Table of contents
- 0. Review of College Algebra4h 43m
- 1. Measuring Angles39m
- 2. Trigonometric Functions on Right Triangles2h 5m
- 3. Unit Circle1h 19m
- 4. Graphing Trigonometric Functions1h 19m
- 5. Inverse Trigonometric Functions and Basic Trigonometric Equations1h 41m
- 6. Trigonometric Identities and More Equations2h 34m
- 7. Non-Right Triangles1h 38m
- 8. Vectors2h 25m
- 9. Polar Equations2h 5m
- 10. Parametric Equations1h 6m
- 11. Graphing Complex Numbers1h 7m
3. Unit Circle
Defining the Unit Circle
Problem 3.53bLial - 12th Edition
Textbook Question
Find the approximate value of s, to four decimal places, in the interval [0, π/2] that makes each statement true.
sin s = 0.4924
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Start by understanding that you need to find the angle such that .
Since is in the interval , you are looking for an angle in the first quadrant where the sine function is positive.
Use the inverse sine function, also known as arcsine, to find . This is written as .
Use a calculator to find the approximate value of in radians, ensuring your calculator is set to the correct mode (radians).
Round the result to four decimal places to get the final approximate value of .
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Sine Function
The sine function, denoted as sin, is a fundamental trigonometric function that relates the angle of a right triangle to the ratio of the length of the opposite side to the hypotenuse. It is periodic and oscillates between -1 and 1. In the context of the unit circle, sin(s) represents the y-coordinate of a point on the circle corresponding to the angle s.
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Graph of Sine and Cosine Function
Inverse Sine Function
The inverse sine function, or arcsin, is used to determine the angle whose sine is a given value. It is denoted as sin⁻¹ or arcsin and is defined for values in the range [-1, 1]. The output of arcsin is restricted to the interval [-π/2, π/2], but when considering the sine function's periodicity, we can find multiple angles that yield the same sine value.
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Inverse Sine
Interval [0, π/2]
The interval [0, π/2] represents the first quadrant of the unit circle, where both sine and cosine functions are positive. In this interval, the sine function is increasing, meaning that as the angle s increases from 0 to π/2, the value of sin(s) also increases from 0 to 1. This property is crucial for finding the angle s that satisfies the equation sin(s) = 0.4924 within the specified range.
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Example 2
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Master Introduction to the Unit Circle with a bite sized video explanation from Callie Rethman
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Related Practice
Textbook Question
In Exercises 1–4, a point P(x, y) is shown on the unit circle corresponding to a real number t. Find the values of the trigonometric functions at t.
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