Textbook Question
Without using a calculator, decide whether each function value is positive or negative. (Hint: Consider the radian measures of the quadrantal angles, and remember that π ≈ 3.14.)
tan 6.29
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3. Unit Circle
Defining the Unit Circle
Problem 3.63
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 = 0.9918
Verified step by step guidance1
Identify the problem: We need to find the angle \( s \) in the interval \([0, \pi/2]\) such that \( \sin s = 0.9918 \).
Recall that the sine function is the ratio of the opposite side to the hypotenuse in a right triangle, and it is also a periodic function with a range of [-1, 1].
Since \( s \) is in the interval \([0, \pi/2]\), we are looking for an angle in the first quadrant where the sine function is positive.
Use the inverse sine function (also known as arcsin) to find \( s \). This can be done using a calculator: \( s = \sin^{-1}(0.9918) \).
Ensure the calculator is set to radians mode, as the interval is given in terms of \( \pi \).
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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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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(x) 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 angles in other intervals, such as [0, π/2].
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Interval Notation
Interval notation is a mathematical notation used to represent a range of values. In this case, the interval [0, π/2] indicates that we are looking for solutions within the closed range from 0 to π/2, inclusive. This is important for determining the valid angles for which the sine function can yield the specified value, ensuring that the solution adheres to the constraints of the problem.
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