Problem 13 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-22: Find the exact value of the expression whenever it is defined. (a) \(\sin \left[\cos ^{-1}\left(-\frac{1}{2}\right)\right]\) (b) \(\cos \left(\tan ^{-1} 1\right)\) (c) \(\tan \left[\sin ^{-1}(-1)\right]\)

Step-by-Step Solution

Answer

(a) \( \frac{\sqrt{3}}{2} \), (b) \( \frac{\sqrt{2}}{2} \), (c) Undefined.

Step 1: Understanding Inverse Trigonometric Functions

Let's first understand that \( \cos^{-1}(x) \), \( \sin^{-1}(x) \), and \( \tan^{-1}(x) \) are the inverse trigonometric functions, meaning they return the angle whose trigonometric function (cosine, sine, or tangent respectively) is \( x \). The output or range is commonly a specific quadrant: for \( \cos^{-1} \), the range is \([0, \pi]\); for \( \sin^{-1} \), the range is \([-\frac{\pi}{2}, \frac{\pi}{2}]\); and for \( \tan^{-1} \), the range is also \([-\frac{\pi}{2}, \frac{\pi}{2}]\).

Step 2: Calculate (a) Using Triangle Understanding

For \( \sin \left[ \cos^{-1} \left(-\frac{1}{2} \right) \right] \), we start by finding \( \theta = \cos^{-1} \left(-\frac{1}{2}\right) \), which means \( \cos(\theta) = -\frac{1}{2} \). The angle \( \theta \) satisfying this is \( \frac{2\pi}{3} \). Now, we need \( \sin(\theta) \) or \( \sin \left( \frac{2\pi}{3} \right) \), which equals \( \frac{\sqrt{3}}{2} \) because \( \sin\left( \frac{2\pi}{3} \right) = \sin \left(\pi - \frac{\pi}{3}\right) = \sin \left(\frac{\pi}{3}\right) \).

Step 3: Calculate (b) Using Special Angles

For \( \cos \left( \tan^{-1}(1) \right) \), find \( \theta = \tan^{-1}(1) \). This means \( \tan(\theta) = 1 \), implying \( \theta = \frac{\pi}{4} \). Then, \( \cos \left( \frac{\pi}{4} \right) \) is \( \frac{\sqrt{2}}{2} \).

Step 4: Calculate (c) Using Inverse Sine

For \( \tan \left[ \sin^{-1}(-1) \right] \), let \( \theta = \sin^{-1}(-1) \). This means \( \sin(\theta) = -1 \), so \( \theta = -\frac{\pi}{2} \). Then, calculate \( \tan \left(-\frac{\pi}{2}\right) \). Since tangent is not defined at odd multiples of \( \frac{\pi}{2} \), \( \tan \left(-\frac{\pi}{2}\right) \) is undefined.

Key Concepts

Trigonometric Functions

Trigonometric functions are fundamental in mathematics, forming the basis for much of trigonometry and its applications. These functions, namely sine (\( \sin \) ), cosine (\( \cos \) ), and tangent (\( \tan \) ), relate the angles of a triangle to the lengths of its sides. Understanding these functions is crucial:

• The sine function (\( \sin \) ) measures the ratio of the opposite side over the hypotenuse in a right-angled triangle.
• The cosine function (\( \cos \) ) measures the adjacent side over the hypotenuse.
• The tangent function (\( \tan \) ) represents the opposite side over the adjacent side.

These functions are periodic, with specific patterns repeating over their domains, which means they oscillate between a set maximum and minimum value as the angle changes. This cyclic nature forms the foundation of their applications in various fields.

Special Angles

Special angles refer to common angle measures, such as \( 0, \ \frac{\pi}{6}, \ \frac{\pi}{4}, \ \frac{\pi}{3}, \ \frac{\pi}{2} \) (radicals often like degrees: \( 0^\circ, 30^\circ, 45^\circ, 60^\circ, 90^\circ \)). These angles are 'special' because their trigonometric function values can be determined without a calculator. With special angles, it’s easier:

• For example, \( \sin \left( \frac{\pi}{4} \right) = \frac{\sqrt{2}}{2} \) and \( \cos \left( \frac{\pi}{4} \right) = \frac{\sqrt{2}}{2} \) .
• These values come from the natural symmetry in unit circles and often show up in simplified forms.

Memorizing these values can simplify calculations involving trigonometric functions and pave the way for tackling more complex problems. It is immensely beneficial in solving problems that involve evaluating inverse trigonometric functions.

Range of Functions

When discussing the range of functions, especially in trigonometry, it's crucial to understand the possible output values. This is particularly true for inverse trigonometric functions, which are designed to return angle measures. The inverse functions each have specific ranges:

• The inverse cosine (\( \cos^{-1} \) ) function gives outputs in the range \( [0, \pi] \) .
• The inverse sine (\( \sin^{-1} \) ) ranges between \( [-\frac{\pi}{2}, \frac{\pi}{2}] \) .
• Similarly, the range for inverse tangent (\( \tan^{-1} \) ) is also \( [-\frac{\pi}{2}, \frac{\pi}{2}] \) .

These ranges guide us on what angle values these functions can provide based on given input. If you align the input with these specified ranges, you can accurately determine the corresponding angles, avoiding any ambiguity in trigonometric calculations.