Consider the angle θin standard position in a unit circle, where 0 ≤θ<π/ 2 or -π/ 2<θ<0 (use bot

step 1 For the given problem, we're going to consider the angle theta in standard position in a unit ci

Consider the angle θin standard position in a unit circle,...

Consider the angle $\theta$ in standard position in a unit circle, where $0 \leq \theta<\pi / 2$ or $-\pi / 2<\theta<0$ (use both figures). a. Show that $|A C|=|\sin \theta|,$ for $-\pi / 2<\theta<\pi / 2$. (Hint: Consider the cases $0 \leq \theta<\pi / 2$ and $-\pi / 2<\theta<0$ separately.) b. Show that $|\sin \theta|<|\theta|,$ for $-\pi / 2<\theta<\pi / 2$. (Hint: The length of arc $A B$ is $\theta,$ if $0 \leq \theta<\pi / 2,$ and $-\theta,$ if $-\pi / 2<\theta<0$ c. Conclude that $-|\theta| \leq \sin \theta \leq|\theta|,$ for $-\pi / 2<\theta<\pi / 2$. d. Show that $0 \leq 1-\cos \theta \leq|\theta|,$ for $-\pi / 2<\theta<\pi / 2$.

Consider the angle $\theta$ in standard position in a unit circle, where $0 \leq \theta<\pi / 2$ or $-\pi / 2<\theta<0$ (use both figures).
a. Show that $|A C|=|\sin \theta|,$ for $-\pi / 2<\theta<\pi / 2$. (Hint:
Consider the cases $0 \leq \theta<\pi / 2$ and $-\pi / 2<\theta<0$ separately.)
b. Show that $|\sin \theta|<|\theta|,$ for $-\pi / 2<\theta<\pi / 2$. (Hint:
The length of arc $A B$ is $\theta,$ if $0 \leq \theta<\pi / 2,$ and $-\theta,$ if $-\pi / 2<\theta<0$
c. Conclude that $-|\theta| \leq \sin \theta \leq|\theta|,$ for $-\pi / 2<\theta<\pi / 2$.
d. Show that $0 \leq 1-\cos \theta \leq|\theta|,$ for $-\pi / 2<\theta<\pi / 2$.

Key Concepts

Trigonometric Inequalities

Inequalities involving trigonometric functions, like bounding the sine and cosine functions by the angle or other expressions, provide important insights into their behavior near the origin. These inequalities are used to compare linear measures, such as the arc length with the chord length or sine value, and they lay the groundwork for further analysis in calculus, particularly in understanding limits and approximations.

Trigonometric Functions

Trigonometric functions such as sine and cosine are defined based on the coordinates of points on the unit circle. Their definitions extend beyond right triangles and allow for a continuous description of periodic behavior. Understanding these functions from both geometric and analytic viewpoints is key to proving relationships and inequalities between the functions and their corresponding angles.

Unit Circle Geometry

The unit circle is a fundamental tool in trigonometry, where each point on the circle corresponds to an angle measured in standard position. Its properties allow for geometric interpretations of trigonometric functions, such as representing the sine of an angle as the vertical coordinate of the corresponding point on the circle, and the cosine as the horizontal coordinate. This framework is essential for connecting algebraic definitions with geometric concepts like chord lengths and arc measurements.

Radian Measure and Arc Length

In the context of a circle, radians provide a natural measure of angles, directly linking the angle size to the length of the arc that it subtends on a circle. For a unit circle, this relationship is particularly simple since the arc length is numerically equal to the angle in radians. This direct proportionality is crucial for deriving inequalities that involve the sine function and the angle itself.

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