Problem 8 - Precalculus with Limits

Question

Fill in the blank to complete the trigonometric identity. \( \sec\left(\dfrac{\pi}{2} - u \right) \)= ________

Step-by-Step Solution

Answer

The equivalent trigonometric expression for \( \sec\left(\dfrac{\pi}{2} - u \right) \) is \( \dfrac{1}{\sin(u)} \)

Step 1: Recognize the trigonometric identity

Here, the function secant \( \sec \) is seen, which is the reciprocal of the cosine function. It's represented by \( \sec(u) = \dfrac{1}{\cos(u)} \)

Step 2: Apply the co-function identity

According to the co-function identity, \( \cos(\dfrac{\pi}{2} - u) = \sin(u) \). This is due to the complementary nature of sine and cosine functions.

Step 3: Substitute in the identity

Finally, substitute the co-function identity in the original identity. Since secant is the reciprocal of cosine, and cosine of \( \dfrac{\pi}{2} - u \) is equal to sine of u, the secant of \( \dfrac{\pi}{2} - u \) becomes the reciprocal of sine of u. Thus, \( \sec\left(\dfrac{\pi}{2} - u \right) = \dfrac{1}{\sin(u)} \)

Key Concepts

Co-function Identity

In trigonometry, the co-function identities are a set of useful relationships between trigonometric functions evaluated at complementary angles. Complementary angles are two angles that sum up to 90 degrees or \frac{\text{\text{\pi}}}{2} radians.

For instance, sine and cosine are co-functions. This can be observed in the identity \text{\( \cos(\frac{\pi}{2} - u) = \sin(u) \)}, where the angle \text{\( u \)} and its complement \text{\( \frac{\pi}{2} - u \)} create this important relationship. Similarly, other trigonometric functions have their co-function pairs: tangent and cotangent, secant and cosecant. Knowing these pairs helps us to simplify problems and evaluate trigonometric expressions more easily.

To apply a co-function identity effectively, remember that switching between sine and cosine, or between their reciprocals, involves using complementary angles. As we saw in the exercise, converting \text{\( \sec\left(\frac{\pi}{2} - u \right) \)} involved recognizing that secant is related to cosine and using the co-function relationship.

Reciprocal Trigonometric Functions

In trigonometry, each primary function has a reciprocal counterpart. These reciprocal relationships are foundational and are used frequently in solving trigonometric problems. For example, the secant function is the reciprocal of the cosine function, defined as \text{\( \sec(u) = \frac{1}{\cos(u)} \)}.

The three primary trigonometric functions—sine, cosine, and tangent—have reciprocals: cosecant, secant, and cotangent, respectively. Here’s the list of these correspondences:

• Sine (\text{\( \sin \)}) and Cosecant (\text{\( \csc \)})
• Cosine (\text{\( \cos \)}) and Secant (\text{\( \sec \)})
• Tangent (\text{\( \tan \)}) and Cotangent (\text{\( \cot \)})

Understanding and using these reciprocal identities is key in simplifying expressions and solving equations in trigonometry. As shown in the provided exercise, recognizing the reciprocal nature of functions enabled us to find the solution seamlessly.

Complementary Angles in Trigonometry

Complementary angles play a significant role in trigonometry, helping establish the fundamental co-function identities. These are pairs of angles that add up to 90 degrees or \text{\( \frac{\pi}{2} \)} radians. When dealing with trigonometric functions of complementary angles, we can use the concept that \text{\( \cos(\alpha) = \sin(\frac{\pi}{2}-\alpha) \)} and vice versa, where \(\alpha\) represents an angle.

This complementary relationship implies that the sine of one angle in a right triangle is equal to the cosine of the other non-right angle, and this property extends to all co-function pairs. Understanding complementary angles is crucial for solving right-angle problems and proves invaluable when working on more complex trigonometric equations or identities, as it allows us to transform functions into their co-function equivalents.