Problem 8 - Algebra 2 and Trigonometry

Question

In \(3-8,\) for each value of \(\theta,\) use half-angle formulas to find a. \(\sin \frac{1}{2} \theta\) b. \(\cos \frac{1}{2} \theta\) c. \(\tan \frac{1}{2} \theta .\) Show all work. $$ \theta=\frac{3 \pi}{2} $$

Step-by-Step Solution

Answer

\(\sin \frac{\theta}{2} = \frac{\sqrt{2}}{2}\), \(\cos \frac{\theta}{2} = \frac{\sqrt{2}}{2}\), \(\tan \frac{\theta}{2} = -1\).

Step 1: Identify the Problem

We need to solve for the half-angle values of sine, cosine, and tangent for \( \theta = \frac{3\pi}{2} \). We'll use the half-angle formulas to find these values.

Step 2: Recall the Half-Angle Formulas

The half-angle formulas we need are:- \( \sin \frac{\theta}{2} = \sqrt{\frac{1 - \cos \theta}{2}} \)- \( \cos \frac{\theta}{2} = \sqrt{\frac{1 + \cos \theta}{2}} \)- \( \tan \frac{\theta}{2} = \frac{\sin \theta}{1 + \cos \theta} \) or alternatively \( \tan \frac{\theta}{2} = \frac{1 - \cos \theta}{\sin \theta} \).

Step 3: Find \(\cos \theta\) and \(\sin \theta\)

For \( \theta = \frac{3\pi}{2} \), we know that:- \( \cos \frac{3\pi}{2} = 0 \)- \( \sin \frac{3\pi}{2} = -1 \).

Step 4: Solve for \(\sin \frac{\theta}{2}\)

Substitute \( \cos \theta = 0 \) into the sine half-angle formula:\[ \sin \frac{\theta}{2} = \sqrt{\frac{1 - 0}{2}} = \sqrt{\frac{1}{2}} = \frac{\sqrt{2}}{2} \].

Step 5: Solve for \(\cos \frac{\theta}{2}\)

Substitute \( \cos \theta = 0 \) into the cosine half-angle formula:\[ \cos \frac{\theta}{2} = \sqrt{\frac{1 + 0}{2}} = \sqrt{\frac{1}{2}} = \frac{\sqrt{2}}{2} \].

Step 6: Solve for \(\tan \frac{\theta}{2}\)

Substitute \( \sin \theta = -1 \) and \( \cos \theta = 0 \) into the tangent half-angle formula:\[ \tan \frac{\theta}{2} = \frac{-1}{1 + 0} = -1 \].

Key Concepts

Trigonometric Identities

Trigonometric identities are essential expressions that express the relationships between trigonometric functions. These identities are based on properties of triangles and angles. Understanding these identities can simplify complex trigonometric problems. In this exercise, we're particularly interested in half-angle identities. Half-angle identities help calculate the trigonometric function values of half an angle, denoted as \( \frac{\theta}{2} \).

These are particularly useful when you need to find functions of nondirect angles. The three main half-angle identities involved here are:

• \( \sin \frac{\theta}{2} = \sqrt{\frac{1 - \cos \theta}{2}} \)
• \( \cos \frac{\theta}{2} = \sqrt{\frac{1 + \cos \theta}{2}} \)
• \( \tan \frac{\theta}{2} = \frac{\sin \theta}{1 + \cos \theta} \)

These identities provide a way to express sine, cosine, and tangent functions of half the original angle in terms of the whole angle's function values.

Sine

The sine function relates a right triangle's opposite side to its hypotenuse. It is one of the primary trigonometric functions.

When using the half-angle formula for sine, \( \sin \frac{\theta}{2} = \sqrt{\frac{1 - \cos \theta}{2}} \), we find the sine of an angle half of \( \theta \). For example, if \( \theta = \frac{3\pi}{2} \), and we know \( \cos \theta = 0 \), substituting in the values gives:

• \( \sin \frac{\theta}{2} = \sqrt{\frac{1 - 0}{2}} = \sqrt{\frac{1}{2}} = \frac{\sqrt{2}}{2} \).

This calculation shows how understanding the relationship between cosine and sine can lead to solving half-angle problems.

Cosine

Cosine, like sine, is a fundamental trigonometric function. It represents the adjacent side over the hypotenuse in a right triangle.

When calculating half the angle's cosine, we utilize the half-angle formula: \( \cos \frac{\theta}{2} = \sqrt{\frac{1 + \cos \theta}{2}} \). With \( \theta = \frac{3\pi}{2} \) and given that \( \cos \theta = 0 \), substituting into the formula results in:

• \( \cos \frac{\theta}{2} = \sqrt{\frac{1 + 0}{2}} = \sqrt{\frac{1}{2}} = \frac{\sqrt{2}}{2} \).

This computation showcases how the cosine function's properties can determine the value of the angle at half the original angle.

Tangent

The tangent function, another key trigonometric function, is the ratio of the sine of an angle to the cosine of that angle. It can also be expressed as the opposite side over the adjacent side in a right triangle.

For the half-angle, the expression for tangent is \( \tan \frac{\theta}{2} = \frac{\sin \theta}{1 + \cos \theta} \). With \( \theta = \frac{3\pi}{2} \), knowing \( \sin \theta = -1 \) and \( \cos \theta = 0 \), we can find:

• \( \tan \frac{\theta}{2} = \frac{-1}{1 + 0} = -1 \).

This illustrates the reduction of complex expressions to manageable numbers and how trigonometric identities bring clarity to solving angular problems.