Textbook Question
In Exercises 23–26, find the exact value of each expression. Do not use a calculator.cos² 𝜋 - tan² 𝜋 4 4
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2. Trigonometric Functions on Right Triangles
Trigonometric Functions on Right Triangles
1:34 minutesProblem 7
Textbook Question
CONCEPT PREVIEW The terminal side of an angle θ in standard position passes through the point (― 3,― I3) Use the figure to find the following values. Rationalize denominators when applicable. r
Verified step by step guidance1
Identify the coordinates of the point through which the terminal side of the angle \( \theta \) passes: \((-3, -\sqrt{3})\).
Calculate the distance \( r \) from the origin to the point using the distance formula: \( r = \sqrt{x^2 + y^2} \).
Substitute the coordinates into the distance formula: \( r = \sqrt{(-3)^2 + (-\sqrt{3})^2} \).
Simplify the expression under the square root to find \( r \).
Rationalize the denominator if necessary, ensuring the final expression for \( r \) is in its simplest form.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Standard Position of an Angle
In trigonometry, an angle is said to be in standard position when its vertex is at the origin of a coordinate system and its initial side lies along the positive x-axis. The terminal side of the angle is determined by rotating the initial side counterclockwise. Understanding this concept is crucial for visualizing angles and their corresponding coordinates in the Cartesian plane.
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Distance Formula
The distance formula is used to calculate the distance 'r' from the origin to a point (x, y) in the Cartesian plane. It is given by r = √(x² + y²). This formula is essential for finding the radius in polar coordinates and is often used in trigonometric problems to relate angles to their corresponding coordinates.
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Rationalizing Denominators
Rationalizing the denominator is a mathematical process used to eliminate any radical expressions from the denominator of a fraction. This is often done by multiplying the numerator and denominator by a suitable value that will simplify the expression. In trigonometry, this technique is important for presenting answers in a standard form, especially when dealing with trigonometric ratios.
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