Problem 37 - Algebra and Trigonometry

Question

35-40 Find \(\sin \frac{x}{2}, \cos \frac{x}{2},\) and \(\tan \frac{x}{2}\) from the given information. \(\csc x=3, \quad 90^{\circ} < x < 180^{\circ}\)

Step-by-Step Solution

Answer

\( \sin \frac{x}{2} = \sqrt{\frac{3 + 2\sqrt{2}}{6}}, \cos \frac{x}{2} = \sqrt{\frac{3 - 2\sqrt{2}}{6}}, \tan \frac{x}{2} = 3 + 2\sqrt{2} \).

Step 1: Identify \\sin x using \\csc x

Given that \( \csc x = 3 \), we know \( \csc x \) is the reciprocal of \( \sin x \). Therefore, \( \sin x = \frac{1}{\csc x} = \frac{1}{3} \).

Step 2: Find \\cos x using \\sin x

Since \( x \) is in the second quadrant \( (90^\circ < x < 180^\circ) \), \( \cos x \) will be negative. Use the Pythagorean identity \( \sin^2 x + \cos^2 x = 1 \) to find \( \cos x \): \( (\frac{1}{3})^2 + \cos^2 x = 1 \), so \( \cos^2 x = \frac{8}{9} \). Hence, \( \cos x = -\frac{2\sqrt{2}}{3} \).

Step 3: Use Half-Angle Formulas to find \\sin \\frac{x}{2}

The half-angle formula for sine is \( \sin \frac{x}{2} = \sqrt{\frac{1 - \cos x}{2}} \). Substitute \( \cos x = -\frac{2\sqrt{2}}{3} \): \( \sin \frac{x}{2} = \sqrt{\frac{1 - (-\frac{2\sqrt{2}}{3})}{2}} = \sqrt{\frac{1 + \frac{2\sqrt{2}}{3}}{2}} \). Simplify to find \( \sin \frac{x}{2} = \sqrt{\frac{3 + 2\sqrt{2}}{6}} \).

Step 4: Use Half-Angle Formulas to find \\cos \\frac{x}{2}

The half-angle formula for cosine is \( \cos \frac{x}{2} = \sqrt{\frac{1 + \cos x}{2}} \). Substitute \( \cos x = -\frac{2\sqrt{2}}{3} \): \( \cos \frac{x}{2} = \sqrt{\frac{1 - \frac{2\sqrt{2}}{3}}{2}} \). Simplify to find \( \cos \frac{x}{2} = \sqrt{\frac{3 - 2\sqrt{2}}{6}} \).

Step 5: Use Half-Angle Formulas to find \\tan \\frac{x}{2}

The half-angle formula for tangent is \( \tan \frac{x}{2} = \frac{1 - \cos x}{\sin x} \). Use the values found: \( \tan \frac{x}{2} = \frac{1 + \frac{2\sqrt{2}}{3}}{\frac{1}{3}} = 3(1 + \frac{2\sqrt{2}}{3}) = 3 + 2\sqrt{2} \).

Key Concepts

Trigonometric Functions

Trigonometric functions are fundamental mathematical functions that relate the angles of a triangle to the lengths of its sides. They are widely used in geometry, physics, engineering, and many other fields. The primary trigonometric functions include sine (\( \sin x \)), cosine (\( \cos x \)), and tangent (\( \tan x \)), along with their reciprocals: cosecant (\( \csc x \)), secant (\( \sec x \)), and cotangent (\( \cot x \)).
These functions are periodic, meaning they repeat their values in regular intervals. They are defined based on a right triangle, where the angle is at one vertex, the opposite and adjacent sides define the lengths relative to the angle, and the hypotenuse is the longest side. Understanding these functions is crucial when solving problems involving right triangles, circular motion, and oscillatory systems.

• The sine function represents the ratio of the length of the opposite side to the hypotenuse.
• The cosine function represents the ratio of the length of the adjacent side to the hypotenuse.
• The tangent function is the ratio of the sine to the cosine of an angle.

Quadrant Analysis

Quadrant analysis involves determining the sign and value of trigonometric functions based on the angle's terminal side position on the coordinate plane. The coordinate plane is divided into four quadrants:

• Quadrant I: angles from 0° to 90° (both sine and cosine are positive).
• Quadrant II: angles from 90° to 180° (sine is positive, cosine is negative).
• Quadrant III: angles from 180° to 270° (both sine and cosine are negative).
• Quadrant IV: angles from 270° to 360° (sine is negative, cosine is positive).

In the given problem, the angle lands in Quadrant II. This quadrant is defined by angles between 90° and 180°, where sine values remain positive, but cosine values turn negative. The trigonometric functions change their signs as they move from one quadrant to another. Quadrant analysis provides essential insights that guide solutions in problems involving trigonometric identities and their equations.
Correctly identifying which quadrant an angle falls in helps determine the signs of trigonometric functions, ensuring accurate calculations.

Pythagorean Identity

The Pythagorean identity is a prevalent and foundational formula within trigonometry, derived from the Pythagorean theorem in a right triangle. It expresses the intrinsic relationship between the sine and cosine functions. For any angle \( x \), the identity is given as follows:\[\sin^2 x + \cos^2 x = 1\]This equation is vital because it allows us to find a connection between the sine and cosine of an angle when one of the values is known.
For example, in this exercise, where \( \sin x \) is known as \( \frac{1}{3} \), you can manipulate the identity to find \( \cos x \) by rearranging:\[\cos^2 x = 1 - \sin^2 x\]Substitute the known sine value:\[\cos^2 x = 1 - (\frac{1}{3})^2 = \frac{8}{9}\]Leading to \( \cos x = -\frac{2\sqrt{2}}{3} \), given that we are dealing with an angle in Quadrant II.
The Pythagorean identity is essential in simplifying expressions and solving trigonometric equations, especially when dealing with half-angle identities and finding precise values of trigonometric functions.