Problem 3 - 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. $$ \theta=480^{\circ} $$

Step-by-Step Solution

Answer

For \( \theta = 480^{\circ} \), \( \sin \frac{1}{2}\theta = \frac{\sqrt{3}}{2} \), \( \cos \frac{1}{2}\theta = \frac{1}{2} \), \( \tan \frac{1}{2}\theta = \sqrt{3} \).

Step 1: Convert Angle to Radians or Reduce the Angle

The angle given is \( \theta = 480^{\circ} \). First, reduce it to a standard position by subtracting \( 360^{\circ} \), since full circle rotations don't change the trigonometric values. \[480^{\circ} - 360^{\circ} = 120^{\circ}.\] This means \( \theta = 120^{\circ} \).

Step 2: Determine Values of trigonometric functions

For \( \theta = 120^{\circ} \), we find:\[\sin 120^{\circ} = \frac{\sqrt{3}}{2}, \quad \cos 120^{\circ} = -\frac{1}{2}, \quad \tan 120^{\circ} = -\sqrt{3}.\] We will use these in half-angle formulas.

Step 3: Use Half-Angle Formula for sine

The half-angle formula for sine is:\[\sin \frac{\theta}{2} = \pm \sqrt{\frac{1 - \cos \theta}{2}}.\]Insert \( \cos 120^{\circ} = -\frac{1}{2} \):\[\sin \frac{120^{\circ}}{2} = \pm \sqrt{\frac{1 - \left(-\frac{1}{2}\right)}{2}} = \pm \sqrt{\frac{1 + \frac{1}{2}}{2}} = \pm \sqrt{\frac{3}{4}} = \pm \frac{\sqrt{3}}{2}.\]Since \( \frac{120^{\circ}}{2} = 60^{\circ} \) and \( \sin 60^{\circ} = \frac{\sqrt{3}}{2} \), we take the positive value.

Step 4: Use Half-Angle Formula for cosine

The half-angle formula for cosine is:\[\cos \frac{\theta}{2} = \pm \sqrt{\frac{1 + \cos \theta}{2}}.\]Insert \( \cos 120^{\circ} = -\frac{1}{2} \):\[\cos \frac{120^{\circ}}{2} = \pm \sqrt{\frac{1 + \left(-\frac{1}{2}\right)}{2}} = \pm \sqrt{\frac{1 - \frac{1}{2}}{2}} = \pm \sqrt{\frac{1}{4}} = \pm \frac{1}{2}.\]Since \( \cos 60^{\circ} = \frac{1}{2} \), we take the positive value.

Step 5: Use Half-Angle Formula for tangent

The half-angle formula for tangent is:\[\tan \frac{\theta}{2} = \pm \sqrt{\frac{1 - \cos \theta}{1 + \cos \theta}}.\]Insert \( \cos 120^{\circ} = -\frac{1}{2} \):\[\tan \frac{120^{\circ}}{2} = \pm \sqrt{\frac{1 - \left(-\frac{1}{2}\right)}{1 + \left(-\frac{1}{2}\right)}}= \pm \sqrt{\frac{1 + \frac{1}{2}}{1 - \frac{1}{2}}} = \pm \sqrt{\frac{3}{1}} = \pm \sqrt{3}.\]Since \( \tan 60^{\circ} = \sqrt{3} \), we take the positive value.

Key Concepts

Understanding Sine with Half-Angle Formulas

In trigonometry, sine functions are essential. They tell us about angles and sides in right triangles. Using half-angle formulas, we can find the sine of half an angle if we know the cosine of the original angle.
When we talk about the angle of \( \theta = 120^{\circ} \), we can use the half-angle formula for sine:

• Formula: \( \sin \frac{\theta}{2} = \pm \sqrt{\frac{1 - \cos \theta}{2}} \)
• Insert \( \cos 120^{\circ} = -\frac{1}{2} \) into the formula.
• Calculate: \( \sin \frac{120^{\circ}}{2} = \pm \sqrt{\frac{3}{4}} = \pm \frac{\sqrt{3}}{2} \)
• Recognize: Since \( \frac{120^{\circ}}{2} = 60^{\circ} \), and \( \sin 60^{\circ} = \frac{\sqrt{3}}{2} \), we choose the positive value.

With half-angle formulas, sine calculations become more manageable, especially with angles like 60 degrees, widely used in geometry and the unit circle.

Delving into Cosine Using Half-Angle Formulas

The cosine function helps us understand the relationship between angles and sides in a triangle through its x-coordinate on the unit circle. With the half-angle formula, working with cosine becomes straightforward.
For any angle such as \( \theta = 120^{\circ} \), the half-angle formula for cosine is:

• Formula: \( \cos \frac{\theta}{2} = \pm \sqrt{\frac{1 + \cos \theta}{2}} \)
• Insert \( \cos 120^{\circ} = -\frac{1}{2} \).
• Calculate: \( \cos \frac{120^{\circ}}{2} = \pm \sqrt{\frac{1}{4}} = \pm \frac{1}{2} \)
• Observation: Since \( \cos 60^{\circ} = \frac{1}{2} \), we opt for a positive value for ring familiarity.

Understanding cosine through half-angle formulas provides accuracy in calculating corners lessthan one full circle, showing that trigonometry operates beyond just integers or straightforward calculations.

Exploring Tangent Via Half-Angle Formulas

Tangent is a trigonometric function that describes the rate of change of a circle and is vital for understanding slopes and angles in geometry. With half-angle formulas, finding the tangent of half an angle becomes a simplified task.
When seeking the tangent for \( \theta = 120^{\circ} \), we use the tangent half-angle formula:

• Formula: \( \tan \frac{\theta}{2} = \pm \sqrt{\frac{1 - \cos \theta}{1 + \cos \theta}} \)
• Plugin \( \cos 120^{\circ} = -\frac{1}{2} \).
• Determine: \( \tan \frac{120^{\circ}}{2} = \pm \sqrt{3} \)
• Note: Given \( \tan 60^{\circ} = \sqrt{3} \), positive is chosen for alignment.

Tangent through half-angle formulas offers systematic ways to make accurate calculations when dealing with degrees and converting them into angles easier to understand in common situations.