Problem 37 - Algebra and Trigonometry

Question

\(37-42\) Find \(\sin \frac{x}{2}, \cos \frac{x}{2},\) and \(\tan \frac{x}{2}\) from the given information. $$ \sin X=\frac{3}{5}, \quad 0^{\circ} < x < 90^{\circ} $$

Step-by-Step Solution

Answer

\(\sin\frac{x}{2} = \frac{\sqrt{10}}{10}, \cos\frac{x}{2} = \frac{3\sqrt{10}}{10}, \tan\frac{x}{2} = \frac{1}{3}\).

Step 1: Determine \\cos{x} Using the Pythagorean Identity

We know that \( \sin{x} = \frac{3}{5} \). To find \( \cos{x} \), use the identity \( \sin^2{x} + \cos^2{x} = 1 \). Substituting in the given values, we have \( \left(\frac{3}{5}\right)^2 + \cos^2{x} = 1 \). Thus, \( \frac{9}{25} + \cos^2{x} = 1 \). Solving for \( \cos^2{x} \) gives \( \cos^2{x} = \frac{16}{25} \), so \( \cos{x} = \frac{4}{5} \). Since \( 0^{\circ} < x < 90^{\circ} \), \( \cos{x} \) is positive.

Step 2: Find \\sin\\frac{x}{2} Using the Half-Angle Formula

The half-angle formula for sine is \( \sin\frac{x}{2} = \sqrt{\frac{1 - \cos{x}}{2}} \). Substituting \( \cos{x} = \frac{4}{5} \), \( \sin\frac{x}{2} = \sqrt{\frac{1 - \frac{4}{5}}{2}} = \sqrt{\frac{1}{10}} = \frac{1}{\sqrt{10}} \). Rationalizing the denominator, \( \sin\frac{x}{2} = \frac{\sqrt{10}}{10} \).

Step 3: Find \\cos\\frac{x}{2} Using the Half-Angle Formula

The half-angle formula for cosine is \( \cos\frac{x}{2} = \sqrt{\frac{1 + \cos{x}}{2}} \). Substituting \( \cos{x} = \frac{4}{5} \), \( \cos\frac{x}{2} = \sqrt{\frac{1 + \frac{4}{5}}{2}} = \sqrt{\frac{9}{10}} = \frac{3}{\sqrt{10}} \). Rationalizing the denominator, \( \cos\frac{x}{2} = \frac{3\sqrt{10}}{10} \).

Step 4: Find \\tan\\frac{x}{2} Using the Half-Angle Formula

The half-angle formula for tangent is \( \tan\frac{x}{2} = \frac{\sin{x}}{1 + \cos{x}} = \frac{1 - \cos{x}}{\sin{x}} \). Using the form \( \tan\frac{x}{2} = \frac{\sin{x}}{1 + \cos{x}} \), we have \( \tan\frac{x}{2} = \frac{\frac{3}{5}}{1 + \frac{4}{5}} = \frac{\frac{3}{5}}{\frac{9}{5}} = \frac{3}{9} = \frac{1}{3} \).

Key Concepts

Half-Angle Formulas

The half-angle formulas are extremely useful in trigonometry. They help compute the trigonometric functions of half-angles through more accessible expressions. For example, the half-angle formula for sine is \( \sin\frac{x}{2} = \sqrt{\frac{1 - \cos{x}}{2}} \).
The cosine half-angle formula is \( \cos\frac{x}{2} = \sqrt{\frac{1 + \cos{x}}{2}} \), and the tangent half-angle formula can take various forms, such as \( \tan\frac{x}{2} = \frac{1 - \cos{x}}{\sin{x}} \).

These formulas can simplify the process of finding sine, cosine, and tangent for angles like \( \frac{x}{2} \) as long as you know either \( \sin \,x \) or \( \cos \,x \).

• The half-angle formulas provide a direct way to calculate trigonometric functions for half-angles.
• They hinge upon the values of the sine and cosine of the original angle.

It is important to remember to rationalize the results, especially when dealing with square root terms, to make them simpler to use.

Pythagorean Identity

The Pythagorean identity is one of the cornerstones of trigonometry. It states that for any angle \(x\), \( \sin^2{x} + \cos^2{x} = 1 \).
This identity allows us to find one trigonometric function given the other. In this particular exercise, knowing \( \sin{x} = \frac{3}{5} \) enabled us to determine that \( \cos^2{x} = \frac{16}{25} \).

The steps involve:

• Substituting the known sine value into the identity.
• Rearranging the equation to solve for \( \cos^2{x} \).
• Taking the square root to determine \( \cos{x} \).

Always consider the quadrant of the angle to determine the sign of \( \cos{x} \). In the first quadrant, both sine and cosine are positive, making \( \cos{x} = \frac{4}{5} \) valid here.

Sine, Cosine, Tangent

Sine, cosine, and tangent, often abbreviated sin, cos, and tan, are the primary trigonometric functions. They are fundamental to understanding trigonometry and relate the angles of a triangle to the lengths of its sides.
For an angle \(x\), in a right-angled triangle:

• \( \sin{x} \) is the ratio of the length of the opposite side to the hypotenuse.
• \( \cos{x} \) is the ratio of the length of the adjacent side to the hypotenuse.
• \( \tan{x} \) is the ratio of the sine of the angle to its cosine \( \left(\tan{x} = \frac{\sin{x}}{\cos{x}}\right) \).

These definitions build a foundation for more complex formulas and identities. In the exercise, knowing \( \sin{x} \) and calculating \( \cos{x} \) allows for deriving \( \tan\frac{x}{2} \) using half-angle formulas.

Rationalizing the Denominator

Rationalizing the denominator is a technique used to eliminate radicals (like square roots) from the denominator of a fraction.
In trigonometry, it often happens in calculations involving half-angle formulas or square roots. To rationalize, you multiply both the numerator and the denominator by the conjugate of the denominator.
For instance, \( \sin\frac{x}{2} = \frac{1}{\sqrt{10}} \) becomes \( \frac{1}{\sqrt{10}} \times \frac{\sqrt{10}}{\sqrt{10}} = \frac{\sqrt{10}}{10} \).

Benefits of rationalizing include:

• Making expressions simpler and easier to interpret.
• Aiding in further calculations, especially those involving addition or subtraction of fractions.

In summary, rationalizing helps us present answers neatly and prepares them for more applied mathematical scenarios.