Problem 33 - Algebra and Trigonometry

Question

\(23-44=\) Find the exact value of the expression, if it is defined. \(\sin ^{-1}\left(\sin \left(\frac{5 \pi}{6}\right)\right)\)

Step-by-Step Solution

Answer

The expression evaluates to \(\frac{\pi}{6}\).

Step 1: Understanding the Inverse Sine Function

The inverse sine function, written as \(\sin^{-1}(x)\), returns the angle whose sine is \(x\). Its range is from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\).

Step 2: Find the Sine Value

Compute \(\sin\left(\frac{5\pi}{6}\right)\). The angle \(\frac{5\pi}{6}\) is equivalent to 150 degrees, which is in the second quadrant where sine is positive. \(\sin\left(\frac{5\pi}{6}\right) = \sin\left(\pi - \frac{\pi}{6}\right) = \sin\left(\frac{\pi}{6}\right) = \frac{1}{2}\).

Step 3: Applying Inverse Sine Function

Now, apply the inverse sine to the value found: \(\sin^{-1}\left(\frac{1}{2}\right)\). Since \(\sin^{-1}(x)\) will return an angle in \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\), and \(\sin\left(\frac{\pi}{6}\right)=\frac{1}{2}\), the function \(\sin^{-1}\left(\frac{1}{2}\right)\) returns \(\frac{\pi}{6}\).

Key Concepts

Trigonometric Functions

Trigonometric functions are essential mathematical functions that relate the angles and sides of a triangle. In particular, they are most commonly used in right-angled triangles but extend to a broader scope in the field of trigonometry, which studies relationships involving lengths and angles of triangles.

The primary trigonometric functions include sine (\(\sin\)), cosine (\(\cos\)), and tangent (\(\tan\)). Each function provides a ratio of two sides of a right triangle relative to an angle. For example:

• The sine of an angle is the ratio of the length of the triangle's opposite side to its hypotenuse.
• The cosine of an angle is the ratio of the adjacent side to the hypotenuse.
• The tangent of an angle is the ratio of the opposite side to the adjacent side.

These functions are easily represented on a unit circle where angles are measured, and side values are identified in a coordinate system.

Understanding these basic trigonometric functions is crucial as they form the foundation for more advanced topics in mathematics, including the study of inverse trigonometric functions.

Inverse Trigonometric Functions

Inverse trigonometric functions are the inverses of the standard trigonometric functions and are used to find angles when given a trigonometric ratio. The most common inverse functions are\(\sin^{-1}(x)\), \(\cos^{-1}(x)\), and \(\tan^{-1}(x)\), denoting the inverse of sine, cosine, and tangent, respectively.

The inverse sine function, \(\sin^{-1}(x)\), returns the angle that corresponds to a given sine value. By definition, this angle is found within the range of \([-\frac{\pi}{2}, \frac{\pi}{2}]\), covering the first and fourth quadrants of the unit circle. This limited range helps ensure each value has a unique angle, maintaining the function's inverse property.

In practical use, especially in exercises like the one provided, if you have \(\sin^{-1}(\sin(a))\), you're asking for the angle \(a\) itself, conditioned by the range of \(\sin^{-1}\). Ensuring that you understand how these ranges overlap and interact is pivotal in effectively using inverse trigonometric functions.

Angle Conversion

Angle conversion is the process of converting between different units of angle measurement, most commonly between degrees and radians. Radians and degrees are the two main units used to measure angles, with:

• Degrees being more familiar and easier to understand for general purposes.
• Radians being commonly used in higher mathematics and calculus due to their natural relation to arc lengths.

The relationship between degrees and radians is defined such that \(180^{\circ}\) corresponds to \(\pi\) radians. Therefore, you can convert degrees to radians by multiplying by \(\frac{\pi}{180}\), and convert radians to degrees by multiplying by \(\frac{180}{\pi}\).

In the provided exercise, knowing how to smoothly switch between radians and degrees allows you to correctly assess the angle \(\frac{5\pi}{6}\). This angle corresponds to 150 degrees, situated in the second quadrant where the sine function is positive, enabling you to simplify the exercise's computations effectively.