Textbook Question
Find each exact function value. See Example 2.
sec 23π/6
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3. Unit Circle
Defining the Unit Circle
Problem 3.35
Textbook Question
Find a calculator approximation to four decimal places for each circular function value. See Example 3. cos (-1.1519)
Verified step by step guidance1
Step 1: Understand the problem. We need to find the cosine of the angle -1.1519 radians.
Step 2: Recall that the cosine function is even, meaning \( \cos(-x) = \cos(x) \). Therefore, \( \cos(-1.1519) = \cos(1.1519) \).
Step 3: Use a calculator to find the cosine of 1.1519 radians. Ensure your calculator is set to radian mode.
Step 4: Enter 1.1519 into the calculator and press the cosine function button to get the approximate value.
Step 5: Round the result 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.
Circular Functions
Circular functions, also known as trigonometric functions, relate the angles of a circle to the ratios of its sides. The primary circular functions include sine, cosine, and tangent, which are defined based on a unit circle. Understanding these functions is essential for evaluating angles and their corresponding values in various contexts, including negative angles.
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Negative Angles
In trigonometry, negative angles are measured in the clockwise direction from the positive x-axis. The values of circular functions for negative angles can be derived using the properties of symmetry in the unit circle. For example, cos(-θ) = cos(θ), which means the cosine function is even, while sin(-θ) = -sin(θ), indicating that the sine function is odd.
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Calculator Approximations
Calculator approximations involve using a scientific calculator to find numerical values of trigonometric functions to a specified degree of accuracy, such as four decimal places. This process typically requires inputting the angle in radians or degrees, depending on the calculator settings. Understanding how to use a calculator effectively is crucial for obtaining precise values for trigonometric functions.
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