Problem 74 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 71-76: Verify the identity. $$ \arccos (-x)=\pi-\arccos x $$

Step-by-Step Solution

Answer

The identity \( \arccos(-x) = \pi - \arccos x \) is verified by using the cosine property \( \cos(\pi - \theta) = -\cos(\theta) \).

Step 1: Understand the Functions

We need to show that the expression \( \arccos(-x) = \pi - \arccos x \) is an identity. Recall the definition of \( \arccos(x) \), which means "the angle (or arc) whose cosine is \( x \)."

Step 2: Analyze the cosine property

Recall the property of cosine: \( \cos(\pi - \theta) = -\cos(\theta) \). This is significant for our identity because we are dealing with the arcsine (inverse cosine) function related to these angles.

Step 3: Express both sides as angles

Let \( \theta = \arccos x \). It follows that \( \cos(\theta) = x \). By the property of cosine, \( \cos(\pi - \theta) = -x \). This suggests that \( \arccos(-x) = \pi - \theta \).

Step 4: Equate the expressions

From Step 3, we obtained \( \arccos(-x) = \pi - \arccos(x) \) by using \( \theta = \arccos x \). This shows that the left-hand side \( \arccos(-x) \) equals the right-hand side \( \pi - \arccos x \), verifying the identity.

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are crucial tools in trigonometry that allow us to find angles when the values of trigonometric functions are known. For the cosine function, the inverse is denoted as \( \arccos(x) \). This function yields the angle whose cosine is \( x \).
It's important to remember that the range of \( \arccos(x) \) is from 0 to \( \pi \). This ensures that for any input within the range of -1 to 1, we get a unique angle value.
The identity given \( \arccos(-x)=\pi-\arccos x \) involves understanding how the inverse function operates with negative inputs. When solving problems involving inverse trigonometric functions, always ensure that the angle solutions are within their defined range.

Cosine Properties

Cosine is a fundamental trigonometric function, often illustrated in the context of right triangles or the unit circle. One significant property is that it is an even function, which means \( \cos(-\theta) = \cos(\theta) \).
This property helps simplify expressions and solve identities. Another key property related to the identity \( \arccos(-x)=\pi-\arccos x \) is that \( \cos(\pi - \theta) = -\cos(\theta) \). This symmetry explains how angles transform under certain operations and is fundamental in verifying the identity.
Understanding these cosine properties can make complex trigonometric proofs less daunting.

Angle Transformations

Angles play a central role in trigonometric functions and their transformations. In our identity, the transformation \( \pi - \theta \) illustrates how angles shift within the unit circle.
This particular transformation flips the angle about the y-axis, effectively taking the cosine value from positive to negative.
When you transform an angle using \( \pi - \theta \), you're leveraging the symmetry of the trigonometric circle, ensuring that properties of trigonometric functions are maintained across different quadrants. This transformation is key to solving our identity as it manipulates the angle \( \theta \) to demonstrate the equivalence of both sides of the equation.

Trigonometry Proofs

Proving trigonometric identities often involves algebraic manipulation, using known identities, and exploiting function properties. The identity \( \arccos(-x) = \pi - \arccos x \) is a great example.
The proof can be broken down into easy steps:

• Start by denoting \( \theta = \arccos x \), so \( \cos(\theta) = x \).
• Apply the cosine property: \( \cos(\pi - \theta) = -x \), meaning \( \pi - \theta \) is the angle for \( \arccos(-x) \).
• Conclude that since these calculations lead to the same output, the identity is verified.

This structured approach is often used in trigonometry to break down proofs into simpler steps, making it easier to verify complex identities.