Textbook Question
In Exercises 17β20, ΞΈ is an acute angle and sin ΞΈ and cos ΞΈ are given. Use identities to find tan ΞΈ, csc ΞΈ, sec ΞΈ, and cot ΞΈ. Where necessary, rationalize denominators.__sin ΞΈ = 6, cos ΞΈ = β137 7
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3. Unit Circle
Trigonometric Functions on the Unit Circle
Problem 6
Textbook Question
In Exercises 5β18, the unit circle has been divided into twelve equal arcs, corresponding to t-values of 0, π, π, π, 2π, 5π, π, 7π, 4π, 3π, 5π, 11π, and 2π. 6 3 2 3 6 6 3 2 3 6 Use the (x,y) coordinates in the figure to find the value of each trigonometric function at the indicated real number, t, or state that the expression is undefined. sin π/3
Verified step by step guidance1
Recall that on the unit circle, the sine of an angle \( t \) corresponds to the y-coordinate of the point on the circle at that angle.
Identify the angle \( t = \frac{\pi}{3} \) on the unit circle. Since the circle is divided into twelve equal arcs, each arc corresponds to \( \frac{2\pi}{12} = \frac{\pi}{6} \).
Determine the coordinates of the point on the unit circle at \( t = \frac{\pi}{3} \). This angle is twice \( \frac{\pi}{6} \), so it corresponds to the second division point.
Recall or use the known exact coordinates for \( \frac{\pi}{3} \) on the unit circle, which are \( \left( \frac{1}{2}, \frac{\sqrt{3}}{2} \right) \).
Since sine corresponds to the y-coordinate, the value of \( \sin \frac{\pi}{3} \) is \( \frac{\sqrt{3}}{2} \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Unit Circle and Radian Measure
The unit circle is a circle with radius 1 centered at the origin of the coordinate plane. Angles on the unit circle are measured in radians, where 2Ο radians correspond to a full rotation (360Β°). Each point on the unit circle corresponds to an angle t, and its coordinates (x, y) represent (cos t, sin t). Understanding this relationship is essential for evaluating trigonometric functions at given radian values.
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Sine Function on the Unit Circle
The sine of an angle t is the y-coordinate of the corresponding point on the unit circle. For example, sin(Ο/3) corresponds to the y-value of the point at an angle of Ο/3 radians. Knowing the exact coordinates of common angles like Ο/3 helps in directly finding sine values without a calculator.
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Special Angles and Their Coordinates
Special angles such as Ο/6, Ο/4, and Ο/3 have well-known sine and cosine values derived from their coordinates on the unit circle. For Ο/3, the coordinates are (1/2, β3/2), so sin(Ο/3) = β3/2. Memorizing these values or understanding how to derive them from the unit circle simplifies solving trigonometric problems.
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