Problem 34 - Algebra and Trigonometry

Question

Use one or more of the six sum and difference identities to solve Exercises \(13-54\) Verify each identity. $$ \sin \left(x+\frac{3 \pi}{2}\right)=-\cos x $$

Step-by-Step Solution

Answer

The given trigonometric identity \(\sin \left(x+\frac{3 \pi}{2}\right)=-\cos x\) is verified successfully.

Step 1: Express the Left Side in Terms of Basic Identities

Let's express the left side of the equation using the identity \(\sin(x+y)\). so, \(\sin \left(x+\frac{3 \pi}{2}\right) = \sin x \cos \left(\frac{3 \pi}{2}\right) + \cos x \sin \left(\frac{3 \pi}{2}\right)\).

Step 2: Simplifying the Resultant Equation

Next, use the values of \(\cos \left(\frac{3 \pi}{2}\right) = 0\), and \(\sin \left(\frac{3 \pi}{2}\right) = -1\) to simplify the equation. Here we got \(\sin x \cos \left(\frac{3 \pi}{2}\right) + \cos x \sin \left(\frac{3 \pi}{2}\right) = \sin x * 0 + \cos x * -1 = -\cos x.\)

Step 3: Verifying the Identity

Finally, we see that the simplified left side is equal to the right side of the given identity, i.e., \(-\cos x = -\cos x\). Thus, we have verified the given trigonometric identity.

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. They play a crucial role in simplifying expressions and solving equations involving trigonometric functions. These identities are based on the inherent properties of angles and the relationship between the sides of a triangle.

One of the most important sets of identities are the sum and difference identities, which express the sine and cosine of the sum or difference of two angles in terms of the sines and cosines of the two angles. The sum identities are defined as:

• \(\sin(\alpha + \beta) = \sin\alpha \cos\beta + \cos\alpha \sin\beta\)
• \(\cos(\alpha + \beta) = \cos\alpha \cos\beta - \sin\alpha \sin\beta\)

Similar identities exist for the difference of two angles. These identities are particularly useful when we deal with compound angles or need to transform products into sums or differences.

Verifying Trigonometric Identities

Verifying trigonometric identities involves proving that both sides of an equation are equal for all values within the domains of the variables. This is often done by transforming one side of the equation to match the other side.

When verifying an identity, you should be familiar with basic identities such as reciprocal identities, quotient identities, and Pythagorean identities. These become the building blocks for manipulation and transformation. The process of verification usually involves the following steps:

• Determining which side of the identity is more complex and simplifying that side to match the simpler side.
• Using known identities to rewrite trigonometric expressions.
• Simplifying expressions by performing algebraic manipulations, like factoring or canceling.
• Working through each expression systematically to avoid making errors.

By following these steps, as seen in the exercise, we verify whether an identity holds by seeing if the simplified result from one side of the equation is identical to the other side.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions is vital to solving trigonometric equations and verifying identities. The goal of simplification is to convert complex expressions into simpler, more recognizable forms by applying trigonometric identities. To simplify a trigonometric expression, one might:

• Use sum and difference identities to break down complex angles.
• Apply reciprocal or quotient identities to express trigonometric functions in terms of sine and cosine, when they are easier to work with.
• Utilize Pythagorean identities to substitute one trigonometric function for another based on their relationships.
• Factor expressions to reveal common terms that can be cancelled or combined.

As illustrated in the original exercise with the identity \(\sin(x + \frac{3\pi}{2})\), by recognizing that certain values of sine and cosine at key angles like \(\frac{\pi}{2}\), \(\pi\), or \(\frac{3\pi}{2}\) are either 0 or \(\pm 1\), we can greatly simplify the process of verifying identities.