Textbook Question
In Exercises 23–26, find the exact value of each expression. Do not use a calculator.cos 2𝜋 sec 2𝜋9 9
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2. Trigonometric Functions on Right Triangles
Trigonometric Functions on Right Triangles
1:10 minutesProblem 10
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. tan θ
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 \( (-3, -\sqrt{3}) \).
Recall that \( \tan \theta = \frac{y}{x} \), where \( x \) and \( y \) are the coordinates of the point on the terminal side of the angle.
Substitute the values of \( x = -3 \) and \( y = -\sqrt{3} \) into the tangent formula: \( \tan \theta = \frac{-\sqrt{3}}{-3} \).
Simplify the fraction by dividing numerator and denominator, and then rationalize the denominator if necessary.
Express the final simplified form of \( \tan \theta \) with a rationalized denominator.
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Video duration:
1mKey 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 the angle based on a point through which its terminal side passes.
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Coordinates and Trigonometric Ratios
The coordinates (x, y) of a point on the terminal side of an angle can be used to find trigonometric ratios. Specifically, tan θ is the ratio of y to x (tan θ = y/x). Knowing the point allows direct calculation of tangent and other trigonometric functions.
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Rationalizing Denominators
Rationalizing the denominator involves eliminating any square roots or irrational numbers from the denominator of a fraction. This is done by multiplying numerator and denominator by a suitable expression, making the expression simpler and more standardized in trigonometry answers.
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