \(y\) with \(D\) in the trigonometric identity established in step 1. This results in \(\sin C - \sin D = 2 \cos \frac{C+D}{2} \sin \frac{C-D}{2}\)

Identify the Correct Option

Comparing the expression obtained in step 2 with the options provided in the question, we observe that option (b) matches the formula we derived.

Key concepts

Introduction to Trigonometry

Trigonometry is a branch of mathematics that revolves around the relationships between the sides and angles of triangles. The word 'trigonometry' itself comes from the Greek 'trigonon' (meaning triangle) and 'metron' (meaning measure). It's a study deeply woven into many fields from astronomy to computer graphics, physics, engineering, and various applications needing measurements of angles and distances.

Central to trigonometry are the functions sine, cosine, and tangent, which relate the angles of a triangle to the ratios of its sides. These functions are useful when it comes to defining the position of a point in polar coordinates relative to another point, the oscillations of waves, and many phenomena that repeat in cycles. Knowing trigonometric formulas is key as they allow the calculation of unknown sides or angles in right-angled triangles and also describe the wave and oscillation patterns.

Understanding the Sine Difference Formula

The sine difference formula is a tool in trigonometry used to express the difference between the sine of two angles in a more manageable form. The formula can be written as \(\sin x - \sin y = 2 \cos \frac{x+y}{2} \sin \frac{x-y}{2}\).

Considering that sines and cosines are fundamental in describing circular motion and wave phenomena, the sine difference formula is especially practical when dealing with wave interference, sound waves, and light waves. When two waves with different frequencies interact, their effective displacement can be found using the sine difference formula, illustrating the formula's significance in physics and engineering.

Importance of Trigonometric Formulas

Trigonometric formulas are the backbone of trigonometry. They consist of various identities and equations that provide a relationship between trigonometric functions and angles. These formulas are widely leveraged in simplifying expressions, solving equations, and transforming one type of trigonometric expression into another. Undoubtedly, they are powerful tools for problem-solving across numerous scientific and mathematical disciplines.

Commonly used trigonometric formulas include the Pythagorean identities, angle sum and difference identities, double angle and half-angle formulas, and product-to-sum formulas. Each of these serves a unique purpose in calculations, and a deep understanding of these formulas allows for efficient and accurate solutions to trigonometric problems.

For example, the sine difference formula applied in the original exercise is a specific instance of the angle difference identities, which is also mirrored in cosine and tangent functions. These identities are key in solving trigonometric equations, modeling periodic processes, and many other practical applications requiring angle measurements.