Problem 27 - Precalculus: Functions and Graphs

Question

Exer. \(25-36:\) Verify the reduction formula. $$\sin \left(x-\frac{5 \pi}{2}\right)=-\cos x$$

Step-by-Step Solution

Answer

The identity \( \sin \left(x - \frac{5\pi}{2}\right) = -\cos x \) is verified using trigonometric identities.

Step 1: Understand the Trigonometric Identity

The problem asks us to verify if \( \sin \left(x - \frac{5\pi}{2}\right) = -\cos x \) can hold true. We need to explore the identity by considering angle reduction or angle addition formulas for sine.

Step 2: Simplify Using Angle Identities

Recall that \( \sin(\theta - 2\pi) = \sin(\theta) \). Thus, rewrite \( x - \frac{5\pi}{2} \) in terms of a familiar angle. Since \( \frac{5\pi}{2} = 2\pi + \frac{\pi}{2} \), rewrite the expression: \( \sin(x - 2\pi - \frac{\pi}{2}) = \sin(x - \frac{\pi}{2}) \).

Step 3: Apply Sine and Cosine Relationship

From trigonometric identities, we know \( \sin(x - \frac{\pi}{2}) = -\cos(x) \). This identity arises because projecting a point on the unit circle \( \frac{\pi}{2} \) radians backwards results in this relationship, based on the behavior of sine and cosine as cofunctions.

Step 4: Verify and Conclude

Substituting back to check if the simplifications hold true: \( \sin(x - \frac{\pi}{2}) = -\cos x \), which matches exactly with the given equation. Therefore, \( \sin \left(x - \frac{5\pi}{2}\right) = -\cos x \) is verified.

Key Concepts

Sine and Cosine Relationship

In trigonometry, the sine and cosine functions have a special cofunction relationship. This means they are interconnected in a way that allows for transforming a sine function into a cosine function and vice versa by shifting angles. A basic understanding starts with the unit circle, which helps in visualizing these functions over different angles.

• When you shift an angle by \(\frac{\pi}{2}\) (90 degrees), the sine of that angle becomes the negative cosine of the original angle. Mathematically, this is expressed as \(\sin\left(x - \frac{\pi}{2}\right) = -\cos(x)\).
• This relationship reflects how the sine function at one angle is directly associated with the cosine function at another angle. This link is crucial for rearranging expressions and solving trigonometric equations.

Understanding this connection not only simplifies complex trigonometric identities but also helps in verifying equations like \(\sin\left(x - \frac{5\pi}{2}\right) = -\cos x\). The sine and cosine relationship is foundational and allows for intuitive manipulation of trigonometric functions.

Angle Reduction

Angle reduction in trigonometry often involves simplifying the expression by compressing angles larger than \(2\pi\) (360 degrees) down to equivalent angles within the first full circle (0 to \(2\pi\)). This simplification uses the periodic nature of trigonometric functions.

• For example, any angle \(\theta - 2\pi\) is equivalent to \(\theta\) since the sine and cosine functions have a period of \(2\pi\).
• This property is applied when reducing \(x - \frac{5\pi}{2}\). Breaking down \(\frac{5\pi}{2}\) as \(2\pi + \frac{\pi}{2}\) shows that \(\sin(x - \frac{5\pi}{2}) = \sin(x - 2\pi - \frac{\pi}{2}) = \sin(x - \frac{\pi}{2})\).
• Reduction makes it easier to compute the sine or cosine of an angle and verify identities.

Understanding how to reduce angles is essential for tackling more involved trigonometric problems and to comprehend cycles, repetition, and identities of trigonometric functions like seen in our original problem.

Trigonometric Functions

Trigonometric functions, including sine and cosine, are fundamental in mathematics. They express the relationship between the angles and sides of a right-angled triangle and are also broadly applicable in modeling periodic phenomena, like waves or sound.

• Sine and cosine are periodic with a period of \(2\pi\), meaning their values repeat every \(2\pi\) units.
• They are defined on the unit circle using the x and y coordinates: \(\cos(\theta)\) is the x-coordinate, and \(\sin(\theta)\) is the y-coordinate of the point where a line at angle \(\theta\) intersects the circle.
• Understanding their behavior on the unit circle aids in verifying equations, solving for unknowns, and simplifying complex expressions.

In our exercise, knowing how these functions relate and transform across different angle reductions is key to reaching the solution effectively and demonstrating identity such as \(\sin \left(x - \frac{5\pi}{2}\right) = -\cos x\). Grasping these concepts empowers students to solve trig equations efficiently.