In Exercises 5–18, 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 guidance

1. Identify the angle t for which you want to find tan(t). Here, t = 0.
2. 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)).
3. From the unit circle diagram, find the coordinates at t = 0. The coordinates are (1, 0).
4. Use the definition of the tangent function in terms of sine and cosine: \(\tan(t) = \frac{\sin(t)}{\cos(t)}\).
5. Substitute the values from the coordinates into the formula: \(\tan(0) = \frac{0}{1}\). Simplify this expression to find the value of tan(0).

Key Concepts

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.

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 ≠ 0. This allows evaluation of trig functions at various angles using the unit circle.

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 = π/2 and 3π/2. Recognizing these points is essential to determine when tangent values do not exist.