![]() ![]() 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 Cos goes opposite from sine, and tangent and cotangent are derived from them so you can always calculate them easily.The above equation satisfies all the points lying on the circle in all four quadrants. ![]() This lesson may look a bit complicated to remember, but it really is not. Consider Sine, Cosine, Tangent, and Cotangent: Following table is very important to remember. ![]() Special angles are angles that have relatively simple values. In one quarter of a circle is $\frac.$$ Trigonometric functions of special angles If you have your number line marked with radians, this is how it would look:įirst, you have a usual unit circle. That means that infinitely many points from number line will fall into same places on a unit circle. ![]() In some point you’ll start your second lap around it, and when you wrap it again, you’ll start third and so on in infinity. You wrap an endless line around a circle. Now that we remembered that, let’s look at our picture. One whole circle has $ 2 \pi$ radians one half of a circle has $\pi$ radians and so on. 1 radian is a part of a circle where length of an arc is equal to the radius. The positive numbers, (up from the origin in the picture) are replicated in a positive mathematical orientation (counterclockwise) and negative (downwards from the origin) are replicated in a negative mathematical orientation (clockwise). It is important that the radius of this circle is equal to 1.Īs you know, you have positive and negative numbers on your number line. The unit circle is a circle with a radius of 1. For every point on our number line, there is exactly one point on a circle. Now what would happen if we would wrap our endless line around a circle with radius 1?Įvery point from the number line will end up on our circle. A number line is a straight endless line with origin and unitary length. Before learning about what a unit circle is, it helps to remember what is a number line. ![]()
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