Problem 63 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the exact values of \(\sin (u / 2), \cos (u / 2),\) and \(\tan (u / 2)\) using the half-angle formulas. $$\csc u=-\frac{5}{3}, \quad \pi

Step-by-Step Solution

Answer

The exact values are \( \sin (u / 2) = \sqrt{9/10}\), \( \cos (u / 2) = - \sqrt{1/10}\), and \( \tan (u / 2) = -3\).

Step 1: Understand What the Cosecant Function Tells Us

The cosecant function is the reciprocal of the sine function, so \(\sin u = -3/5\). The negative sign tells us that the angle \(u\) is in either the third or fourth quadrant, considering that sine is negative in those quadrants. Since we know from the problem that \(\pi < u < 3\pi / 2\), we can determine that \(u\) is in the third quadrant.

Step 2: Calculate the Cosine of the Angle

We don't have the cosine of the angle directly, but we can infer it from the sine using the Pythagorean identity \(\sin^2 u + \cos^2 u = 1\). Substituting the given sine value yields, \((-3/5)^2 + \cos^2 u = 1\), which simplifies to \(\cos^2 u = 1 - 9/25 = 16/25\), so \(\cos u\) can be \(\pm 4/5\). But since we're in the third quadrant where cosine is also negative, we have \(\cos u = -4/5\).

Step 3: Apply the Half-Angle Formulas for Sine and Cosine

Using the half-angle formulas \(\sin (u / 2) = \pm \sqrt{(1 - \cos u)/2}\) and \(\cos (u / 2) = \pm \sqrt{(1 + \cos u)/2}\). Substituting the value of \(\cos u = -4/5\) into these formulas, we get \(\sin (u / 2) = \pm \sqrt{(1 - (-4/5))/2} = \pm \sqrt{9/10}\) and \(\cos (u / 2) = \pm \sqrt{(1 + (-4/5))/2} = \pm \sqrt{1/10}\). We choose the signs based on the quadrant in which \(u/2\) lies. Since \(u\) is in the third quadrant, \(u/2\) will be in the second quadrant. Here, sine is positive and cosine is negative, hence \(\sin (u / 2) = \sqrt{9/10}\) and \(\cos (u / 2) = - \sqrt{1/10}\).

Step 4: Apply the Half-Angle Formulas for Tangent

The half-angle formula for the tangent is \(\tan (u / 2) = \sin (u / 2) / \cos (u / 2)\). Substituting \(\sin (u / 2) = \sqrt{9/10}\) and \(\cos(u / 2) = - \sqrt{1/10}\), we get \(\tan (u / 2) = \sqrt{9/10} / - \sqrt{1/10} = - \sqrt{9}\) which simplifies to \(\tan (u / 2) = -3\).

Key Concepts

Cosecant Function

The cosecant function, often denoted by csc, is one of the basic trigonometric functions. It is defined as the reciprocal of the sine function, meaning that \(csc(\theta) = \frac{1}{\sin(\theta)}\). The cosecant function tells us about the ratio of the length of the hypotenuse to the opposite side in a right-angled triangle.

When we encounter a problem involving the cosecant function, such as \(csc(u) = -\frac{5}{3}\), we're dealing with a scenario where this ratio is negative, which indicates that the angle in question is situated in a quadrant where the sine function is negative. Specifically, for the given range \(\pi < u < \frac{3\pi}{2}\), angle \( u\) resides in the third quadrant where both sine and cosine values are negative. Knowing this helps determine the signs of other trigonometric functions derived from \( u\).

• The cosecant is rarely used on its own; it's often more practical to consider the sine and work from there.
• Understanding the relationships between cosecant and other trigonometric functions is essential for solving trigonometric equations.
• The sign of csc(\theta) helps identify the quadrant in which the angle lies and guides the determination of other functions' signs.

Pythagorean Identity

The Pythagorean identity is a fundamental aspect of trigonometry, which relates the squares of the sine and cosine functions of an angle to the number 1. The basic form of the identity is expressed as \(\sin^2(\theta) + \cos^2(\theta) = 1\). This equation arises from the Pythagorean theorem applied to the unit circle.

To leverage this identity in solving trigonometric problems, one can use known values of either the sine or cosine functions to find the other. For example, given \(\sin u = -\frac{3}{5}\), the Pythagorean identity allows us to calculate \(\cos u\) by rearranging the equation to \(\cos^2 u = 1 - \sin^2 u\) and solving for \(\cos u\). In the context of half-angle formulas, this identity paves the way to find the exact values of \(\sin (u / 2)\) and \(\cos (u / 2)\), as seen in the given solution.

In practice, remember that:

• The Pythagorean identity is invaluable for verifying trigonometric identities and solving equations.
• It’s particularly useful when given limited information about an angle.
• The identity's application extends beyond right-angled triangles to any context involving sine and cosine, including waves and oscillations.

Trigonometric Functions

Trigonometric functions are a suite of functions that relate the angles of a triangle to the ratios of its sides. They play a crucial role in both pure and applied mathematics, with sine, cosine, and tangent being the three primary functions most commonly used.

These functions are defined for angles—which can be more than 90 degrees when considering their application to the unit circle. This means that they are valid not just for acute angles in right triangles, but for any angle:

• \textbf{Sine (sin)} represents the ratio of the opposite side to the hypotenuse.
• \textbf{Cosine (cos)} denotes the ratio of the adjacent side to the hypotenuse.
• \textbf{Tangent (tan)} is defined as the ratio of the sine to the cosine, or the opposite side to the adjacent side.

In the exercise provided, the half-angle formulas are used, which are derived from the primary functions to calculate values for \(\sin (u / 2)\), \(\cos (u / 2)\), and \(\tan (u / 2)\) given \(\sin u\) and \(\cos u\). This is crucial for solving problems that involve angles not directly given or where only partial information is provided. The ability to manipulate and understand these functions is key to grasping the landscape of trigonometry and its applications in a wide array of scientific fields, such as physics, engineering, and astronomy.