Problem 74 - Precalculus

Question

In Exercises 73 and 74 , determine whether each statement is true or false. $$\csc \left(x-\frac{\pi}{2}\right)=\sec x$$

Step-by-Step Solution

Answer

The statement is false.

Step 1: Recall Definitions

Start by recalling the definitions of the cosecant and secant functions. The cosecant function is defined as \( \csc \theta = \frac{1}{\sin \theta} \), and the secant function is defined as \( \sec \theta = \frac{1}{\cos \theta} \).

Step 2: Simplify the Given Identity

The expression to verify is \( \csc \left(x - \frac{\pi}{2}\right) = \sec x \). Using the definition of cosecant, we can rewrite the left-hand side: \( \csc \left(x - \frac{\pi}{2}\right) = \frac{1}{\sin\left(x - \frac{\pi}{2}\right)} \).

Step 3: Use Angle Transformation for Sine

Apply the angle transformation identity for sine: \( \sin\left(x - \frac{\pi}{2}\right) = -\cos x \). This comes from the identity \( \sin(x - \frac{\pi}{2}) = \cos(x) \) reflecting around the x-axis. Thus, \( \sin(x - \frac{\pi}{2}) = -\cos x \).

Step 4: Substitute and Simplify

Substitute \( \sin\left(x - \frac{\pi}{2}\right) \) in the reciprocal function to get \( \csc\left(x - \frac{\pi}{2}\right) = \frac{1}{-\cos x} = -\sec x \).

Step 5: Compare Both Sides

Compare the expression \(-\sec x\) with \(\sec x\). Since \(-\sec x eq \sec x\) for non-zero angles, the expressions are not equal under normal conditions.

Key Concepts

Cosecant Function

The cosecant function is one of the six fundamental trigonometric functions. It is represented as \( \csc \theta \).
This function is the reciprocal of the sine function, which can be expressed by the formula:

• \( \csc \theta = \frac{1}{\sin \theta} \).

Observing the unit circle can help one understand the cosecant function better. In a right-angled triangle, the sine of an angle is the ratio of the opposite side to the hypotenuse. Therefore, the cosecant, being the reciprocal, relates the hypotenuse to the opposite side.
The cosecant function is undefined when the sine of the angle is zero because division by zero is not possible.
This occurs at angles like \( \theta = n\pi \), where \( n \) is an integer. The graph of the cosecant function features vertical asymptotes at these multiples of \( \pi \), providing a unique characteristic where it diverges to infinity.

Secant Function

The secant function, denoted as \( \sec \theta \), is another essential trigonometric function and is the reciprocal of the cosine function. Its formula is:

• \( \sec \theta = \frac{1}{\cos \theta} \).

In the context of a right-angled triangle, the cosine of an angle represents the adjacent side over the hypotenuse. Therefore, the secant is the reciprocal of this ratio, indicating the hypotenuse over the adjacent side.
Similar to the cosecant function, the secant function is also undefined when its base function, cosine, equals zero. This occurs at \( \theta = \frac{(2n+1)\pi}{2} \), where \( n \) is any integer.
The secant function graph features vertical asymptotes at these points, which show discontinuities similar to the cosecant function. Insight into secant and its relationship with cosine helps in transformations and solving identities involving trigonometric functions.

Angle Transformation

Angle transformation refers to the manipulation of angles to achieve a desired trigonometric value.
One popular transformation is the use of angle subtraction identities. An important identity used in transformations is:

• \( \sin(x - \frac{\pi}{2}) = -\cos x \)

This handy identity is derived from the idea that subtracting \( \frac{\pi}{2} \) (or 90 degrees) from any angle results in a rotational transformation on the unit circle, essentially reflecting the function across the x-axis.
These transformations are vital in solving trigonometric identities, where understanding how angles relate to each other can simplify complex problems.
For example, in the original problem, applying angle transformation allowed conversion of \( \sin(x - \frac{\pi}{2}) \) into \(-\cos x \), showing that the original equation was false due to negative signs obtained through this transformation.