Problem 4 - Algebra and Trigonometry

Question

Find the exact value of each expression. $$ \sin ^{-1} \frac{\sqrt{3}}{2} $$

Step-by-Step Solution

Answer

\(\sin^{-1} \frac{\sqrt{3}}{2} = \pi/3\)

Step 1: Understanding the problem

Understanding the problem means to recognize that \(\sin^{-1}\) means asking for which angle the sine is \(\frac{\sqrt{3}}{2}\). This is a value that should be known as a basic math fact from the unit circle, where all sine and cosine values should be familiar.

Step 2: Recall the unit circle

Recall from the unit circle in trigonometry, that the sine of an angle is equal to the y-coordinate of the point where the line through the origin making that angle with the positive x-axis intersects the unit circle. The number \(\sqrt{3}/2\) corresponds to angle \(\pi/3\) or \(60^\circ\) in the first quadrant where sine is positive.

Step 3: Find the exact value

From the unit circle, you then identify that the angle whose sine is \(\frac{\sqrt{3}}{2}\) is \(\pi/3\). So, \(\sin^{-1} \frac{\sqrt{3}}{2} = \pi/3\).

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry that helps us visualize and understand trigonometric functions. It is simply a circle with a radius of one, centered at the origin of a coordinate plane. The unit circle allows us to find the exact values of sine, cosine, and tangent for key angles.

When working with the unit circle, each point \(x, y\) on the circle represents the cosine and sine values for a given angle. The angle is measured from the positive x-axis in a counter-clockwise direction.
The unit circle is helpful because it provides us with a consistent reference for key reference angles like \(rac{\pi}{6}\), \(rac{\pi}{4}\), \(rac{\pi}{3}\), and others. These angles correspond to common ratios you frequently encounter with angles in trigonometry.

• The angle \(rac{\pi}{3}\) or 60° is located in the first quadrant, where all trigonometric functions like sine and cosine are positive.
• The sine value at this angle is the y-coordinate of the corresponding point on the unit circle, which is \(\frac{\sqrt{3}}{2}\).

Understanding the unit circle can greatly simplify the learning process for inverse trigonometric functions, as you can visually locate the angle for a given trigonometric function value.

Sine Function

The sine function, often written as \(\sin(\theta)\), represents the y-coordinate of a point on the unit circle corresponding to an angle \(\theta\). In trigonometry, the sine function is a periodic function, meaning it repeats its values in regular intervals.

Typically, the sine function is defined for all real numbers, but when using inverse trigonometric functions, we are often interested in angles between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\). This is known as the principal value range for the sine inverse.

• For \(\sin^{-1} \), we are asked to determine the angle whose sine value matches a given ratio. For example, \(\sin^{-1}(\frac{\sqrt{3}}{2})\) asks what angle has a sine of \(\frac{\sqrt{3}}{2}\).
• This value is well-known from the unit circle as \(\frac{\pi}{3}\), or 60°.

Though you can use a calculator for sine values, knowing basic unit circle values by heart is crucial. Memorizing these helps solve problems quickly and simplifies understanding inverse functions.

Trigonometric Identities

Trigonometric identities are equations that hold true for all values of the variable where the functions are defined. They are essential tools for simplifying and solving trigonometric equations.

One of the most fundamental trigonometric identities is the Pythagorean identity, which can be derived from the unit circle:\[ \sin^2(\theta) + \cos^2(\theta) = 1 \]
This identity shows the relationship between sine and cosine for any angle \(\theta\). Additionally, the Pythagorean identity helps confirm values, such as verifying the given problem:

• Given \(\sin(\theta) = \frac{\sqrt{3}}{2},\theta = \frac{\pi}{3}\), you can find that \(\cos(\theta) = \frac{1}{2}\).

Other useful identities include angle sum and difference identities, double angle identities, and others. But these identities primarily assist in transforming complex expressions in simpler trigonometric problems.

Understanding these identities will aid in mastering problems involving inverse trigonometric functions. These provide a foundation for deriving angles and values based on initial trigonometric information.