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Questions and Answers
What is the radius of the unit circle?
In a right triangle, which side is opposite the given acute angle?
What is the period of the sine function?
Which trigonometric function relates the ratio of the opposite side to the hypotenuse in a right triangle?
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Which trigonometric function is the inverse of the cosine function?
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What is the value of $\sin(\pi/6)$ on the unit circle?
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Which of the following correctly describes a property of the cosine function?
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If $\tan(\theta) = 3/4$, what is the value of $\theta$ in the range $[0, 2\pi)$?
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What is the range of the inverse sine function, $\sin^{-1}(x)$?
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If $f(x) = \sin(2x)$, what is the period of $f(x)$?
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Study Notes
Trigonometry and Functions
Introduction
Trigonometry is a branch of mathematics that deals with the relationships between angles and their corresponding side lengths in triangles. It has numerous applications across various fields, including engineering, physics, astronomy, and navigation. In trigonometry, we primarily focus on right triangles due to their unique properties. The six primary functions used in trigonometry are sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), and cosecant (csc). This article explores these concepts and their subtopics: right triangle trigonometry, unit circle, trigonometric functions, graphing trigonometric functions, and inverse trigonometric functions.
Right Triangle Trigonometry
A right triangle is a triangle with one angle equal to 90°. Its sides can be defined as the hypotenuse (the longest side opposite the right angle), the adjacent side (next to the right angle and shortest side), and the opposite side (opposite the given angle). Using the ratios of these sides, we define the trigonometric functions. For example, the sine function relates the ratio of the length of the opposite side to the length of the hypotenuse for an acute angle in the right triangle.
Unit Circle
The unit circle is a circle centered at the origin with a radius of 1. It plays a crucial role in defining the trigonometric functions. By measuring the radian values along the circumference of the unit circle, we can find the trigonometric function values for different angles.
Trigonometric Functions
Trigonometric functions are defined using the ratios of sides of right triangles:
- Sine (sin): sinθ = opposite side/hypotenuse
- Cosine (cos): cosθ = adjacent side/hypotenuse
- Tangent (tan): tanθ = opposite side/adjacent side
These functions represent the relationships between angles and their corresponding side lengths in right triangles.
Graphing Trigonometric Functions
The graphs of trigonometric functions (sin(x), cos(x), tan(x)) are sinusoidal, meaning they repeat periodically with distinct properties such as amplitude, midline, and period. Amplitude refers to the height of the wave; midline is the horizontal line around which the wave oscillates; and period represents the length of one complete cycle.
Inverse Trigonometric Functions
Inverse trigonometric functions allow us to find the angles corresponding to given side lengths. For example, sin⁻¹(x) returns the angle whose sine is x. These inverse functions have specific ranges and domains determined by their graphical behavior.
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Description
Explore the fundamentals of trigonometry including right triangle trigonometry, unit circle, trigonometric functions, graphing trigonometric functions, and inverse trigonometric functions. Learn about the relationships between angles and side lengths in triangles, and how to apply trigonometric functions in various fields like engineering and physics.
