Problem 41 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the fundamental identities to simplify the expression. Use the table feature of a graphing utility to check your result numerically. $$\sin \left(\frac{\pi}{2}-x\right) \csc x$$

Step-by-Step Solution

Answer

The simplified form of the expression \( \sin \left(\frac{\pi}{2}-x\right) \csc x \) is \( \cot x \).

Step 1: Use of Co-function Identity

The co-function identity for sine is \( \sin(\frac{\pi}{2} - x) = \cos x \). Thus, the expression will change into: \( \cos x \csc x \)

Step 2: Express csc in terms of cos and sin

Recall that the cosecant \(csc x\) equals \( \frac{1}{\sin x}\), so the expression becomes \( \cos x * \frac{1}{\sin x} = \frac{\cos x}{\sin x} = \cot x\)

Step 3: Verification with Graphing Utility

Lastly, using a graphing calculator or tool to compare the graphs of \( \sin \left(\frac{\pi}{2}-x\right) \csc x \) and \( \cot x \). If all steps have been performed accurately, the two graphs should coincide.

Key Concepts

Fundamental Trigonometric Identities

Trigonometry is a branch of mathematics that deals with relationships between the sides and angles of triangles. Within its scope, there are several equalities known as the fundamental trigonometric identities. These identities are crucial for simplifying trigonometric expressions, solving trigonometric equations, and verifying the equality of trigonometric functions.

• The Pythagorean identity tells us that for any angle x, the equation \( \sin^2 x + \cos^2 x = 1 \) holds true.
• The reciprocal identities state that \( \csc x = \frac{1}{\sin x} \) (cosecant), \( \sec x = \frac{1}{\cos x} \) (secant), and \( \cot x = \frac{1}{\tan x} \) (cotangent) are the reciprocals of the sine, cosine, and tangent functions, respectively.
• The quotient identity indicates that \( \tan x = \frac{\sin x}{\cos x} \) and its reciprocal, \( \cot x = \frac{\cos x}{\sin x} \) are the ratios of sine to cosine and cosine to sine, respectively.

Using these identities is key to simplifying complex trigonometric expressions into more manageable forms. In our exercise, we see the application of reciprocal identities to transform \(\csc x \) into \(\frac{1}{\sin x}\), which plays a pivotal role in simplifying the given expression.

Co-function Identities

The concept of co-function identities is grounded in the complementary relationships between certain pairs of trigonometric functions. In essence, the co-function identities illustrate that the sine of an angle is equal to the cosine of its complement (and vice versa), which applies to other trigonometric pairs as well.

• For the sine and cosine co-function, the identity is \( \sin(\frac{\pi}{2} - x) = \cos x \) and \( \cos(\frac{\pi}{2} - x) = \sin x \).
• For the tangent and cotangent co-function, the identity is \( \tan(\frac{\pi}{2} - x) = \cot x \) and \( \cot(\frac{\pi}{2} - x) = \tan x \).

In our exercise, the sine co-function identity is exploited in step 1 to convert \(\sin(\frac{\pi}{2} - x)\) into \(\cos x\). This transformation is essential to progress with the simplification process. Recognizing co-function identities not only enables the simplification of expressions but also deepens the understanding of the symmetrical properties of trigonometric functions.

Graphing Calculator Verification

Verification using a graphing calculator serves as a powerful tool in confirming the accuracy of our algebraic manipulations in trigonometry. By plotting the initial and the simplified trigonometric expressions, we can visually ascertain whether we simplified the expression correctly; overlapping graphs indicate the expressions are equivalent.

When using a graphing calculator for verification:

• Ensure that the calculator is in the correct mode (radians or degrees) corresponding to the problem.
• Input the original and simplified expressions into separate functions to plot their graphs.
• Observe if the graphs of the two functions are identical — this suggests that the trigonometric identities were applied correctly.

In our case, step 3 involves using a graphing calculator to compare \(\sin(\frac{\pi}{2} - x) \csc x\) with \(\cot x\). If both functions produce the same graph, it validates the simplification process covered in previous steps. This visual confirmation is incredibly satisfying and reaffirms the student's understanding of the trigonometric concepts involved.