Problem 37 - Algebra 2

Question

Verify each identity. $$ \sin (A-B)=\sin A \cos B-\cos A \sin B $$

Step-by-Step Solution

Answer

Since the given expression exactly matches with the sine subtraction identity, hence it is verified that \(\sin(A - B) = \sin A \cos B - \cos A \sin B\).

Step 1: Recall the trigonometric identities

Firstly, remember the addition and subtraction identities for sine. They are formulated as follows: \(\sin(A + B) = \sin A \cos B + \cos A \sin B\) and \(\sin(A - B) = \sin A \cos B - \cos A \sin B\). These identities hold true for all angles A and B.

Step 2: Look at the given exercise

Our aim is to verify the identity \(\sin(A - B) = \sin A \cos B - \cos A \sin B\), which is the subtraction identity of sine.

Step 3: Verify the identity

From the identities recalled in Step 1, it is clear that the given expression \(\sin(A - B) = \sin A \cos B - \cos A \sin B\) holds true. It exactly matches with the subtraction identity of sine.

Key Concepts

Sine Subtraction Identity

The Sine Subtraction Identity is an essential element in trigonometry. It helps to simplify expressions involving angles. The identity is written as \( \sin(A - B) = \sin A \cos B - \cos A \sin B \). Here, the sine of the difference between two angles \(A\) and \(B\) is represented as a combination of the sine and cosine of the individual angles. This formula is particularly useful because it allows us to break down a complex expression into simpler parts.

By using this identity, you can convert a subtraction problem involving two angles into a problem that only involves multiplication and subtraction. This is very handy when working with trigonometric problems that require simplification. In practice, remember that both sine and cosine functions are involved, and they alternately swap places between addition and subtraction, contributing to a neat calculation.

Angle Addition and Subtraction

The concepts of angle addition and subtraction provide a framework for working with angles in trigonometry. They relate to how angles can be combined or separated to simplify calculations. The addition and subtraction identities for sine are given by:

• For addition: \( \sin(A + B) = \sin A \cos B + \cos A \sin B \)
• For subtraction: \( \sin(A - B) = \sin A \cos B - \cos A \sin B \)

If you notice, these identities have similar structures but differ in the operation: plus for addition and minus for subtraction. This similarity is not coincidental but rather follows from how sine and cosine functions work together to balance each other's changes.

Understanding these identities allows you to solve equations where an angle is expressed as the sum or difference of other angles, making complex trigonometric calculations more manageable.

Trigonometric Functions

Trigonometric functions such as sine, cosine, and tangent are fundamental in analyzing angles and triangles. These functions help in establishing relationships between the angles and side lengths of triangles. The sine function, in particular, helps in calculating the ratio of the opposite side to the hypotenuse in a right triangle.

The three main trigonometric functions are:

• Sine (sin): This is the ratio of the opposite side to the hypotenuse.
• Cosine (cos): This is the ratio of the adjacent side to the hypotenuse.
• Tangent (tan): This is the ratio of the opposite side to the adjacent side.

In the realm of identities, these functions interrelate nicely, allowing transformations and simplifications, such as with the sine subtraction identity. Trigonometric functions are crucial not just for simplifying angles but are also used in wave functions, circles, and even in various fields like physics and engineering.