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Chapter 10: Problem 37

Find all of the angles which satisfy the given equation. $$ \sin (\theta)=-1 $$

Short Answer

Expert verified
The angles are \( \theta = \frac{3\pi}{2} + 2k\pi \), \( k \in \mathbb{Z} \).

Step by step solution

01

Understanding the Sine Function

The sine function, \( \sin(\theta) \), gives the y-coordinate of a unit circle at an angle \( \theta \). This means \( \sin(\theta) = -1 \) corresponds to a point on the unit circle where the y-coordinate is -1.
02

Identifying the Angle

The angle where \( \sin(\theta) = -1 \) occurs at the bottom of the unit circle. This angle is \( \theta = \frac{3\pi}{2} \) radians or \( 270^\circ \).
03

General Solution for Sine Function

Since the sine function has a period of \( 2\pi \), the general solution for \( \sin(\theta) = -1 \) can be written as \( \theta = \frac{3\pi}{2} + 2k\pi \), where \( k \) is any integer.

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Key Concepts

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

Sine Function
The sine function is a basic trigonometric function, usually expressed as \( \sin(\theta) \). It's a function that relates an angle \( \theta \) to the y-coordinate of a point on the unit circle. The sine function takes an angle as input and provides a value ranging from -1 to 1 as output. This value represents the vertical position (or y-coordinate) of a point on the unit circle.

When applying the sine function, it's crucial to understand that it tells us how far an angle opens up vertically. For angles where \( \sin(\theta) = -1 \), the point is located at the lowest position on the unit circle, meaning the y-coordinate is at its minimum possible value.
Unit Circle
The unit circle is a powerful concept in trigonometry. It is a circle with a radius of 1, centered at the origin (0,0) of a coordinate system.
  • Any point on the unit circle can be described with coordinates \( (\cos(\theta), \sin(\theta)) \).
  • As a point moves around the unit circle, the angle \( \theta \) changes, and so do the coordinates, but the radius always remains 1.

For trigonometric functions, the unit circle is essential because it provides a geometric way to visualize them. As the angle \( \theta \) changes, the coordinates at any point on the circle demonstrate how the sine and cosine functions behave.
Periodicity of Sine
The sine function is periodic, which means it repeats its values in regular intervals. The period of the sine function is \( 2\pi \), or 360 degrees.

This periodicity enables us to find all possible solutions for equations involving sine.

For example, when we know that \( \sin(\theta) = -1 \) at \( \theta = \frac{3\pi}{2} \) radians, we can find other angles that satisfy this equation by adding multiples of \( 2\pi \) to this angle.
Thus, the general solution is \( \theta = \frac{3\pi}{2} + 2k\pi \), where \( k \) is any integer, indicating that the negative extrema of the sine function occurs at regular intervals on the unit circle.
Angle Measurement
Angles can be measured in degrees or radians, the two most common units of measurement. In trigonometry, radians are often used because they relate directly to the unit circle.

To convert between degrees and radians, you can use:
  • \( 180^\circ = \pi \) radians
  • Thus, to convert degrees to radians, multiply by \( \frac{\pi}{180} \)
  • To convert radians to degrees, multiply by \( \frac{180}{\pi} \)
The angle \( \frac{3\pi}{2} \) radians converts to 270 degrees.
It's important to be comfortable switching between these units, as different equations and contexts might use either one.

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Most popular questions from this chapter

Rewrite the quantity as algebraic expressions of \(x\) and state the domain on which the equivalence is valid. $$ \cos (2 \arcsin (4 x)) $$

Assume that the range of arcsecant is \(\left[0, \frac{\pi}{2}\right) \cup\left(\frac{\pi}{2}, \pi\right]\) and that the range of arccosecant is \(\left[-\frac{\pi}{2}, 0\right) \cup\left(0, \frac{\pi}{2}\right]\) when finding the exact value. \(\operatorname{arccsc}\left(\csc \left(\frac{9 \pi}{8}\right)\right)\)

Rewrite the quantity as algebraic expressions of \(x\) and state the domain on which the equivalence is valid. $$ \cos (\arcsin (x)+\arctan (x)) $$

Find the domain of the given function. Write your answers in interval notation. $$ f(x)=\arccos \left(\frac{3 x-1}{2}\right) $$

find the exact value or state that it is undefined. \(\cot (\operatorname{arccsc}(\sqrt{5}))\)
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