Problem 63 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the properties of inverse functions to find the exact value of the expression, if possible. \(\sin ^{-1}\left(\tan \frac{5 \pi}{4}\right)\)

Step-by-Step Solution

Answer

\(\sin ^{-1}\left(\tan \frac{5 \pi}{4}\right) = \frac{\pi}{2}\

Step 1: Calculate \(\tan \frac{5 \pi}{4}\)

The first thing you need to do is to calculate the value of \(\tan \frac{5 \pi}{4}\). Using the unit circle and recalling that tangent is defined as the ratio of the coordinates (y/x), you will find that \(\tan \frac{5 \pi}{4} = 1\). So \(\tan \frac{5 \pi}{4} = 1\).

Step 2: Recognize range of arcsine function

Now the task is divided into finding the value of \(\sin^{-1}(1)\). It's important to first ensure that '1' is within the range of the arcsine function. The range of \(\sin^{-1}(x)\) is [-1, 1]. Hence, '1' does lie within the range.

Step 3: Evaluate \(\sin^{-1}(1)\)

The last step is to evaluate \(\sin^{-1}(1)\). You need to find an angle such that the sin of that angle is 1. In the unit circle, that angle is \(\frac{\pi}{2}\). Hence, \(\sin^{-1}(1) = \frac{\pi}{2}\).

Key Concepts

Tangent Function

The tangent function, represented as \( \tan \theta \), is one of the three primary trigonometric functions. It is defined as the ratio of the

• sine function to the cosine function, \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)
• In terms of coordinates on the unit circle, this ratio can be expressed as \( \tan \theta = \frac{y}{x} \), where \( (x, y) \) are coordinates of a point on the circle corresponding to angle \( \theta \).

The tangent function is periodic with a period of \( \pi \), meaning it repeats its values every \( \pi \) radians. This behavior affects how we calculate angles. For example, if you find that \( \tan \theta = 1 \), the angle can be \( \frac{\pi}{4}, \frac{5\pi}{4}, \frac{9\pi}{4} \), and so on.
The function also has vertical asymptotes where \( \cos \theta = 0 \), leading to undefined values at those points. These occur at odd multiples of \( \frac{\pi}{2} \), such as \( \frac{\pi}{2}, \frac{3\pi}{2} \), etc.

Arcsine Function

The arcsine function, denoted as \( \sin^{-1}(x) \), is the inverse of the sine function. This function is used to determine the angle whose sine is a given value.

• The range of the arcsine function is the interval [-1, 1], and its principal value is within the interval \( \left[ -\frac{\pi}{2}, \frac{\pi}{2} \right] \).
• Therefore, \( \sin^{-1}(y) \) gives an angle \( \theta \) such that \( \sin \theta = y \) and \( -\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2} \).

When evaluating expressions like \( \sin^{-1}(1) \), you need to determine which angle within the determined range has a sine of 1. On the unit circle, this angle is \( \frac{\pi}{2} \). It's essential to consider the range to ensure the correct values are used.

Unit Circle

The unit circle is a fundamental concept in trigonometry and serves as a tool for understanding and solving trigonometric functions. A unit circle is a circle with a radius of 1 centered at the origin of a coordinate plane.

• Every point \((x, y)\) on the unit circle satisfies the equation \(x^2 + y^2 = 1\).
• The angle \(\theta\) in the unit circle is measured from the positive x-axis, in a counterclockwise direction.

When evaluating trigonometric functions, angles on the unit circle correspond to points whose coordinates are the values of the cosine and sine of the angle.

• For angle \(\frac{5 \pi}{4}\), the coordinates are \((\frac{-\sqrt{2}}{2}, \frac{-\sqrt{2}}{2})\) leading to \( \tan \theta = 1 \) because \( \frac{-\sqrt{2}/2}{-\sqrt{2}/2} = 1 \).
• Understanding the unit circle allows you to visualize the behavior of inverse trigonometric functions and compute exact values for various trigonometric expressions.