Textbook Question
In Exercises 1β4, a point P(x, y) is shown on the unit circle corresponding to a real number t. Find the values of the trigonometric functions at t.
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3. Unit Circle
Trigonometric Functions on the Unit Circle
1:58 minutesProblem 10
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.
tan 0
Verified step by step guidance1
Identify the angle t for which you want to find tan(t). Here, t = 0.
Recall that on the unit circle, the coordinates of a point corresponding to an angle t are given by (x, y) = (cos(t), sin(t)).
From the unit circle diagram, find the coordinates at t = 0. The coordinates are (1, 0).
Use the definition of the tangent function in terms of sine and cosine: \(\tan(t) = \frac{\sin(t)}{\cos(t)}\).
Substitute the values from the coordinates into the formula: \(\tan(0) = \frac{0}{1}\). Simplify this expression to find the value of tan(0).
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Video duration:
1mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Unit Circle and Coordinates
The unit circle is a circle with radius 1 centered at the origin of the coordinate plane. Each point on the circle corresponds to an angle t measured in radians from the positive x-axis. The coordinates (x, y) of each point represent the cosine and sine of the angle t, respectively.
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Introduction to the Unit Circle
Trigonometric Functions on the Unit Circle
The sine, cosine, and tangent functions can be defined using the coordinates of points on the unit circle. For an angle t, cos(t) is the x-coordinate, sin(t) is the y-coordinate, and tan(t) is the ratio y/x, provided x β 0. This allows evaluation of trig functions at various angles using the unit circle.
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Undefined Values of Tangent
Tangent is undefined when the cosine of the angle is zero because tan(t) = sin(t)/cos(t). On the unit circle, this occurs at points where the x-coordinate is zero, such as t = Ο/2 and 3Ο/2. Recognizing these points is essential to determine when tangent values do not exist.
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