Question

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

Short answer

Question: Prove the trigonometric identity $$\cos 2\sin^{-1}x = 1 - 2x^2$$ and find the values of x for which the identity is true. Answer: The trigonometric identity is proven through the following steps: 1. Recalling the double angle formula for cosine and the definition of inverse sine. 2. Creating a right triangle to find the relation between angle and sides. 3. Finding the value of sin(2θ). 4. Applying the double angle formula for cosine. 5. Replacing the angles and simplifying. 6. Finding the values of x for which the identity is true. The identity is true for all values of x in the interval \([-1, 1]\).

Step-by-step solution

Recall the double angle formula for cosine and the definition of inverse sine

First, we need to recall the double angle formula for cosine, which is given by: $$\cos 2\alpha = 1 - 2\sin^2\alpha$$ And we should also recall the definition of the inverse sine function (also known as arcsin) for an angle: $$\sin^{-1}x = \alpha$$ That is, if the sine of angle \(\alpha\) is equal to x, then the inverse sine of x is equal to \(\alpha\).

Create a right triangle to find the relation between angle and sides

Let's create a right triangle with angle θ such that: $$\sin\theta = x$$ According to the definition of sine for a right triangle, we have: $$\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$$ Let's set the opposite side to a and the hypotenuse to b. Then we have: $$\frac{a}{b} = x$$ By the Pythagorean theorem, we can find the adjacent side c as follows: $$c = \sqrt{b^2 - a^2}$$

Find the value of sin(2θ)

Now we need to find the value of \(\sin (2\theta)\). We use the sine double angle formula: $$\sin 2\theta = 2\sin\theta\cos\theta$$ Using our triangle, we have: $$\sin 2\theta = 2\left(\frac{a}{b}\right)\left(\frac{c}{b}\right) = 2\frac{ac}{b^2}$$

Apply the double angle formula for cosine

Now, let's apply the double angle formula for cosine: $$\cos 2\theta = 1 - 2\sin^2\theta$$ Using the triangle, the sine of the angle θ is equal to x, so we have: $$\cos 2\theta = 1 - 2x^2$$

Replace the angles and simplify

Now we replace the angles and simplify. Since \(\sin ^{-1}x=\theta\), we have: $$\cos 2\theta = \cos 2\sin ^{-1} x$$ Now, using the relationship found in step 4, we have: $$\cos 2\sin ^{-1} x = 1 - 2x^2$$

Find the values of x for which the identity is true

For this identity to hold true, x must be in the domain of \(\sin^{-1}x\), which is \(-1 \leq x \leq 1\). Therefore, the given identity is true for all values of x in the interval \([-1, 1]\).

Key concepts

Double Angle Formula

The double angle formula is a powerful tool in trigonometry that simplifies expressions involving angles that are twice as large as a given angle. It provides a way to express trigonometric functions of double angles in terms of single angles, simplifying complex problems. In this exercise, we use the cosine double angle formula:\[ \cos 2\alpha = 1 - 2\sin^2\alpha \]
This formula allows us to switch between twined angles and regular trigonometric expressions. By understanding this transformation, you solve expressions more efficiently. It's especially useful in proving identities and solving equations involving trig functions.
Keep in mind that similar double angle formulas exist for sine, tangent, and other trigonometric functions, each with their own unique identity.

Inverse Trigonometric Functions

Inverse trigonometric functions, such as arcsin, arccos, and arctan, help us find angles when we know the value of the trigonometric function. The expression \[ \sin^{-1}x = \alpha \]
means that the angle \(\alpha\) is such that its sine value equals \(x\). These functions reverse the process of the regular trigonometric functions to obtain the angle from the ratio.

While regular trigonometric functions take angles and return values, inverse trig functions take these values and return angles. One thing to remember is that each inverse function has a specific defined range. For arcsin, this range is from \(-\pi/2\) to \(\pi/2\), allowing us to find precise angle measurements within these limits, ensuring the results are unique. This is essential for both solving triangle problems and understanding unit circle properties.

Pythagorean Theorem

The Pythagorean Theorem is a foundational principle of geometry, especially applicable to right-angled triangles. It states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. Mathematically: \[ c^2 = a^2 + b^2 \]
In this exercise, the theorem aids in finding the third side of a right triangle, given two sides. Suppose you know the lengths of the opposite and hypotenuse: \(a\) and \(b\) respectively. To find the adjacent side \(c\), rearrange to:\[ c = \sqrt{b^2 - a^2} \]
By applying this theorem, you can determine any missing side of a right triangle, crucial for further calculations involving trigonometric identities and solving real-world geometry problems like building design. Understanding this theorem bridges basic algebra and geometry with more complex trigonometric relations.

Right Triangle Trigonometry

Right triangle trigonometry is about understanding the relationships between the angles and sides of a right triangle. The primary trigonometric functions—sine, cosine, and tangent—are defined based on these relationships:

• \( \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} \)
• \( \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} \)
• \( \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \)

These definitions help translate geometric relations into trigonometric language, crucial for solving angles and lengths within a triangle. In the exercise, right triangle trigonometry assists in expressing the sine and cosine in terms of side lengths. Establishing these ratios allows you to apply identities and transformations. It's the foundation of solving many practical problems, such as calculating heights of buildings or lengths of shadows using angles of elevation. Basic right triangle concepts extend to larger concepts in trigonometry, like the unit circle and angle transformations.