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Chapter 4: Problem 38

Evaluate the trigonometric function using its period as an aid. $$ \sin \frac{9 \pi}{4} $$

Short Answer

Expert verified
\(\sin \frac{9 \pi}{4} = \frac{\sqrt{2}}{2}\)

Step by step solution

01

Recognizing period of sine function

The sine function, \(\sin(x)\), is a periodic function with a period of \(2\pi\). This means that every \(2\pi\) the function will repeat its values, or in other words, \(\sin(x + 2n\pi) = \sin(x)\), where \(n\) is any integer.
02

Calculation of an equivalent angle in range [0, 2蟺]

The angle \(\frac{9 \pi}{4}\) is more than \(2\pi\) so it is out of the primary period. An equivalent angle within the range [0, \(2\pi\)] can be found by subtracting \(2\pi\) (or multiples thereof) from the original angle until it falls within the desired range. In terms of quarters of Pi, \(2\pi\) is the same as \(8\pi/4\), so \(\frac{9 \pi}{4} - 2\pi = \frac{9 \pi}{4} - \frac{8 \pi}{4} = \frac{\pi}{4}\).
03

Evaluating sine of the calculated angle

Now the problem is reduced to the well known trigonometric value of \(\sin(\frac{\pi}{4})\). This value is well known and equals \(\frac{\sqrt{2}}{2}\).

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Key Concepts

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

Period of Sine Function
The sine function is one of the most fundamental concepts in trigonometry, repeating its pattern in a predictable cycle known as its period. The period of the sine function, denoted as \( 2\pi \), is the length of one complete cycle of the wave. In simpler terms, it's just like the circular track of a Ferris wheel where, after one full rotation, the seats return to their initial position.

Understanding this concept is crucial for evaluating trigonometric functions, especially when dealing with angles greater than \( 2\pi \) or less than \( -2\pi \). By leveraging the period, we can \

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Most popular questions from this chapter

Sketch the graph of the function. Include two full periods. $$ y=3 \csc 4 x $$

Use a graphing utility to graph the two equations in the same viewing window. Use the graphs to determine whether the expressions are equivalent. Verify the results algebraically. $$ y_{1}=\tan x \cot ^{2} x, \quad y_{2}=\cot x $$

Use the graph of the function to determine whether the function is even, odd, or neither. Verify your answer algebraically. $$ f(x)=\sec x $$

Use a graphing utility to graph the function and the damping factor of the function in the same viewing window. Describe the behavior of the function as \(x\) increases without bound. $$ f(x)=2^{-x / 4} \cos \pi x $$

Evaluate the expression without using a calculator. $$ \tan ^{-1} 0 $$
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