Problem 147 - Precalculus

Question

For the following exercises, evaluate the expression without using a calculator. Give the exact value. $$ \frac{\sin ^{-1}\left(\frac{1}{2}\right)-\cos ^{-1}\left(\frac{\sqrt{2}}{2}\right)+\sin ^{-1}\left(\frac{\sqrt{3}}{2}\right)-\cos ^{-1}(1)}{\cos ^{-1}\left(\frac{\sqrt{3}}{2}\right)-\sin ^{-1}\left(\frac{\sqrt{2}}{2}\right)+\cos ^{-1}\left(\frac{1}{2}\right)-\sin ^{-1}(0)} $$

Step-by-Step Solution

Answer

The expression evaluates to 1.

Step 1: Evaluate each inverse trigonometric function

1. \( \sin^{-1}\left(\frac{1}{2}\right) \) is the angle in the range \([-\frac{\pi}{2}, \frac{\pi}{2}]\) whose sine value is \(\frac{1}{2}\). This angle is \(\frac{\pi}{6}\).2. \( \cos^{-1}\left(\frac{\sqrt{2}}{2}\right) \) is the angle in the range \([0, \pi]\) whose cosine value is \(\frac{\sqrt{2}}{2}\). This angle is \(\frac{\pi}{4}\).3. \( \sin^{-1}\left(\frac{\sqrt{3}}{2}\right) \) is the angle in the range \([-\frac{\pi}{2}, \frac{\pi}{2}]\) whose sine value is \(\frac{\sqrt{3}}{2}\). This angle is \(\frac{\pi}{3}\).4. \( \cos^{-1}(1) \) is the angle in the range \([0, \pi]\) whose cosine value is \(1\). This angle is \(0\).

Step 2: Substitute values into the numerator

Now substitute the evaluated angles into the expression for the numerator:\[\frac{\pi}{6} - \frac{\pi}{4} + \frac{\pi}{3} - 0\].

Step 3: Simplify the numerator

To simplify, find a common denominator for the fractions. The least common multiple of \(6, 4, \) and \(3\) is \(12\):\[\frac{\pi}{6} = \frac{2\pi}{12}, \quad \frac{\pi}{4} = \frac{3\pi}{12}, \quad \frac{\pi}{3} = \frac{4\pi}{12}\].So, the simplified numerator is:\[\frac{2\pi}{12} - \frac{3\pi}{12} + \frac{4\pi}{12}\ = \frac{3\pi}{12}\ = \frac{\pi}{4}\].

Step 4: Evaluate denominator

First, let's find each inverse trigonometric function value for the denominator:1. \(\cos^{-1}\left(\frac{\sqrt{3}}{2}\right) = \frac{\pi}{6}\).2. \(\sin^{-1}\left(\frac{\sqrt{2}}{2}\right) = \frac{\pi}{4}\).3. \(\cos^{-1}\left(\frac{1}{2}\right) = \frac{\pi}{3}\).4. \(\sin^{-1}(0) = 0\).

Step 5: Substitute values into the denominator

Substitute the evaluated angles into the expression for the denominator:\[\frac{\pi}{6} - \frac{\pi}{4} + \frac{\pi}{3} - 0\].

Step 6: Simplify the denominator

Using common denominators, convert the fractions:\[\frac{\pi}{6} = \frac{2\pi}{12}, \quad \frac{\pi}{4} = \frac{3\pi}{12}, \quad \frac{\pi}{3} = \frac{4\pi}{12}\].Substituting these into the expression, we have:\[\frac{2\pi}{12} - \frac{3\pi}{12} + \frac{4\pi}{12}\ = \frac{3\pi}{12}\ = \frac{\pi}{4}\].

Step 7: Divide the simplified numerator by the simplified denominator

Now, divide the simplified numerator by the simplified denominator:\[\frac{\frac{\pi}{4}}{\frac{\pi}{4}} = 1.\]

Step 8: Result

The evaluated expression equals \(1\).

Key Concepts

Trigonometric Identities

Trigonometric identities relate the angles and sides of triangles in various forms. They form the foundation for understanding inverse trigonometric functions. An inverse trigonometric function helps us find the angle that corresponds to a given trigonometric value. For instance, \( \sin^{-1} \) and \( \cos^{-1} \) are used to find angles when the sine and cosine values are known.

Understanding these identities in terms of angles:

• \( \sin^{-1}\left(\frac{1}{2}\right) \) refers to an angle where the sine is \( \frac{1}{2} \). This means the angle is \( \frac{\pi}{6} \).
• \( \cos^{-1}\left(\frac{\sqrt{2}}{2}\right) \) points to an angle whose cosine is \( \frac{\sqrt{2}}{2} \), arriving at \( \frac{\pi}{4} \).

Inverse trigonometric functions are crucial for solving equations where the angles are unknown, allowing us to reconstruct angles from given values. This relationship proves essential in simplifying complex expressions in trigonometry.

Angle Measurement

Angles are measured in radians or degrees, with radians being more prevalent in calculus and advanced mathematics. One full rotation around a circle is \( 2\pi \) radians or 360 degrees. When evaluating inverse trigonometric functions, it is essential to understand the range of angles they can yield.

For example:

• The range of \( \sin^{-1} \) is \([-\frac{\pi}{2}, \frac{\pi}{2}] \), focusing on angles from negative half to positive half of the unit circle.
• The range for \( \cos^{-1} \) is \([0, \pi] \), providing angles from the top half of the unit circle.

Knowing these ranges helps predict the possible angles without a calculator, deducing values such as \( \sin^{-1}(0) = 0 \) and \( \cos^{-1}(1) = 0 \). Understanding how to measure and evaluate angles enhances our capacity to simplify and solve trigonometric problems.

Fraction Simplification

Fraction simplification in trigonometric computations often involves finding a common denominator for combining or comparing fractional expressions.

In this exercise, simplifying both the numerator and denominator required converting fractions of various denominators to a uniform base:

• \( \frac{\pi}{6} = \frac{2\pi}{12} \).
• \( \frac{\pi}{4} = \frac{3\pi}{12} \).
• \( \frac{\pi}{3} = \frac{4\pi}{12} \).

Once fractions are converted to have the same denominator, arithmetic operations like addition or subtraction become straightforward, as shown by:\[\frac{2\pi}{12} - \frac{3\pi}{12} + \frac{4\pi}{12} = \frac{3\pi}{12} = \frac{\pi}{4}. \]Simplifying fractions in trigonometric calculations not only makes solving the problem more manageable but also ensures results are in their simplest form.