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Chapter 3: Problem 1

Find the exact values of the following sums or differences. \(\pi+\frac{\pi}{3}\)

Short Answer

Expert verified
The exact value is \(\frac{4\text{蟺}}{3}\).

Step by step solution

01

Identify the terms

The problem provides the terms \(\text{蟺}\) and \(\frac{\text{蟺}}{3}\). Recognize that both terms are multiples of \(\text{蟺}\).
02

Express as Common Fraction

Rewrite \(\text{蟺}\) as \(\frac{3\text{蟺}}{3}\), so that both terms have a common denominator. This makes it easier to add the fractions.
03

Add the Fractions

Add \(\frac{3\text{蟺}}{3}\) to \(\frac{\text{蟺}}{3}\) by combining the numerators over the common denominator: \(\frac{3\text{蟺}}{3} + \frac{\text{蟺}}{3} = \frac{4\text{蟺}}{3}\)
04

Simplify the Expression

The final expression is already simplified. So, the exact value of the sum \(\text{蟺} + \frac{\text{蟺}}{3}\) is \(\frac{4\text{蟺}}{3}\).

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Key Concepts

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

trigonometric functions
In this exercise, understanding the role of \(\text{蟺}\) is crucial because \(\text{蟺}\) is a fundamental constant in trigonometry. Trigonometric functions such as sine, cosine, and tangent are often evaluated at angles that are multiples of \(\text{蟺}\) or \(\frac{\text{蟺}}{3}\). Recognize that angles can be measured in radians. One full rotation around a circle is equivalent to an angle of \(\text{2蟺}\) radians.
When we add or subtract multiples of \(\text{蟺}\), we are essentially working with angles on the unit circle, which is central to the study of trigonometry.
fractions
To solve trigonometric addition problems like \(\text{蟺} + \frac{\text{蟺}}{3}\), it is necessary to work with fractions. Identify the fractions in the problem and work towards getting a common denominator.

Steps to work with fractions:
  • Write each term in the problem with a common denominator
  • Combine the numerators
  • Simplify (if possible)
In our example, rewriting \(\text{蟺}\) as \(\frac{3\text{蟺}}{3}\) makes it easier to add it to \(\frac{\text{蟺}}{3}\). This common denominator allows us to combine the numerators, leading to \(\frac{4\text{蟺}}{3}\).
simplification
Simplification is the process of reducing an expression to its simplest form. It makes the problem easier to understand and solve.

Steps for simplification:
  • Combine like terms
  • Reduce fractions if necessary
In the final step of our exercise, we check if the expression \(\frac{4\text{蟺}}{3}\) can be simplified further. Since it is already in its simplest form, no further simplification is needed. Recognizing when an expression is simplified is essential to solving and understanding trigonometric problems efficiently.

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Most popular questions from this chapter

Prove that each equation is an identity. \(\cos 2 y=\frac{1-\tan ^{2} y}{1+\tan ^{2} y}\)

Determine the period, asymptotes, and range for the function \(y=2 \sec (x / 4)\)

Simplify \(\frac{1}{1+\sin (-x)}+\frac{1}{1+\sin (x)}\)

Match each expression with an equivalent expression from \((a)-(h)\) Do not use a calculator: a. \(\cos (0)\) b. \(-\cos \left(44^{\circ}\right)\) c. \(-\tan \left(44^{\circ}\right)\) d. \(\cot \left(\frac{5 \pi}{14}\right)\) e. \(-\cos \left(46^{\circ}\right)\) f. \(\csc \left(\frac{\pi-2}{2}\right)\) g. \(\sin \left(46^{\circ}\right)\) h. \(\sin \left(44^{\circ}\right)\) $$ \tan \left(\frac{\pi}{7}\right) $$

Prove that each equation is an identity. \((\sin \alpha-\cos \alpha)^{2}=1-\sin 2 \alpha\)
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