Problem 117 - Algebra and Trigonometry Real Mathematics, Real People

Question

Prove the identity arcsin \(x+\arccos x=\frac{\pi}{2}\)

Step-by-Step Solution

Answer

The identity \(\arcsin(x) + \arccos(x) = \frac{\pi}{2}\) is proven true by applying the trigonometric identity for sum of sine and cosine functions and simplifying the equation.

Step 1: Write down the identity

First, write down the identity we want to prove, which is \(\arcsin (x) + \arccos(x) = \frac{\pi}{2}\).

Step 2: Take sine on both sides of the equation

Now, apply the sine function to both sides of the equation, yielding \(\sin(\arcsin (x) + \arccos(x)) = \sin(\frac{\pi}{2})\).

Step 3: Apply trigonometric identity

Apply the trigonometric identity for sine of sum of two angles on the left-hand side. This identity says that \(\sin(a+b) = \sin(a) \cdot \cos(b) + \cos(a) \cdot \sin(b)\). Substituting \(a=\arcsin(x)\) and \(b=\arccos(x)\) yields \(\sin(\arcsin(x)) \cdot \cos(\arccos(x)) + \cos(\arcsin(x)) \cdot \sin(\arccos(x)) = \sin(\frac{\pi}{2})\).

Step 4: Simplify the equation

\(\sin(\arcsin(x))\) is \(x\) and \(\cos(\arccos(x))\) is also \(x\) as they are inverse functions. We also know that \(\cos(\arcsin(x))\) is \(\sqrt{1-x^2}\) and \(\sin(\arccos(x))\) also equals to \(\sqrt{1-x^2}\). The equation simplifies to \(x^2 + (1-x^2) = \sin(\frac{\pi}{2})\). This simplifies further to \(x^2 + 1-x^2 = 1\).

Step 5: Prove the identity

The left-hand side simplifies to 1, which is equal to \(\sin(\frac{\pi}{2})\) on the right-hand side. Thus, the original identity is proven.

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions, such as \(\arcsin(x)\) and \(\arccos(x)\), are the inverse processes of the sine and cosine functions respectively. These functions are crucial in defining angles when we know the values of the sine or cosine of those angles.
Their main property is the ability to 'undo' the effect of a trigonometric function, allowing us to recover the original angle from its sine or cosine value.

• \(\arcsin(x)\) is the inverse function of sine, giving an angle whose sine is \(x\).
• \(\arccos(x)\) is the inverse function of cosine, giving an angle whose cosine is \(x\).

These functions are commonly found in solving trigonometric equations and proving identities involving angles.

Sine and Cosine Relationships

The trigonometric functions sine and cosine share an interconnected relationship. This relationship is evident in identities such as \(\sin^2(x) + \cos^2(x) = 1\). This particular identity illustrates how both sine and cosine values for the same angle interrelate to sum to unity.
Under the arcsin and arccos, the connections are as follows:

• \(\sin(\arcsin(x)) = x\)
• \(\cos(\arccos(x)) = x\)
• \(\cos(\arcsin(x)) = \sqrt{1-x^2}\)
• \(\sin(\arccos(x)) = \sqrt{1-x^2}\)

Understanding these relationships helps in converting between angles and their trigonometric values, a skill essential for problem-solving in trigonometry.

Proof Techniques

Proving identities in trigonometry often involves using algebraic manipulation along with known trigonometric identities. A common technique is to transform one side of an identity into the form of the other side through legitimate operations.
Steps typically include:

• Applying known trigonometric identities, like \(\sin(a+b) = \sin(a)\cos(b) + \cos(a)\sin(b)\).
• Simplifying expressions using operations such as addition, multiplication, or square roots.
• Ensuring both sides of the identity match after simplification.

These techniques can extend beyond proving and into verifying and simplifying expressions in various trigonometric problems.

Educational Algebra and Trigonometry

Algebra and trigonometry education encompasses learning how these two branches interact and support one another. The knowledge of algebra is extensively applied in trigonometry to manipulate equations and expressions.
Here's how they connect:

• Using algebraic formulas to re-arrange trigonometric identities for easier simplification.
• Employing algebraic operations like factoring and distributing to facilitate proving equations.
• Recognizing patterns common in both algebra and trigonometry to simplify problem-solving.

Mastery of both subjects is crucial to seamlessly solving complex mathematical problems and understanding the foundational concepts of geometry and calculus.