Starting from 0° and progressing through 90°, cos(0°)=1=. This pattern repeats periodically for the respective angle measurements, and we can identify the values of sin(θ) based on the position of θ in the unit circle, taking the sign of sine into consideration: sine is positive in quadrants I and II and negative in quadrants III and IV.Ī similar memorization method can be used for cosine. The values of sine from 0° through -90° follows the same pattern except that the values are negative instead of positive since sine is negative in quadrant IV. The subsequent values, sin(30°), sin(45°), sin(60°), and sin(90°) follow a pattern such that, using the value of sin(0°) as a reference, to find the values of sine for the subsequent angles, we simply increase the number under the radical sign in the numerator by 1, as shown below. Starting from 0° and progressing through 90°, sin(0°) = 0 =. One method that may help with memorizing the common trigonometric values is to express all the values of sin(θ) as fractions involving a square root. At any of these angles, sin(θ) or cos(θ) has a value of –1, 0, or 1. The other angles on the unit circle to remember are those whose terminal sides lie on the x- or y-axis: 0° or 0 (which has equivalent sine and cosine values as 360° or 2π), 90° or, 180° or π and, 270° or. Therefore, remembering these three values and how they correspond to multiples of 30°, 45° and 60° will enable you to fill in all the values on the unit circle. Because of the nature of the unit circle, these values are the same for their respective angles in different quadrants on the unit circle, with the only difference being their signs based on the quadrant the angle is in. AngleĪs can be seen from the table or the unit circle above, there are three values to remember. Below is a table of the values of these angles, as well as a figure of the values on a unit circle. In radians, they correspond to respectively. While we can find trigonometric values for any angle, some angles are worth remembering because of how frequently they are used in trigonometry. Based on this, we can determine the definitions of the rest of the trigonometric functions, as shown in the table below. ![]() Using the fact that the radius of the unit circle is 1 (and therefore the hypotenuse of the right triangle is equal to 1), we can use the right triangle definitions of the trigonometric functions to find that, and. Together with θ, the angle formed between the initial side of an angle along the positive x-axis and the terminal side of the angle formed by rotating the ray counter-clockwise, we can form a right triangle. In the figure above, point A has coordinates of (x, y). Below is a figure showing all of the trigonometric relationships as they relate to the unit circle. The unit circle is often used in the definition of trigonometric functions. Unit circle definitions of trigonometric functions This is true for all points on the unit circle, not just those in the first quadrant, and is useful for defining the trigonometric functions in terms of the unit circle. Based on the Pythagorean Theorem, the equation of the unit circle is therefore: ![]() The hypotenuse of the right triangle is equal to the radius of the unit circle, so it will always be 1. ![]() When a ray is drawn from the origin of the unit circle, it will intersect the unit circle at a point (x, y) and form a right triangle with the x-axis, as shown above. It is commonly used in the context of trigonometry. Home / trigonometry / unit circle Unit CircleĪ unit circle is a circle with radius 1 centered at the origin of the rectangular coordinate system.
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