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Chapter 5: Problem 87

Use words to describe the formula for: the sine of half an angle.

Short Answer

Expert verified
The formula for the sine of half an angle is \( \sqrt{\frac{1 - \cos(\theta)}{2}} \). The operations involve taking the cosine of the given angle, subtracting it from 1, dividing by 2, and finally taking the square root.

Step by step solution

01

Define the half-angle formula for sine

The formula for sine of half an angle, denoted as \( \sin(\frac{\theta}{2}) \), is given by \( \sqrt{\frac{1 - \cos(\theta)}{2}} \).
02

Explain each part of the formula

In this formula, \( \theta \) represents the given angle, \( \cos(\theta) \) is the cosine of the given angle, and \(\sqrt{} \) is the square root function.
03

Describe the operations

The calculation follows these steps: take your given angle and calculate its cosine, subtract the result from 1, divide by 2, and finally, take the square root of the entire expression to get \( \sin(\frac{\theta}{2}) \).

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Key Concepts

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

Understanding Sine
The sine function is a fundamental concept in trigonometry that helps us understand angles and their relationships to triangles. When you hear someone talk about the sine of an angle, they are referring to the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle. This ratio varies as the angle changes.

A special scenario occurs when dealing with half-angle formulas. These are particularly useful for reducing the size of the angle when simplifying expressions. For the sine of half an angle, represented as \( \sin\left(\frac{\theta}{2}\right) \), the formula is \( \sqrt{\frac{1 - \cos(\theta)}{2}} \).

Key points about sine and its half-angle formula:
  • The value of sine changes as the angle increases or decreases.
  • The half-angle formula of sine helps manage more complex trigonometric calculations by breaking them into simpler parts.
  • Sine is always concerned with the opposite side and hypotenuse in geometry context.
Exploring Cosine
Cosine is another crucial trigonometric function alongside sine. It is defined as the ratio of the adjacent side to the hypotenuse in a right-angled triangle. This means that while sine deals with the opposite side, cosine focuses on the adjacent side.

Cosine is integral to the half-angle formula for sine. In the formula \( \sin\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 - \cos(\theta)}{2}} \), the term \( \cos(\theta) \) is central. Knowing the cosine of an angle helps us calculate the sine of its half.

Important insights about cosine:
  • The cosine value will help determine the sine of half an angle through our formula.
  • Understanding cosine is vital because it's used frequently with other trigonometric identities and formulas.
  • Just like sine, cosine values are crucial in various applications, such as solving triangle problems and modeling periodic phenomena.
Getting to Know Trigonometry
Trigonometry is not just about memorizing formulas but understanding relationships within triangles and angles. It鈥檚 the branch of mathematics that studies the relationships between the lengths and angles of triangles, encapsulated by functions like sine, cosine, and tangent.

In the context of trigonometry, half-angle formulas like the one for sine are essential in simplifying computations involving angles. They allow us to break down complex trigonometric functions into more manageable calculations.

Why trigonometry is important:
  • It provides tools to solve triangles, essential for physics, engineering, and even computer graphics.
  • Trigonometric functions describe periodic phenomena such as sound and light waves.
  • Understanding concepts like sine and cosine deeply connects you to practical uses across various scientific fields.
Developing a strong grasp of trigonometry can significantly enhance problem-solving skills in mathematical and real-life challenges.

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Most popular questions from this chapter

Use this information to solve Exercises \(129-130 .\) Our cycle of normal breathing takes place every 5 seconds. Velocity of air flow, y, measured in liters per second, after \(x\) seconds is modeled by $$ y=0.6 \sin \frac{2 \pi}{5} x $$ Velocity of air flow is positive when we inhale and negative when we exhale. Within each breathing cycle, when are we inhaling at a rate of 0.3 liter per second? Round to the nearest tenth of a second.

Determine the amplitude and period of \(y=3 \cos 2 \pi x\) Then graph the function for \(-4 \leq x \leq 4\) (Section 4.5, Example 5)

In Exercises \(121-126,\) solve each equation on the interval \([0,2 \pi)\) $$3 \cos ^{2} x-\sin x=\cos ^{2} x$$

Use the power-reducing formulas to rewrite \(\sin ^{6} x\) as an equivalent expression that does not contain powers of trigonometric functions greater than 1

Exercises \(166-168\) will help you prepare for the material covered in the first section of the next chapter. Solve each equation by using the cross- products principle to clear fractions from the proportion: $$ \text { If } \frac{a}{b}=\frac{c}{d}, \text { then } a d=b c .(b \neq 0 \text { and } d \neq 0) $$ Round to the nearest tenth. $$\text { Solve for } a: \frac{a}{\sin 46^{\circ}}=\frac{56}{\sin 63^{\circ}}$$
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