Question

Right-triangle relationships Draw a right triangle to simplify the given expressions. Assume \(x>0.\) $$\sin \left(2 \cos ^{-1} x\right)(\text { Hint: Use } \sin 2 \theta=2 \sin \theta \cos \theta$$

Short answer

Question: Simplify the expression \(\sin\left(2 \cos^{-1} x\right)\), assuming x is positive. Answer: $$\sin\left(2 \cos ^{-1} x\right) = 2x\sqrt{1-x^2}$$

Step-by-step solution

Create the right triangle and assign values

Since we are given \(\cos^{-1} x\), we'll create a right triangle with an angle \(\theta\) such that \(\cos\theta = x\). Label the adjacent side to angle \(\theta\) as `A`, the opposite side as `B`, and the hypotenuse as `H`. We have: - \(A = x\cdot H\) - \(B = \sqrt{H^2 - A^2}\) Notice that \(\cos\theta = \frac{A}{H}\), and \(\sin\theta = \frac{B}{H}\).

Find the value of \(\sin\left(2\cos^{-1} x\right)\) using the given hint

By using the trigonometric identity \(\sin(2\theta)=2\sin\theta\cos\theta\), we need to find expressions for \(\sin\theta\) and \(\cos\theta\). Since we already have \(\cos\theta=x\), we need to find out \(\sin\theta\). From step 1, we know that: - \(\sin\theta = \frac{B}{H} = \frac{\sqrt{H^2 - A^2}}{H} = \frac{\sqrt{H^2 - (x\cdot H)^2}}{H} = \frac{\sqrt{H^2 - x^2\cdot H^2}}{H} = \frac{\sqrt{(1-x^2)H^2}}{H}\)

Substitute in the trigonometric identity

Now we have expressions for both \(\sin\theta\) and \(\cos\theta\), we can substitute these into the trigonometric identity: \(\sin(2 \cos^{-1} x) = 2\sin\cos^{-1} x \cos\cos^{-1} x = 2 \cdot \frac{\sqrt{(1-x^2)H^2}}{H} \cdot x\)

Simplifying the expression

Now, we simplify the expression from step 3: \(\sin(2 \cos^{-1} x) = \frac{2x\sqrt{(1-x^2)H^2}}{H}\) Since we are asked to assume \(x > 0\) and \(H>0\), we can conclude that the final simplified expression for the given problem is: $$\sin\left(2 \cos ^{-1} x\right) = 2x\sqrt{1-x^2}$$

Key concepts

Right Triangle

A right triangle is a simple yet powerful geometric figure. It plays a crucial role in trigonometry, especially for simplifying expressions. A right triangle is characterized by having one angle equal to 90 degrees. The sides of a right triangle have specific roles: the longest side opposite the right angle is called the hypotenuse, while the other two sides are known as the legs.
To simplify trigonometric expressions using a right triangle, we often label the sides based on the trigonometric functions involved:

• The adjacent side is the side closest to the angle in question, excluding the hypotenuse.
• The opposite side is across from the angle.
• The hypotenuse is always the longest side.

Understanding these relationships is crucial when working with trigonometric identities and functions, as in the exercise provided. By creating a right triangle for the expression \( \cos^{-1} x \), you can easily assign values to the sides based on this function, leading to simplification of the problem.

Inverse Trigonometric Functions

Inverse trigonometric functions are used to determine the angle that corresponds to a given trigonometric ratio. In this exercise, the function \( \cos^{-1} x \) (also known as arccosine) is utilized. This function yields an angle \( \theta \) whose cosine is \( x \).
The inverse cosine function is particularly useful when dealing with angles and triangles because it allows us to do the following:

• Translate a given trigonometric ratio back into an angle.
• Facilitate the construction of right triangles for simplification processes.

In practical applications, once you find the angle using \( \cos^{-1} x \), you can apply other trigonometric identities and functions to solve complex problems. For example, knowing \( \theta \) allows us to find \( \sin \theta \) and use other identities like \( \sin(2\theta) = 2 \sin \theta \cos \theta \). This transforms complex expressions into simpler forms, making calculations more manageable.

Trigonometric Simplification

Trigonometric simplification involves rewriting complex trigonometric expressions into simpler forms using known identities. This is particularly useful when dealing with expressions involving angles, functions, or inverse functions.
In our exercise, simplification enables the conversion of the expression \( \sin(2\cos^{-1} x) \) into a form that's easier to evaluate. We achieve this by:

• Recognizing and using known identities, like \( \sin(2\theta) = 2 \sin \theta \cos \theta \), which are crucial for simplification.
• Substituting known values for \( \sin \theta \) and \( \cos \theta \) derived from the right triangle relationships.
• Reducing the expressions mathematically to express them in simplest terms: \( 2x\sqrt{1-x^2} \).

These strategies make trigonometric simplification a powerful technique in solving real-world problems where direct calculations may not be straightforward. It allows us to handle expressions of various complexities with ease and derive solutions that are insightful and useful.