Question

Find each value without using a calculator $$ \cos \left[2 \sin ^{-1}\left(-\frac{2}{3}\right)\right] $$

Short answer

The value is \( \frac{1}{9} \).

Step-by-step solution

Understand the Inverse Sine Function

The inverse sine function, \( \sin^{-1}(x) \), gives an angle \( \theta \) such that \( \sin(\theta) = x \). Therefore, if \( \sin^{-1}(-\frac{2}{3}) = \theta \), then \( \sin(\theta) = -\frac{2}{3} \). The range of \( \sin^{-1}(x) \) is \([ -\frac{\pi}{2}, \frac{\pi}{2} ]\), which means the angle is in the fourth or first quadrant.

Use the Double Angle Identity for Cosine

We need to find \( \cos(2\theta) \) because the expression to solve is \( \cos[2\sin^{-1}(-\frac{2}{3})] \). Using the cosine double angle identity, \( \cos(2\theta) = 1 - 2\sin^2(\theta) \).

Solve for \( \sin^2(\theta) \)

Since \( \sin(\theta) = -\frac{2}{3} \), we can calculate \( \sin^2(\theta) = \left(-\frac{2}{3}\right)^2 = \frac{4}{9} \).

Substitute in the Double Angle Identity

Now substitute \( \sin^2(\theta) = \frac{4}{9} \) into the double angle identity: \[ \cos(2\theta) = 1 - 2\left(\frac{4}{9}\right) = 1 - \frac{8}{9} = \frac{1}{9} \].

Conclude the Result

The value of the given expression \( \cos[2\sin^{-1}(-\frac{2}{3})] \) is \( \frac{1}{9} \).

Key concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are the keys to unlocking angles from given trigonometric values. For instance, the inverse sine function, denoted as \( \sin^{-1}(x) \), finds an angle \( \theta \) where \( \sin(\theta) = x \). This means the angle you find will fall in the range of \([-\frac{\pi}{2}, \frac{\pi}{2}]\). This range corresponds to angles located in the first and fourth quadrants of the unit circle.
Understanding this is vital when dealing with expressions such as \( \sin^{-1}(-\frac{2}{3}) \). Here, we seek an angle whose sine value is \(-\frac{2}{3}\), leaving us within the specified quadrant range.
These functions help convert trigonometric values into comprehensible angles, enabling further calculations and manipulations for solving geometrical and algebraic problems.

Double Angle Formula

The double angle formulas are very useful in trigonometry, simplifying complex expressions by relating them to single-angle calculations. For the cosine function, the double angle formula is \( \cos(2\theta) = 1 - 2\sin^2(\theta) \), which is derived from fundamental trigonometric identities.
This formula is crucial when faced with the task of finding double angle values such as \( \cos(2\theta) \), where direct calculation might be cumbersome. In the given problem, we use the double angle identity after determining \( \sin(\theta) \) as \(-\frac{2}{3}\).
Double angle formulas provide a shortcut, helping to break down complex expressions into manageable terms that utilize well-known trigonometric identities and aids in simplifying calculations.

Cosine Function

The cosine function relates the angle in a right triangle to the ratio of the adjacent side over the hypotenuse. In the unit circle context, cosine represents the horizontal coordinate of a point on the circle.
The double angle identity \( \cos(2\theta) = 1 - 2\sin^2(\theta) \) is derived using the properties of the cosine function in conjunction with trigonometric identities. This identity is particularly effective for reducing formulas involving \( 2\theta \) to a function of \( \theta \) itself.
In practical applications, cosine is essential for understanding periodic behaviors, modeling wave forms, and has numerous applications in physics and engineering that necessitate precision beyond simple arithmetic calculations.

Trigonometry

Trigonometry is a branch of mathematics that studies relationships involving lengths and angles of triangles. It forms the backbone of various mathematical concepts used to solve real-world problems across different scientific fields.
Understanding basic functions like sine, cosine, and tangent, along with their respective inverse functions, allows individuals to decipher complex geometric configurations. Trigonometric identities, such as the double angle formulas, are tools that enable simplification, transformation, and resolution of trigonometric expressions.
Through trigonometry, we can explore not only geometric aspects but also provide essential tools for calculus, signal processing, and even computer graphics. Mastery of these concepts provides critical analytical skills beneficial for both academic pursuits and practical applications.