Problem 7 - Algebra 2

Question

Find each value. Write degree measures in radians. Round to the nearest hundredth. \(\sin \left(\sin ^{-1} \frac{3}{4}\right)\)

Step-by-Step Solution

Answer

\(\sin(\sin^{-1}(\frac{3}{4})) = 0.75\).

Step 1: Recognize the Inverse Sine Function

Understand that the expression \( \sin^{-1} \left( \frac{3}{4} \right) \) represents the angle whose sine value is \( \frac{3}{4} \). This function is also known as arcsine.

Step 2: Apply Sine on the Result

Since \( \sin^{-1} \left( \frac{3}{4} \right) \) gives the angle \( \theta \) such that \( \sin(\theta) = \frac{3}{4} \), applying sine to this angle simply returns the original sine value: \( \sin \left( \sin^{-1} \left( \frac{3}{4} \right) \right) = \frac{3}{4} \).

Step 3: Understanding Result is Already Evaluated

The problem essentially asks for the sine of the angle which has the sine value \( \frac{3}{4} \). The application of \( \sin \left( \sin^{-1} \left( \frac{3}{4} \right) \right) \) reverses each other, yielding the same value \( \frac{3}{4} \).

Step 4: Verify Problem Rounding Requirement

The value \( \frac{3}{4} \) is a fraction. As a decimal, it is \(0.75\), which to the nearest hundredth is already \(0.75\). No further rounding is needed.

Key Concepts

Sine Function

The sine function, often denoted as \( \sin \theta \), is one of the fundamental trigonometric functions. It is commonly used to relate the angles of a right triangle to the ratios of two of its sides. Specifically, the sine of an angle is the ratio of the length of the opposite side to the hypotenuse in a right triangle.

• Sine function is periodic with a period of \(2\pi\).
• Its range is from \(-1\) to \(1\), meaning that regardless of the angle, the sine value will be within these bounds.
• Common angles for sine are \(0^\circ\), \(30^\circ\), \(45^\circ\), \(60^\circ\), and \(90^\circ\); for example, \( \sin(30^\circ) = \frac{1}{2} \).

Understanding the sine function is crucial as it frequently appears in physics, engineering, and mathematics to solve various real-world problems.

Arcsine

Arcsine, represented as \(\sin^{-1} x\), is the inverse of the sine function. It aims to find the angle \(\theta\) whose sine is a given number \(x\).

• The range of arcsine is from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\), or equivalently from \(-90^\circ\) to \(+90^\circ\).
• Arcsine returns a principal value for the angle, meaning it chooses the result that is closest to zero on the number line.
• Since it's the inverse, when you take the sine of \(\sin^{-1} x\), it returns \(x\), provided \(x\) is within the valid range \([-1, 1]\).

In the context of the original exercise, \( \sin^{-1} \left( \frac{3}{4} \right) \) finds the angle whose sine is \( \frac{3}{4} \), and taking the sine of this angle simply yields \( \frac{3}{4} \) again.

Trigonometric Identities

Trigonometric identities are mathematical statements that relate different trigonometric functions to one another. These equations are true for all values of the involved angles or numbers, assuming they are within the valid ranges.

• One of the most essential identities is \( \sin^2 \theta + \cos^2 \theta = 1 \), known as the Pythagorean identity.
• Other important identities include angle addition and subtraction, double-angle formulas, and half-angle formulas.
• The identity used in the exercise is \(\sin(\sin^{-1}(x)) = x\), assuming \(x\) is within the interval \([-1, 1]\). This shows how sine and arcsine functions are inverses of each other.

Using trigonometric identities, we can solve complex mathematical problems and prove relationships within triangle geometry and wave functions.