Problem 35 - Algebra and Trigonometry

Question

Find the exact value of each expression, if possible. Do not use a calculator. $$ \sin ^{-1}\left(\sin \frac{5 \pi}{6}\right) $$

Step-by-Step Solution

Answer

The exact value of \(\sin ^{-1}\left(\sin \frac{5 \pi}{6}\right)\) is \( \frac{5 \pi}{6}\).

Step 1 Title

Identify the given expression: We are given \( \sin ^{-1}\left(\sin \frac{5 \pi}{6}\right)\). The inverse sine function \( \sin^{-1}(x) \) is defined for -1 ≤ x ≤ 1, where x is the sine of an angle in the interval from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\).

Step 2 Title

Relate the angle to the valid range of \(\sin^{-1}(x)\): The given argument of the \(\sin^{-1}(x)\) function is \(\sin \frac{5 \pi}{6}\). Because the angle \(\frac{5 \pi}{6}\) lies within the range \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\), \(\sin \frac{5 \pi}{6}\) is valid directly.

Step 3 Title

Substitute the sine function to the angle: Since the angle is valid, the inverse sine of the sine of an angle is the angle itself. Thus, \(\sin ^{-1}\left(\sin \frac{5 \pi}{6}\right)\) is simply equal to \( \frac{5 \pi}{6}\).

Key Concepts

Angle Measurement

Angles are a fundamental part of trigonometry and can be measured in different units, the most common being degrees and radians. Understanding angle measurement is crucial in solving trigonometric problems.
Degrees divide a circle into 360 equal parts, but in higher mathematics, radians are often preferred.
This is because they provide a more natural way of measuring angles.

• A full circle in radians is expressed as \(2\pi\).
• A right angle is \(\frac{\pi}{2}\) radians.

Grasping why we use radians, especially in calculus and trigonometry, makes many concepts easier to comprehend.

Sine Function

The sine function is one of the primary trigonometric functions that relates angles to ratios of sides in a right triangle. It’s typically given by \( \sin(\theta) = \frac{\text{opposite side}}{\text{hypotenuse}} \).
In the unit circle definition:

• The sine of an angle is the y-coordinate of the corresponding point on the unit circle.
• It ranges from -1 to 1 for all real numbers.

Understanding the sine function helps in situations like finding the height of an object or analyzing wave patterns.

Radian Measure

The radian measure is another way of counting the sizes of angles using the radius of a circle. In simpler terms, a radian is the angle created when the arc length is the same as the radius length.

• There are \(2\pi\) radians in one complete circle.
• \(\frac{\pi}{2}\) radians make up a right angle.

The decision to use radian measure comes from its many mathematical conveniences, particularly in calculus. It makes trigonometric integrals and derivatives easier to handle.

Inverse Sine

Inverse sine, denoted as \( \sin^{-1}(x) \), is the function that reverses what the sine function does. It finds the angle whose sine is a given number. The range for inverse sine is restricted to ensure that each output is unique.

• It is defined for \(-1 \leq x \leq 1\).
• The output angle is in the interval \([-\frac{\pi}{2}, \frac{\pi}{2}]\).

In the problem \( \sin^{-1}(\sin \frac{5 \pi}{6}) \), you apply this concept to find the real angle, which is \( \frac{5 \pi}{6} \), ensuring it's within the acceptable range.