Problem 3 - Algebra and Trigonometry

Question

Find the exact value of each expression, if it is defined. \(\begin{array}{lll}{\text { (a) } \sin ^{-1} \frac{1}{2}} & {\text { (b) } \cos ^{-1}\left(-\frac{\sqrt{3}}{2}\right)} & {\text { (c) } \tan ^{-1}(-1)}\end{array}\)

Step-by-Step Solution

Answer

(a) \(\frac{\pi}{6}\), (b) \(\frac{5\pi}{6}\), (c) \(-\frac{\pi}{4}\)

Step 1: Understand the Problem

To find the exact value of the inverse trigonometric functions: (a) \(\sin^{-1}\left(\frac{1}{2}\right)\), (b) \(\cos^{-1}\left(-\frac{\sqrt{3}}{2}\right)\), and (c) \(\tan^{-1}(-1)\), we must identify angles whose trigonometric ratios correspond to the given values. These angles must be within the range of the respective inverse functions.

Step 2: Solve for \(\sin^{-1}\left(\frac{1}{2}\right)\)

The inverse sine function, \(\sin^{-1}\), outputs angles in the range \([-\frac{\pi}{2}, \frac{\pi}{2}]\). We need an angle \(\theta\) such that \(\sin(\theta) = \frac{1}{2}\). This occurs at \(\theta = \frac{\pi}{6}\) or 30 degrees.

Step 3: Solve for \(\cos^{-1}\left(-\frac{\sqrt{3}}{2}\right)\)

The inverse cosine function, \(\cos^{-1}\), outputs angles in the range \([0, \pi]\). We need an angle \(\theta\) such that \(\cos(\theta) = -\frac{\sqrt{3}}{2}\). This happens at \(\theta = \frac{5\pi}{6}\) or 150 degrees.

Step 4: Solve for \(\tan^{-1}(-1)\)

The inverse tangent function, \(\tan^{-1}\), outputs angles in the range \((-\frac{\pi}{2}, \frac{\pi}{2})\). We need an angle \(\theta\) such that \(\tan(\theta) = -1\). This occurs at \(\theta = -\frac{\pi}{4}\) or -45 degrees.

Key Concepts

Understanding Trigonometric Ratios

Trigonometric ratios are fundamental in understanding angles and triangles.When dealing with right-angled triangles:

• Sine ( \( ext{sin}\)): This ratio is the length of the opposite side divided by the hypotenuse.
• Cosine ( \( ext{cos}\)): This ratio is the length of the adjacent side divided by the hypotenuse.
• Tangent ( \( ext{tan}\)): This ratio is the length of the opposite side divided by the adjacent side.

Inverse trigonometric functions, such as \( ext{sin}^{-1}\), \( ext{cos}^{-1}\), and \( ext{tan}^{-1}\), help us find angles when these ratios are known.Each inverse function will provide us with an angle that corresponds to a specific trigonometric ratio.

Angle Measurement in Inverse Trigonometric Functions

Inverse trigonometric functions return angle measurements, typically in radians or degrees. The importance of degrees and radians is highlighted when the inverse functions are used. It's beneficial to grasp the conversions:

• 360 degrees = 2π radians
• 180 degrees = π radians
• 90 degrees = π/2 radians

For example, in our problem, knowing that \(rac{ ext{π}}{6}\) radians equals 30 degrees helps us understand precisely what angle we are dealing with.Inverse sine and cosine functions have limited ranges:

• \( ext{sin}^{-1}\) outputs angles in \([−\frac{\text{π}}{2},\frac{\text{π}}{2}]\) or [−90, 90] degrees.
• \( ext{cos}^{-1}\) outputs angles in \([0,\text{π}]\) or [0, 180] degrees.
• \( ext{tan}^{-1}\) outputs angles in \((-\frac{\text{π}}{2},\frac{\text{π}}{2})\) or [−90, 90) degrees.

Recognizing these ranges ensures that we obtain the correct angle from an inverse function.

Function Ranges in Inverse Trigonometry

The range of an inverse trigonometric function is the set of possible outputs, i.e., angles, it can produce. Each inverse function has its peculiar range due to the nature of the trigonometric ratio's relationship to the angle.
The inverse sine function, \( \sin^{-1} \), maps input ratios to angles within the range \([−\frac{\pi}{2},\frac{\pi}{2}]\). This means it finds angles only in the right half-circle, encompassing negative and positive outputs.
The \( \cos^{-1} \) function, conversely, returns angles within \([0, \pi]\). It covers the top half-circle, finding all relevant angles.Lastly, \( \tan^{-1} \) has the range \((-\frac{\pi}{2}, \frac{\pi}{2})\), covering half of the full circle range, with continuous values but excluding the boundaries.These ranges are set for convenience, ensuring that each input has one unique output, avoiding confusion over multiple possible angles that could have the same trigonometric ratio.