Problem 33 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

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

Step 1: Recall Sum Identity

Recall the sum identity for sine, which is \(\sin (A + B) = \sin A \cos B + \cos A \sin B\)

Step 2: Apply Sum Identity to Problem

Use this identity to rewrite \(\sin \left(x+\frac{\pi}{2}\right)\) . The result is \(\sin x \cos \frac{\pi}{2} + \cos x \sin \frac{\pi}{2}\)

Step 3: Simplify the Result

Remember that \(\cos \frac{\pi}{2} = 0\) and \(\sin \frac{\pi}{2} = 1\), so the expression simplifies to \(0 \cdot \sin x + 1 \cdot \cos x = \cos x\).

Key Concepts

Trigonometric Identities

Understanding trigonometric identities is essential for solving a variety of problems in trigonometry. These identities are equations that are true for all values of the variables involved. They provide a way to simplify complex trigonometric expressions and to prove other mathematical properties.

There are many types of trigonometric identities, but they all stem from the unit circle and the relationships between the sides of a right triangle. For example, the Pythagorean identities that connect the sine, cosine, and tangent functions are derived from the Pythagorean theorem. The reciprocal identities show the relationship between the basic trigonometric functions and their reciprocals - cosecant, secant, and cotangent.

Trigonometric identities are used not only in academic settings but also in real-world applications such as engineering, physics, and computer science. Mastery of these identities allows for the analysis of waves, oscillations, and circular motion among other phenomena.

Sine Sum Identity

The sine sum identity is a fundamental relationship in trigonometry that allows us to rewrite the sine of a sum of two angles in terms of the sine and cosine of the individual angles. The identity is expressed as:

\[\begin{equation}\sin(A + B) = \sin A \cos B + \cos A \sin B\end{equation}\]
For instance, if you wanted to find the sine of the sum of angle x and \frac{}\pi}{2}] , you would use this identity. The sine sum identity helps in breaking down more complex expressions into simpler terms, making calculations easier to perform.

In the context of solving trigonometric problems, the sine sum identity is particularly useful. It enables you to handle expressions where the argument of the sine function is not simply a single angle but a combination of angles, as seen in the given exercise. By understanding this identity, students can approach such problems with confidence.

Verifying Trigonometric Identities

Verifying trigonometric identities involves proving that two trigonometric expressions are equivalent regardless of the value of the variable involved. It is a critical skill in trigonometry, often used to confirm that a trigonometric equation holds true for all possible values.

To verify an identity, one would typically manipulate one side of the equation using known identities until it matches the other side. During this process, it's crucial to be familiar with the fundamental trigonometric identities and to apply algebraic manipulations such as factoring, distributing, and combining like terms.

For example, in the exercise provided, you verify the identity by transforming the left side of the equation using the sine sum identity. The process of simplification using the values of sine and cosine at specific angles leads to the anticipated result, thus verifying the identity. Approaching verification with a systematic process can make it a much less daunting task.