Textbook Question
CONCEPT PREVIEW Name the corresponding angles and the corresponding sides of each pair of similar triangles.
Ch. 1 - Trigonometric Functions
Lial12th EditionTrigonometryISBN: 9780136552161
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Table of contents
- Ch. R - Algebra Review381
- Ch. 1 - Trigonometric Functions188
- Ch. 2 - Acute Angles and Right Triangles168
- Ch. 3 - Radian Measure and The Unit Circle155
- Ch. 4 - Graphs of the Circular Functions100
- Ch. 5 - Trigonometric Identities238
- Ch. 6 - Inverse Circular Functions and Trigonometric Equations158
- Ch. 7 - Applications of Trigonometry and Vectors148
- Ch. 8 - Complex Numbers, Polar Equations, and Parametric Equations15
Chapter 2, Problem 7
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. Here, the point is given as \( (-3, -\sqrt{3}) \).
Recall that \( r \) represents the distance from the origin to the point \( (x, y) \) on the terminal side of the angle. This distance \( r \) is the hypotenuse of the right triangle formed by the x-axis and the point.
Use the distance formula to find \( r \): \(\n\[\n\)\( r = \sqrt{x^2 + y^2} \) \(\n\]\nSubstitute\) \( x = -3 \) and \( y = -\sqrt{3} \) into the formula.
Calculate \( r \) by squaring each coordinate, adding them, and then taking the square root: \(\n\)\(\n\)\( r = \sqrt{(-3)^2 + (-\sqrt{3})^2} = \sqrt{9 + 3} \).
Simplify the expression under the square root to get \( r = \sqrt{12} \). Then, simplify \( \sqrt{12} \) by factoring out perfect squares and rationalize the denominator if necessary.
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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
An angle is in standard position when its vertex is at the origin and its initial side lies along the positive x-axis. The terminal side is the ray that rotates from the initial side to form the angle θ. Understanding this helps locate points on the terminal side and relate coordinates to trigonometric values.
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Distance Formula and Radius (r)
The radius r is the distance from the origin to the point (x, y) on the terminal side, calculated using r = √(x² + y²). This value is essential for defining trigonometric functions like sine, cosine, and tangent based on the coordinates of the point.
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6:36Quadratic Formula
Trigonometric Ratios from Coordinates
Given a point (x, y) on the terminal side, sine, cosine, and tangent of θ are defined as sin θ = y/r, cos θ = x/r, and tan θ = y/x. These ratios allow calculation of trigonometric values directly from the point's coordinates, facilitating problem solving in trigonometry.
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05:32Intro to Polar Coordinates
Related Practice
Textbook Question
CONCEPT PREVIEW Name the corresponding angles and the corresponding sides of each pair of similar triangles.
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