Problem 110 - Algebra and Trigonometry

Question

Find the exact value of each expression. Do not use a calculator. $$ \sin ^{2}\left(\frac{1}{2} \cos ^{-1} \frac{3}{5}\right) $$

Step-by-Step Solution

Answer

The exact value of \( \sin ^{2}\left(\frac{1}{2} \cos ^{-1} \frac{3}{5}\right)\) is 0.2

Step 1: Understand the inverse cosine function

\(\cos^{-1}\) of a number gives the angle whose cosine is that number. This means that \(\cos^{-1} \frac{3}{5}\) is the angle whose cosine value is \(\frac{3}{5}\). We don't need to find the exact angle to solve the problem, because we need to find the sine of half of this angle.

Step 2: Use the half angle formula for sine squared

The half angle formula for sine-squared of an angle \( \alpha \) is \( \sin ^{2}(\alpha /2) = \frac{1 -\cos \alpha}{2}\). This allows us to calculate the sine-squared of half an angle if we know the cosine of that angle.

Step 3: Plug in the cosine of the angle

From step 1, we know that the cosine of the angle is \(\frac{3}{5}\). Plugging this into the formula from step 2 we get \( \sin ^{2}\left(\frac{1}{2} \cos ^{-1} \frac{3}{5}\right) = \frac{1-\frac{3}{5}}{2}\)

Step 4: Simplify the fraction

To simplify the fraction \( \frac{1-\frac{3}{5}}{2}\), first simplify the numerator to get \(\frac{2}{5}\). Therefore, \( \frac{2}{5} / 2 = \frac{1}{5} = 0.2\)

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are used to determine the angle that corresponds to a specific trigonometric value. These functions are the inverses of the sine, cosine, and tangent functions and are denoted as \(\sin^{-1}\), \(\cos^{-1}\), and \(\tan^{-1}\) or arcsin, arccos, and arctan, respectively. For instance, \(\cos^{-1} \left( \frac{3}{5} \right)\) finds the angle whose cosine is \(\frac{3}{5}\). It's important to note that these functions will return an angle within certain ranges, which are the principal values, to ensure the function gives a single value. For \(\cos^{-1}\), the range is from 0 to \(\pi\).

Understanding inverse trigonometric functions is crucial when dealing with problems where the angle needs to be determined from a known trigonometric value. So, always remember that the inverse trigonometric functions are about finding angles, not ratios.

Sine Squared

The term 'sine squared' is denoted as \(\sin^{2}(x)\) and represents the sine of an angle squared. It's a common expression in trigonometry, especially when dealing with trigonometric identities and equations. It is important to understand that \(\sin^{2}(x)\) is not the same as \(\sin(x^2)\), which would imply taking the sine of a squared angle. Instead, \(\sin^{2}(x)\) means taking the sine of an angle \((x)\) and then squaring the result.

This expression often appears in various trigonometric identities, including half-angle and double-angle formulas. You will frequently encounter it when simplifying trigonometric expressions or solving equations, so becoming comfortable with it is essential for mastering trigonometry.

Exact Trigonometric Values

Exact trigonometric values refer to the trigonometric ratios of special angles (like \(\frac{\pi}{6}\), \(\frac{\pi}{4}\), \(\frac{\pi}{3}\), and \(\frac{\pi}{2}\)) for which the sine, cosine, and tangent can be expressed as exact fractions, square root expressions, or integers, rather than decimal approximations. Learning these values is essential as they allow for the precise calculation of trigonometric functions without the need for a calculator.

These exact values are derived from the geometric properties of right-angled triangles and the unit circle. Knowing them well will help in solving trigonometric problems much more quickly and accurately, particularly when dealing with angles that are multiples or fractions of the special angles.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. These identities are used to simplify trigonometric expressions, to solve trigonometric equations, and to prove statements in trigonometry. There are several basic identities, such as the Pythagorean identity \((\sin^{2}(x) + \cos^{2}(x) = 1\)), and angle sum and difference identities. Among these, half-angle formulas, like the one used in our example exercise, allow us to express functions of half angles in terms of the whole angle.

For example, the half-angle formula for sine squared \((\sin^{2}(\alpha/2) = \frac{1 - \cos(\alpha)}{2}\)) helped us transform the given problem into a simpler expression that could be easily solved without actually determining the angle. Trigonometric identities are powerful tools in trigonometry; knowing them not only aids in solving problems but also deepens the understanding of how trigonometric functions behave and relate to one another.