Question

Identity proofs Prove the following identities and give the values of x for which they are true. $$\sin \left(2 \sin ^{-1} x\right)=2 x \sqrt{1-x^{2}}$$

Short answer

Question: Prove the following trigonometric identity and determine the range of x values for which it holds true: \(\sin(2\sin^{-1}(x)) = 2x\sqrt{1-x^2}\). Answer: The given trigonometric identity is true and holds for values of x in the range \(-1 \le x \le 1\).

Step-by-step solution

Recall double-angle identity

The double-angle identity for sine is given by: $$\sin(2\theta) = 2\sin(\theta)\cos(\theta)$$

Apply double-angle identity to the given expression

In the given problem, we have \(\sin(2\sin^{-1}(x))\). We can use the identity as follows: $$\sin(2\sin^{-1}(x)) = 2\sin(\sin^{-1}(x))\cos(\sin^{-1}(x))$$

Simplify the left side of the identity

Since \(\sin(\sin^{-1}(x)) = x\), $$\sin(2\sin^{-1}(x)) = 2x\cos(\sin^{-1}(x))$$

Replace the cosine term with an equivalent expression

Using the Pythagorean identity, \(\sin^2\theta + \cos^2\theta = 1\), we can express cosine in terms of sine: $$\cos^2\theta = 1 - \sin^2\theta$$ Thus, \(\cos\theta = \sqrt{1 - \sin^2\theta}\) In our case, \(\theta = \sin^{-1}(x)\). Therefore, we have: $$\cos(\sin^{-1}(x)) = \sqrt{1-x^2}$$

Simplify the given identity

Now, we substitute the value of \(\cos(\sin^{-1}(x))\) in the equation from Step 3: $$\sin(2\sin^{-1}(x)) = 2x\sqrt{1-x^2}$$ This confirms that the given identity is true.

Determine the range of x values for the identity

The inverse sine function has a domain of \([-1, 1]\) and a range of \([-\frac{\pi}{2}, \frac{\pi}{2}]\). Thus, the valid values of x for this identity are: $$-1 \le x \le 1$$

Key concepts

Double-Angle Identity

Understanding double-angle identities is crucial when dealing with trigonometric expressions involving angles that are multiples of a basic angle. Specifically, the double-angle identity for sine states that \(\sin(2\theta) = 2\sin(\theta)\cos(\theta)\). It's derived from the sum of angles identity, which is itself a fundamental building block in trigonometry.

When you encounter an expression like \(\sin(2\sin^{-1}(x))\), the double-angle identity allows you to break it down into two parts—thus simplifying complex problems involving trigonometric functions. The key to applying this identity successfully is to match the given angle to the form of the double-angle identity, as was demonstrated in the exercise's solution. Pay careful attention to the way the original angle \(2\sin^{-1}(x)\) was deconstructed into \(2\sin(\sin^{-1}(x))\cos(\sin^{-1}(x))\).

However, remember that this identity only applies to sine and cosine functions, and there are different double-angle identities for tangent and other trigonometric functions. This reflects the versatility of trigonometric identities in simplifying and solving a wide range of mathematical problems.

Inverse Trigonometric Functions

Inverse trigonometric functions, sometimes called arc functions, are the inverses of the trigonometric functions, which allow you to find the angle that yields a specific trigonometric value. For example, \(\sin^{-1}(x)\), also known as arcsin(x), gives us the angle whose sine is x. These functions are central when we need to switch from an angle's trigonometric value back to the angle itself.

Domains and Ranges

The exercise highlighted the importance of the domain and range of these functions. For instance, the domain of \(\sin^{-1}(x)\) is \([-1, 1]\), meaning the function will only accept values within this range. Its range, or the possible output angles, is limited to \([-\frac{\pi}{2}, \frac{\pi}{2}]\), the principal values for arcsine.

When solving trigonometry problems, it's essential to ensure that the values you work with are within the function's domain and that the result falls within the expected range. Ignoring these constraints could lead to incorrect solutions or undefined results.

Pythagorean Identity

The Pythagorean identity is another cornerstone of trigonometry that arises from the Pythagorean theorem and the unit circle. It provides a relationship between the square of the sine and cosine of an angle: \(\sin^2\theta + \cos^2\theta = 1\). This identity is very useful in transforming expressions from sine to cosine or vice versa, as it was used in the exercise to express \(\cos(\sin^{-1}(x))\) in terms of x.

By rearranging the identity, \(\cos^2\theta = 1 - \sin^2\theta\), we can solve for one trigonometric function using the other, which is particularly helpful when an expression involves both sine and cosine. It's a valuable tool in simplifying trigonometric expressions and solving trigonometric equations.

The beauty of this identity lies in its ability to tie together the trigonometric functions, making it possible to transition between them fluidly, an approach that was effectively employed in proving the given identity in the exercise.