Problem 7 - Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

\(\cos u\)

Step 1: Identifying Trigonometric Identity

The task requires to complete the trigonometric identity. The identity given is of the form \( \sin \left(\dfrac{\pi}{2} - u\right) \). This form is recognized as a standard co-function identity in trigonometry, where the sin of (90 degrees - angle) or in this case \( \dfrac{\pi}{2} - u \) is equal to cos of the same angle.

Step 2: Applying Trigonometric Identity

Applying the co-function identity to the given problem, the fill in the blank is \( \cos u \). Therefore, the completed trigonometric identity is \( \sin \left(\dfrac{\pi}{2} - u\right) = \cos u\).

Key Concepts

Trigonometric Identity

Trigonometric identities are like mathematical rules that relate different trigonometric functions to one another. These identities hold true for any angle and are crucial tools in simplifying complex expressions and solving trigonometry problems.
Some of the most common trigonometric identities include the Pythagorean identities, reciprocal identities, and co-function identities. Co-function identities, specifically, describe the relationship between complementary angles.
In this particular exercise, we are looking at a co-function identity:

• For any angle \( u \), the co-function identity states that \( \sin \left(\frac{\pi}{2} - u\right) = \cos u \).

These identities are especially helpful when you need to switch between sine and cosine functions to simplify mathematical expressions or solve equations. Understanding and remembering these basic identities gives you a solid foundation in trigonometry.

Sine Function

The sine function, often represented as \( \sin \theta \), is a fundamental concept in trigonometry. It measures the ratio of the length of the side of a right triangle opposite the angle to the length of the hypotenuse.
The sine of an angle in a right triangle is a very useful function in a variety of contexts, ranging from theoretical mathematics to practical applications such as physics and engineering.
Key properties of the sine function include:

• It's a periodic function, repeating its values every \( 2\pi \) radians or 360 degrees.
• Its range is between -1 and 1, inclusive.
• Sine is an odd function, meaning that \( \sin(-\theta) = -\sin \theta \).

Recognizing the sine function's patterns and relationships with other trigonometric functions, like in the co-function identity seen here, enhances your ability to solve problems involving angles.

Cosine Function

The cosine function, denoted by \( \cos \theta \), also plays a central role in trigonometry, similar to the sine function. It's defined as the ratio of the length of the adjacent side of a right triangle to the length of the hypotenuse.
This function helps in modeling a wide range of phenomena, from calculating the trajectory of objects to analyzing electrical circuits.
Here are a few important properties of the cosine function:

• It is periodic, like sine, with a period of \( 2\pi \) radians or 360 degrees.
• The cosine function is even, which means \( \cos(-\theta) = \cos \theta \).
• Its value ranges from -1 to 1.

In the exercise above, understanding the cosine function helps solve the co-function identity \( \sin \left(\frac{\pi}{2} - u\right) = \cos u \). Recognizing this equivalence simplifies the understanding and use of these functions in solving problems.