Problem 94 - Precalculus with Limits

Question

In Exercises 91 - 94, determine whether the statement is true or false. Justify your answer. \( \sin\left(x - \dfrac{\pi}{2}\right) = - \cos x \)

Step-by-Step Solution

Answer

The statement \( \sin\left(x - \dfrac{\pi}{2}\right) = - \cos x \) is true.

Step 1: Consider the co-function identities

Recall the co-function identities, especially the identity \(\sin\left(x - \dfrac{\pi}{2}\right) = \cos\left(\dfrac{\pi}{2} - x\right)\), this identity will be used next step to make a substitution.

Step 2: Make a substitution

Using the identity from Step 1, replace \(\sin\left(x - \dfrac{\pi}{2}\right)\) with \(\cos\left(\dfrac{\pi}{2} - x\right)\) in the given equation. Thus, the given equation will become \(\cos\left(\dfrac{\pi}{2} - x\right) = - \cos x\).

Step 3: Consider the properties of cosine function

Recall that cosine is an even function, which means \(\cos(-x)=\cos x\). Hence \( -\cos x = \cos(-x)\), so replace \(-\cos x\) with \(\cos(-x)\) in the equation. Thus the equation will become \(\cos\left(\dfrac{\pi}{2} - x\right) = \cos(-x)\). Checking this using the function of cosine, we find that this is actually true when \(x\) is within \(0 ≤ x ≤ π\). This clarifies that the given statement is indeed true.

Key Concepts

Co-function Identities

Co-function identities are a key concept in trigonometry that show the relationship between sine and cosine functions. They help us to transform angles by showing how sine and cosine can relate to angles that sum up to a right angle, or \(\frac{\pi}{2}\).

This identity \(\sin\left(x - \frac{\pi}{2}\right) = \cos\left(\frac{\pi}{2} - x\right)\) is derived from the fact that sine and cosine are complementary. These identities are valuable for simplifying trigonometric expressions or equations because they allow substitutions that can reveal new perspectives or simpler forms.

• Co-function identities ensure that we can interchange functions based on their complementary behavior.
• Key co-function pairs include: \(\sin(x) = \cos\left(\frac{\pi}{2} - x\right)\) and \(\cos(x) = \sin\left(\frac{\pi}{2} - x\right)\).
• Understanding and applying these identities can greatly assist you in solving complex trigonometric problems.

The example above uses a co-function identity to transform the given expression, providing a pathway to testing if the statement is true.

Cosine Function

The cosine function is one of the primary trigonometric functions, often associated with the adjacent side over hypotenuse in right triangles. It's a periodic function, meaning it repeats its values in regular intervals. This periodicity is essential in understanding how cosine behaves with negative and positive inputs.

The cosine function is expressed as \(\cos(x)\), and it is periodic with a period of \(2\pi\), meaning it repeats every \(2\pi\) units. This property is beneficial when solving trigonometric equations or transformations, as it helps predict the function’s behavior over different intervals.

• The graph of the cosine function is a wave-like pattern, which alternates between -1 and 1.
• Cosine is especially useful in modeling rotational and wave patterns in physics and engineering.
• Key identity to remember: \(\cos(x) = \cos(2\pi - x)\) because of its periodic nature.

In the solution, recognizing how cosine can be expressed equivalently with different angles helps in verifying the trigonometric statement.

Even Functions

In mathematics, an even function is a function where \(f(x) = f(-x)\). This symmetry means the function mirrors itself around the vertical axis, making even functions easier to work with for certain transformations and evaluations.

Cosine is a classic example of an even function. When performing transformations or substitutions, knowing that \(\cos(-x) = \cos(x)\) simplifies many problems and makes equations more manageable.

• An even function is symmetrical along the y-axis, which helps in verifying equations.
• The property \(\cos(-x) = \cos(x)\) allows for substituting negative angles without changing the function’s value.
• This characteristic is vital for problems involving reflections and oscillations.

In the given solution, recognizing cosine as an even function enables the substitution \(-\cos(x) = \cos(-x)\), leading to the conclusion that the original statement is true.