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Chapter 6: Problem 71

In Exercises \(71-76,\) find all the solutions of the equation. $$\sin t=1$$

Short Answer

Expert verified
Answer: The general solution for the equation \(\sin t = 1\) is \(t = \frac{\pi}{2} + 2 \pi n\), where \(n\) is an integer.

Step by step solution

01

Find the principal solution

The principal solution is the smallest positive angle that satisfies the equation. Since \(\sin t = 1\), we know that the angle must be in the first quadrant. Recall that the sine function is positive in the first and second quadrants. The smallest positive angle in the first quadrant whose sine equals 1 is \(t = \frac{\pi}{2}\).
02

Find all solutions in the first cycle

Recall that the sine function is periodic with a period of \(2 \pi\). Therefore, for every integer \(n\), the angles \(t = \frac{\pi}{2} + 2 \pi n\) will also have sine equal to 1. So the general solution of the equation \(\sin t = 1\) can be written as: $$t = \frac{\pi}{2} + 2 \pi n$$ where \(n\) is an integer. The equation has infinitely many solutions since \(n\) can be any integer.

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Key Concepts

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

Principal Solution
Identifying the principal solution is a fundamental step in solving trigonometric equations. The principal solution refers to the smallest positive angle for which the trigonometric function (in this case, the sine function) satisfies the given equation. It's like finding the starting point in a cyclic pattern before recognizing the entire loop.

For the equation \(\sin t = 1\), the principal solution is \(t = \frac{\pi}{2}\) since it is the smallest angle in radians at which the sine function reaches its maximum value of 1. This occurs at the peak of the unit circle, corresponding to a 90-degree angle if you are thinking in degrees. Understanding the concept of principal solutions is crucial, as it allows one to anchor additional solutions in a structured way, ensuring that no solutions within the defined range are missed.
Sine Function
The sine function is one of the primary trigonometric functions and is essential in describing periodic phenomena. It relates an angle of a right-angled triangle to the ratio of the length of the opposite side to the length of the hypotenuse. When picturing the sine function within the unit circle, it represents the y-coordinate of a point on the circle's circumference.

In the unit circle, the sine function's value ranges from -1 to 1, where \(\sin t = 0\) occurs at angles \(t = 0\) and \(t = \pi\), and the function's maximum, \(\sin t = 1\), occurs at \(t = \frac{\pi}{2}\). In dealing with trigonometric equations, recognizing the sine function's behavior can substantially simplify the process of finding solutions. It's also important to remember that the sine function is defined not only for acute angles but also for obtuse and reflex angles, expanding its range of application.
Periodicity of Trigonometric Functions
Trigonometric functions, including the sine function, exhibit periodicity, meaning they repeat their values in regular intervals along the angle measure. The curve of these functions repeats itself after a specific interval known as the 'period'. For the sine function, the period is \(2\pi\) radians or 360 degrees.

This periodic nature implies that once the principal solution is known, additional solutions can be found by adding or subtracting multiples of the period. In the context of our example \(\sin t = 1\), the general solution takes the form of \(t = \frac{\pi}{2} + 2\pi n\), where \(n\) is any integer, positive or negative. By grasping the concept of periodicity, students can solve trigonometric equations over any interval, not just within the first cycle, expanding their toolkit for mathematics and its many applications where such patterns are evident.

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Most popular questions from this chapter

Provide further examples of functions with different graphs, whose graphs appear identical in certain viewing windows. Approximating trigonometric functions by polynomials. For each odd positive integer \(n,\) let \(f_{n}\) be the function whose rule is $$ f_{n}(t)=t-\frac{t^{3}}{3 !}+\frac{t^{5}}{5 !}-\frac{t^{7}}{7 !}+\cdots-\frac{t^{n}}{n !} $$ since the signs alternate, the sign of the last term might be \+ instead of \(-,\) depending on what \(n\) is. Recall that \(n !\) is the product of all integers from 1 to \(n\); for instance, \(5 !=1 \cdot 2 \cdot 3 \cdot 4 \cdot 5=120\) (a) Graph \(f_{7}(t)\) and \(g(t)=\sin t\) on the same screen in a viewing window with \(-2 \pi \leq t \leq 2 \pi .\) For what values of \(t\) does \(f_{7}\) appear to be a good approximation of \(g ?\) (b) What is the smallest value of \(n\) for which the graphs of \(f_{n}\) and \(g\) appear to coincide in this window? In this case, determine how accurate the approximation is by finding \(f_{n}(2)\) and \(g(2)\)

Show that the given function is periodic with period less than \(2 \pi\). [Hint: Find a positive number \(k\) with \(k<2 \pi \text { such that } f(t+k)=f(t) \text { for every t in the domain of } f .]\) $$f(t)=\sin (\pi t)$$

Graph the function over the interval \([0,2 \pi)\) and determine the location of all local maxima and minima. [This can be done either graphically or algebraically.] $$g(t)=2 \sin (2 t / 3-\pi / 9)$$

In Exercises \(61-64\), use graphs to determine whether the equa. tion could possibly be an identity or is definitely not an identity. $$\frac{\sec t+\csc t}{1+\tan t}=\csc t$$

Graph the function. Does the function appear to be periodic? If so, what is the period? $$g(t)=|\sin t|$$
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