Problem 57 - Precalculus with Limits

Question

In Exercises 57 - 60, write the trigonometric expression as an algebraic expression. \( \sin\left(\arcsin x + \arccos x\right) \)

Step-by-Step Solution

Answer

The algebraic expression of \( \sin\left(\arcsin x + \arccos x\right) \) is 1.

Step 1: Recall trigonometric identity

Firstly, remember the fundamental trigonometric identity which states that the sum of \(\arcsin x\) and \(\arccos x\) is \( \pi/2 \) or 90 degrees.

Step 2: Substitute the trigonometric identity

Following the identity from step 1, replace \( \arcsin x + \arccos x \) with \( \pi/2 \). So our expression is now \( \sin(\pi/2) \).

Step 3: Evaluate the sin function

Finally, since the sine of \( \pi/2 \) or 90 degrees is 1, our algebraic expression is 1.

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the involved variables. One of the most well-known identities involves the sine and cosine functions. The sum of the angles for these inverse trigonometric functions, \( \arcsin x \) and \( \arccos x \), is a fundamental identity. Specifically, this identity tells us that:

• \( \arcsin x + \arccos x = \frac{\pi}{2} \) or 90 degrees.

This identity is crucial because it allows us to simplify complex expressions. By substituting \( \arcsin x + \arccos x \) with \( \pi/2 \), it helps in reducing expressions to a more manageable form. As a result, we are equipped to transform trigonometric expressions into simpler, often algebraic, ones.

Inverse Trigonometric Functions

Inverse trigonometric functions are the inverses of trigonometric functions like sine, cosine, and tangent. These functions help us determine the angle that corresponds to a given trigonometric value. For instance, \( \arcsin x \) represents the angle whose sine is \( x \), and \( \arccos x \) represents the angle whose cosine is \( x \).

• \( \arcsin x \) is defined for \( -1 \leq x \leq 1 \) and returns values in the range \(-\frac{\pi}{2} \) to \( \frac{\pi}{2} \).
• \( \arccos x \) is defined for \( -1 \leq x \leq 1 \) and returns values in the range \( 0 \) to \( \pi \).

These inverse functions are essential for solving trigonometric equations, especially when working backwards from known values to find the associated angles. They are commonly used in various fields such as physics, engineering, and computer graphics.

Evaluation of Trigonometric Functions

The evaluation of trigonometric functions involves finding the numerical value of functions like sin, cos, and tan given angles or expressions. For example, the sine of a right angle, \( \frac{\pi}{2} \), plays an important role in many calculations, particularly due to its value:

• The sine of \( \frac{\pi}{2} \) is equal to 1.

When evaluating the function \( \sin\left(\arcsin x + \arccos x\right) \) using identities, we substitute the expression with \( \sin(\frac{\pi}{2}) \). From our previous knowledge, this results in evaluating to 1. Understanding how to find these values is key in transforming and simplifying expressions into their algebraic forms, making complex trigonometric problems more approachable.