Multiple Choice
How many sides does a regular polygon have if each interior angle measures ?
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2. Trigonometric Functions on Right Triangles
Special Right Triangles
1:36 minutesProblem 52
Textbook Question
Give the exact value of each expression. See Example 5. cos 30°
Verified step by step guidance1
Recall that 30° is one of the special angles in trigonometry, and its cosine value is well-known from the unit circle or special right triangles.
Recognize that the cosine of 30° corresponds to the adjacent side over the hypotenuse in a 30°-60°-90° right triangle.
In a 30°-60°-90° triangle, the sides are in the ratio 1 (opposite 30°) : \(\sqrt{3}\) (adjacent 30°) : 2 (hypotenuse).
Therefore, the cosine of 30° is the length of the adjacent side over the hypotenuse, which can be written as \(\cos 30^\circ = \frac{\sqrt{3}}{2}\).
Write down the exact value using the simplified radical form without decimal approximation.
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1mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Unit Circle and Special Angles
The unit circle is a circle with radius 1 centered at the origin, used to define trigonometric functions for all angles. Special angles like 30°, 45°, and 60° have well-known sine and cosine values derived from equilateral and right triangles, which help in finding exact trigonometric values.
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Introduction to the Unit Circle
Cosine Function Definition
Cosine of an angle in a right triangle is the ratio of the adjacent side to the hypotenuse. On the unit circle, cosine corresponds to the x-coordinate of the point where the terminal side of the angle intersects the circle, allowing exact values to be determined for standard angles.
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Exact Values of Cosine for 30°
The exact value of cos 30° is derived from the 30°-60°-90° triangle, where the sides are in the ratio 1:√3:2. Cos 30° equals √3/2, representing the adjacent side over the hypotenuse, providing a precise, simplified radical form rather than a decimal approximation.
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5:08Sine, Cosine, & Tangent of 30°, 45°, & 60°
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