Problem 49 - Precalculus with Calculus Previews

Question

Use the given information to find (a) \(\cos (x / 2)\), (b) \(\sin (x / 2)\), and (c) \(\tan (x / 2)\). $$ \sec x=\frac{3}{2}, 0

Step-by-Step Solution

Answer

(a) \( \cos \frac{x}{2} = \sqrt{\frac{5}{6}} \); (b) \( \sin \frac{x}{2} = \sqrt{\frac{1}{6}} \); (c) \( \tan \frac{x}{2} = \sqrt{\frac{1}{5}} \).

Step 1: Determine Cosine of x

Given \( \sec x = \frac{3}{2} \), we know that \( \sec x = \frac{1}{\cos x} \). Therefore, \( \cos x = \frac{2}{3} \).

Step 2: Use Half-Angle Formula for Cosine

The half-angle identity for cosine is \( \cos\frac{x}{2} = \sqrt{\frac{1 + \cos x}{2}} \). Substitute \( \cos x = \frac{2}{3} \) into the formula: \( \cos\frac{x}{2} = \sqrt{\frac{1 + \frac{2}{3}}{2}} = \sqrt{\frac{5}{6}} \).

Step 3: Use Half-Angle Formula for Sine

The half-angle identity for sine is \( \sin\frac{x}{2} = \sqrt{\frac{1 - \cos x}{2}} \). Substitute \( \cos x = \frac{2}{3} \) into the formula: \( \sin\frac{x}{2} = \sqrt{\frac{1 - \frac{2}{3}}{2}} = \sqrt{\frac{1}{6}} \).

Step 4: Use Half-Angle Formula for Tangent

The half-angle identity for tangent is \( \tan\frac{x}{2} = \frac{\sin\frac{x}{2}}{\cos\frac{x}{2}} \). Substituting the values calculated: \( \tan\frac{x}{2} = \frac{\sqrt{\frac{1}{6}}}{\sqrt{\frac{5}{6}}} = \sqrt{\frac{1}{5}} \).

Key Concepts

Half-Angle Formulas

Half-angle formulas are key tools in trigonometry, used to find the trigonometric functions of half an angle. They are particularly useful when solving problems that involve angles divided by two, enhancing understanding of angle transformations. These formulas are derived from the double angle identities and are highlighted as follows:

• The half-angle formula for cosine is \( \cos\frac{x}{2} = \sqrt{\frac{1 + \cos x}{2}} \).
• For sine, it is \( \sin\frac{x}{2} = \sqrt{\frac{1 - \cos x}{2}} \).
• Finally, for tangent, the formula is \( \tan\frac{x}{2} = \frac{\sin\frac{x}{2}}{\cos\frac{x}{2}} \).

These formulas allow us to find the trigonometric values of angles that are otherwise difficult to compute. When using half-angle formulas, it is essential to consider the quadrant in which the angle lies, as this can affect the sign of the calculated trigonometric values.

Secant Function

The secant function, often denoted as \( \sec x \), is one of the six fundamental trigonometric functions. It is the reciprocal of the cosine function, defined as \( \sec x = \frac{1}{\cos x} \). This means that secant tells us how much larger the hypotenuse of a right triangle is compared to its adjacent side. Here are some important notes about the secant function:

• Secant is undefined when the cosine of the angle is zero, as division by zero is undefined.
• The secant function has a range of values from \([-\infty, -1] \cup [1, \infty]\).
• It is periodic with a period of \(360^\circ\) or \(2\pi\) radians.

Understanding secant helps in analyzing problems where direct cosine values are elusive, simplifying the approach by considering reciprocal identities.

Degree Measurement

Degree measurement in trigonometry refers to the method of measuring angles using degrees, where a full circle is divided into 360 equal parts or degrees. This system is widely used because it is simple to understand and apply. Key aspects of degree measurement include:

• One entire circle equals \(360^\circ\), and a right angle is \(90^\circ\).
• Degrees can be converted to radians for advanced calculations, where \(180^\circ = \pi\) radians.
• In the context of trigonometric functions, the sign of an angle's cosine, sine, and tangent changes depending on the angle's quadrant.

Grasping the concept of degree measurement is fundamental in precalculus and calculus, providing a basis for solving more complex trigonometric problems such as the one discussed in the exercise.

Precalculus

Precalculus is a foundational mathematical course that prepares students for calculus. It combines algebra and trigonometry to equip learners with essential skills needed for understanding limits, derivatives, and integrals. Key components of precalculus include:

• Trigonometric identities and functions, such as sine, cosine, tangent, and secant.
• Functions and their properties, including linear, polynomial, exponential, and logarithmic functions.
• Analyzing and graphing functions, illustrating transformations and translations.

Encountering concepts like half-angle formulas within precalculus provides a crucial stepping stone for understanding the behavior and properties of functions that will be further explored in calculus. Precalculus therefore acts as a bridge, easing the transition to more advanced mathematical studies.