Problem 3 In $3-8,$ for each value of \(... [FREE SOLUTION]

Chapter 12: Problem 3

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. $$ \theta=480^{\circ} $$

Short Answer

Expert verified

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 by step solution

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} $.

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.

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.

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.

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

91影视

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. $$ \theta=480^{\circ} $$

Short Answer

Step by step solution

Convert Angle to Radians or Reduce the Angle

Determine Values of trigonometric functions

Use Half-Angle Formula for sine

Use Half-Angle Formula for cosine

Use Half-Angle Formula for tangent

Key Concepts

Understanding Sine with Half-Angle Formulas

Delving into Cosine Using Half-Angle Formulas

Exploring Tangent Via Half-Angle Formulas

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