Problem 61 - Precalculus with Limits

Question

In Exercises 61 - 70, prove the identity. \( \sin \left(\dfrac{\pi}{2} - x\right) = \cos x \)

Step-by-Step Solution

Answer

The given identity is proven by applying the Co-Function Identity from Trigonometry, which states that \(\sin\left(\dfrac{\pi}{2} - x\right) = \cos x\).

Step 1: Set Up the Identity

We start with the left side of the equation: \(\sin\left(\dfrac{\pi}{2} - x\right)\). This is our starting point. Now the aim is to use the known identity to transform this expression into the form on the right side of the equation, which is \(\cos x\)

Step 2: Apply the Co-Function Identity

The Co-Function Identity says that \(\sin\left(\dfrac{\pi}{2} - x\right) = \cos x\). This directly proves our identity, as the left side of the equation is now equal to the right side of the equation.

Step 3: Finalize the Proof

Having applied the Co-Function Identity and transformed the left side into the right side, the proof of the identity is finished. Therefore, \(\sin\left(\dfrac{\pi}{2} - x\right)\) is indeed equal to \(\cos x\) as was to be shown.

Key Concepts

Co-Function Identity

The co-function identity is a fundamental concept in trigonometry that relates paired trigonometric functions. Specifically, it shows how the sine and cosine of complementary angles are related. An angle in trigonometric terms is often represented as a part of a right triangle, and complementary angles add up to 90 degrees or \( \frac{\pi}{2} \) radians.

For sine and cosine, the co-function identity is particularly interesting because it states that:

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

This means that the sine of an angle is equal to the cosine of its complement, and vice versa.

Using these identities can help in simplifying expressions and solving trigonometric equations. They're especially useful in calculus and other advanced mathematical topics where transformations of angles are required.

Sine

The sine function is one of the fundamental trigonometric functions. It relates an angle of a right triangle with the ratio of the length of the opposite side to the hypotenuse.

Mathematically, for an angle \(\theta\) in a right triangle:

• \( \sin\theta = \frac{\text{Opposite side}}{\text{Hypotenuse}} \)

Sine functions are periodic and have specific properties, such as:

• Its range is between -1 and 1 inclusive.
• It has a period of \(2\pi\), meaning it repeats every \(2\pi\) radians.

Knowing the sine of angles such as \(0, \frac{\pi}{4}, \frac{\pi}{2}\), and their respective sine values can make solving trigonometric equations easier.

Sine is also used in the realm of wave functions for physics and engineering, where it describes oscillating behaviors such as sound and light waves.

Cosine

The cosine function complements the sine function in trigonometry. It relates an angle in a right triangle with the ratio of the length of the adjacent side to the hypotenuse.

Mathematically, for an angle \(\theta\) in a right triangle:

• \( \cos\theta = \frac{\text{Adjacent side}}{\text{Hypotenuse}} \)

Cosine shares many characteristics with sine, such as being periodic:

• Its range is also between -1 and 1 inclusive.
• It has a period of \(2\pi\), meaning it repeats every \(2\pi\) radians.

The cosine function is closely related to the unit circle, a concept crucial for understanding angles in the coordinate system.

In the context of the co-function identity, the cosine function's significance becomes clear. It's incredibly useful in solving problems where converting between sine and cosine can simplify equations or proofs.