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Chapter 6: Problem 62

Expand \(\cos (\alpha+\beta) .[5.2]\)

Short Answer

Expert verified
The expanded form of \(\cos (\alpha+\beta)\) according to the cosine addition formula, is \(\cos \alpha \cos \beta - \sin \alpha \sin \beta\).

Step by step solution

01

Recall the Addition Formula for Cosine

The first step is to remember the addition formula for cosine, which is given as: \(\cos (\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta\)
02

Apply the Addition Formula

Given \(\cos (\alpha+\beta)\), apply the addition formula: \(\cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta\)
03

Complete Expansion

After applying the addition formula, the final expanded form of \(\cos (\alpha+\beta)\) becomes: \(\cos \alpha \cos \beta - \sin \alpha \sin \beta\). This is the expanded form which was required in the expression.

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Key Concepts

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

Addition Formula for Cosine
The addition formula for cosine is a fundamental trigonometric identity. It allows us to express the cosine of the sum of two angles in terms of the cosine and sine of the individual angles. This identity is written as:\[\cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta\]This formula is incredibly useful in simplifying and evaluating trigonometric expressions. It breaks down a complex expression involving the cosine of a sum into simpler components that are easier to manipulate and understand.
  • \(\cos(\alpha)\) and \(\cos(\beta)\) are separate cosine functions.
  • \(\sin(\alpha)\) and \(\sin(\beta)\) are separate sine functions.
  • The subtraction sign in between shows how each pair combines the angle's sine and cosine values.
By mastering this formula, one can predict how different angles affect each other when added, which is particularly useful in fields such as physics and engineering.
Cosine Function
The cosine function is one of the primary functions in trigonometry. It is represented as \(\cos(\theta)\), where \(\theta\) is an angle. The cosine of an angle can be interpreted in a few different ways:
  • On the unit circle, \(\cos(\theta)\) represents the x-coordinate of a point on the circle corresponding to an angle \(\theta\).
  • In a right-angled triangle, \(\cos(\theta)\) is the ratio of the length of the adjacent side to the hypotenuse.
  • The cosine function is periodic with a period of \(2\pi\), meaning it repeats its values every \(2\pi\) radians.
Another aspect of the cosine function is its even nature, meaning \(\cos(-\theta) = \cos(\theta)\). This property is significant when solving trigonometric equations and simplifying expressions. The graph of the cosine function is a wave starting at 1 when \(\theta = 0\). Understanding these basics helps when dealing with more complex trigonometric identities such as the addition formula.
Sine Function
The sine function, another cornerstone of trigonometry, is typically expressed as \(\sin(\theta)\), with \(\theta\) representing an angle. Similar to cosine, it has several interpretations:
  • In the unit circle, \(\sin(\theta)\) corresponds to the y-coordinate of the point associated with the angle \(\theta\).
  • In a right triangle, it is the ratio of the opposite side to the hypotenuse, offering a straightforward geometric interpretation.
  • The sine function is also periodic with a period of \(2\pi\), which means it completes a full cycle every \(2\pi\) radians.
The sine function is odd, presented by the identity \(\sin(-\theta) = -\sin(\theta)\), which is crucial in simplifying expressions and solving problems. Starting at zero and peaking at 1 at \(\frac{\pi}{2}\), its graph complements the cosine graph. The sine function plays a key role in the addition formula for cosine, interrelated yet distinct from its cosine counterpart.

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Suppose \(a_{n}\) and \(b_{n}\) are two sequences such that \(a_{1}=4\) \(a_{n}=b_{n-1}+5\) and \(b_{1}=2, b_{n}=a_{n-1}+1 .\) Show that \(a_{n}\) and \(b_{n}\) are arithmetic sequences. Find \(a_{100}\)

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