This relation is used to convert angles from radians to degrees.Ī unit circle is divided into four quadrants making an angle of 90°, 180°, 270°, and 360° (in degrees) or π/2, π. The entire unit circle represents a complete angle of 360°, which is equal to 2π when expressed in radians. Unit Circle Table Preparing the Complete Unit Circle Chart The values of the special angles of all the trigonometric functions are represented in the form of a table given below. A reference angle always uses an x-axis as its frame of reference. Their value is always between 0 and 90° when measured in degrees or 0 and π/2 when measured in radians. These angles, called special angles, are used for simplifying the calculations related to trigonometric functions with different angles. Similarly, we can find the values of the trigonometric functions for some specific values of θ such as 30°, 60°, and 90°. Let us find their values for θ = 0° and θ = 90°.įor θ = 0°, the x-coordinate is 1 and the y-coordinate is 0. In the unit circle, we have cosine as the x-coordinate and sin as the y-coordinate. These angles are called reference angles. It is important to remember the values of sin, cos, and tan for some specific angles. The value of θ can be expressed in both degree and in radians. Refer our Value of Angles in Trigonometric Functions Identifying the reference angles will help us identify a pattern in these values. They take on the value of 0 as well as positive and negative values of three numbers √3/2, √2/2, and ½. If we take a close look at the unit circle, we will find that the sin and cos values of angles fluctuate between -1 and 1. Finally, in quadrant IV, ‘Class’ only cosine is positive. In quadrant III, ‘Trig’ only tangent is positive. In quadrant II, ‘Smart’, only sine is positive. In quadrant I, which is ‘A’ all of the trigonometric functions are positive. To help remember which of the trigonometric functions are positive in each quadrant, we can use the mnemonic phrase ‘ All Students Take Calculus’ or All Sin Tan, Cos (ASTC).Įach of the four words in the phrase corresponds to one of the four quadrants, starting with quadrant I and rotating in counterclock-wise manner. The sign of a trigonometric function depends on the quadrant that the angle is found. Sin 2 θ + cos 2 θ = 1 Sign of Trigonometric Functions Since, the equation of a unit circle is given by x 2 + y 2 = 1, where x = cos θ and y = sin θ, we get an important relation: Applying this values in trigonometry, we getĬosec θ = 1/sin θ = Hypotenuse/ Altitude = 1/y Thus we have a right triangle with sides measuring 1, x, y. The lengths of the two legs (base and altitude) have values x and y respectively. The radius of the unit circle is the hypotenuse of the right triangle, which makes an angle θ with the positive x-axis. By drawing the radius and a perpendicular line from the point P to the x-axis we will get a right triangle placed in a unit circle in the Cartesian-coordinate plane. Being a unit circle, its radius ‘r’ is equal to 1 unit, which is the distance between point P and center of the circle. Let us take a point P on the circumference of the unit circle whose coordinates be (x, y). Here we will use the Pythagorean Theorem in a unit circle to understand the trigonometric functions. We can calculate the trigonometric functions of sine, cosine, and tangent using a unit circle. Finding the Angles of Trigonometric Functions Using a Unit Circle: Sin, Cos, Tan The above equation satisfies all the points lying on the circle in all four quadrants.
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