Textbook Question
Find the exact value of s in the given interval that has the given circular function value.
[ 0, π/2] ; cos s = √2/2
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3. Unit Circle
Defining the Unit Circle
Problem 3.61
Textbook Question
Find the approximate value of s, to four decimal places, in the interval [0, π/2] that makes each statement true.
tan s = 0.2126
Verified step by step guidance1
Identify the trigonometric function involved: \( \tan(s) = 0.2126 \).
Recall that the tangent function, \( \tan(s) \), is the ratio of the opposite side to the adjacent side in a right triangle.
To find \( s \), use the inverse tangent function: \( s = \tan^{-1}(0.2126) \).
Ensure that the solution is within the given interval \([0, \frac{\pi}{2}]\).
Use a calculator to find \( s \) to four decimal places, ensuring the calculator is set to the correct mode (radians).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Tangent Function
The tangent function, denoted as tan, is a fundamental trigonometric function defined as the ratio of the opposite side to the adjacent side in a right triangle. It can also be expressed in terms of sine and cosine as tan(s) = sin(s)/cos(s). Understanding the behavior of the tangent function, especially within specific intervals, is crucial for solving equations involving tan.
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Inverse Trigonometric Functions
Inverse trigonometric functions, such as arctan, are used to find angles when the value of a trigonometric function is known. For example, if tan(s) = 0.2126, then s can be found using s = arctan(0.2126). These functions are essential for determining angles in various contexts, particularly when working with specific values of trigonometric ratios.
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Quadrants and Angle Ranges
Trigonometric functions have different signs in different quadrants of the unit circle. The interval [0, π/2] corresponds to the first quadrant, where both sine and cosine are positive, and thus tangent is also positive. Recognizing the appropriate quadrant is important for determining the correct angle that satisfies the given trigonometric equation.
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