Problem 24 - Algebra 2

Question

Given \(\cos \theta=-\frac{15}{17}\) and \(180^{\circ}<\theta<270^{\circ}\) , find the exact value of each expression. $$ \cos \frac{\theta}{2} $$

Step-by-Step Solution

Answer

\(\cos{\theta/2} = -\sqrt{1/17}\)

Step 1: Identify the quadrant

In this exercise, the given angle \(\theta\) is between 180 and 270 degrees which lies in the third quadrant.

Step 2: Apply the half-angle formula

The formula to calculate \(\cos{\theta/2}\) is \(\sqrt{(1+\cos{\theta})/2}\). Plugging in the given value of \(\cos{\theta} = -15/17\), we get \(\cos{\theta/2} = \sqrt{(1 - 15/17)/2}\).

Step 3: Simplify the expression

Simplify the expression under the square root to get \(\cos{\theta/2} = \sqrt{(2/17)/2}\). Therefore, \(\cos{\theta/2} = \sqrt{1/17}\).

Step 4: Consider the sign of the answer

Because we're in the third quadrant (180-270), cosine function is negative, so our final answer should be \(-\sqrt{1/17}\).

Key Concepts

Half-Angle Formulas

Half-angle formulas are a type of trigonometric identity that helps us find the values of trigonometric functions for half angles, like \(\frac{\theta}{2}\). These formulas are especially useful when the original angle is known, but the corresponding half angle are not straightforward to compute.

The half-angle formula for cosine is given by:

• \[\cos\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 + \cos \theta}{2}}\]

This formula uses the basic value of \(\cos \theta\) to find \(\cos\left(\frac{\theta}{2}\right)\). The plusses and minuses represent that the result can be positive or negative depending on the quadrant.

Application of these formulas can significantly simplify complex trigonometric calculations, reducing them to simple algebraic operations.

Cosine Function

The cosine function, one of the fundamental trigonometric functions alongside sine and tangent, measures the adjacent side's length over the hypotenuse length in a right triangle. It is commonly represented as \(\cos(x)\).

Some significant properties of the cosine function include:

• It varies between \(-1\) and \(1\).
• It has a period of \(360^{\circ}\) (or \(2\pi\) radians).
• Cosine is an even function, thus symmetric about the y-axis, which implies \(\cos(-x) = \cos(x)\).

These attributes help us determine the cosine values for varied angles throughout the different quadrants, enhancing our understanding of the sign and magnitude for each quadrant.

Quadrants in Trigonometry

In trigonometry, the coordinate plane is divided into four quadrants to help determine the sign and value of trigonometric functions like sine, cosine, and tangent. This is crucial when working with angles greater than \(90^{\circ}\).

The four quadrants are:

• Quadrant I: where both sine and cosine are positive.
• Quadrant II: where sine is positive, while cosine is negative.
• Quadrant III: where both sine and cosine are negative.
• Quadrant IV: where sine is negative, while cosine is positive.

When \(\theta\) is between \(180^{\circ} \text{and} \270^{\circ}\), it falls into the third quadrant. Hence, both sine and cosine values, including their half-angle counterparts, are negative.

Trigonometric Expressions

Trigonometric expressions are mathematical expressions involving trigonometric functions like sine, cosine, tangent, and their reciprocals. These expressions can be simplified, evaluated, or manipulated using various trigonometric identities and formulas.

Some common techniques used with trigonometric expressions include:

• Using specific identities such as Pythagorean identities, sum and difference formulas, and double or half-angle formulas.
• Rationalizing trigonometric algebraic fractions.
• Changing expressions using reciprocal, quotient, and co-function identities.

By applying these techniques, particularly in exercises like finding \(\cos\left(\frac{\theta}{2}\right)\), we can solve problems accurately and efficiently, often simplifying intricate trigonometric equations into more manageable forms.