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.)
cos 6
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3. Unit Circle
Defining the Unit Circle
Problem 3.62
Textbook Question
Find the approximate value of s, to four decimal places, in the interval [0, π/2] that makes each statement true.
cos s = 0.7826
Verified step by step guidance1
Step 1: Understand the problem. We need to find the angle \( s \) in the interval \([0, \frac{\pi}{2}]\) such that \( \cos(s) = 0.7826 \).
Step 2: Use the inverse cosine function to find \( s \). The inverse cosine function, denoted as \( \cos^{-1} \), will help us find the angle whose cosine is 0.7826.
Step 3: Calculate \( s = \cos^{-1}(0.7826) \). This will give us the angle in radians.
Step 4: Ensure that the calculated angle \( s \) is within the specified interval \([0, \frac{\pi}{2}]\).
Step 5: Round the value of \( s \) to four decimal places as required by the problem.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Cosine Function
The cosine function, denoted as cos, is a fundamental trigonometric function that relates the angle of a right triangle to the ratio of the length of the adjacent side to the hypotenuse. It is defined for all real numbers and is periodic with a period of 2π. In the context of the unit circle, cos(s) represents the x-coordinate of a point on the circle corresponding to the angle s.
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Inverse Cosine (arccos)
The inverse cosine function, or arccos, is used to find the angle whose cosine is a given value. It is defined for values in the range [-1, 1] and returns angles in the interval [0, π]. This function is essential for solving equations involving cosine, such as finding the angle s when cos(s) = 0.7826.
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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 s must be between 0 and π/2, inclusive. Understanding this notation is crucial for determining the valid solutions for s, as it restricts the possible angles to the first quadrant where cosine values are positive.
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