Further within the unit circle, we can also prove the other two Pythagorean identities. Thus we have successfully proved the first identity using the Pythagoras theorem.
Also in a unit circle, we have, x = cosθ, and y = sinθ, and applying this in the above statement of the Pythagoras theorem, we have, cos 2θ + sin 2θ = 1. Applying Pythagoras theorem we have x 2 + y 2 = 1 which represents the equation of a unit circle. Let us take x and y as the legs of the right-angled triangle having a hypotenuse 1 unit.
Here we shall try to prove the first identity with the help of the Pythagoras theorem. The three Pythagorean identities in trigonometry are as follows. The Pythagoras theorem states that in a right-angled triangle the square of the hypotenuse is equal to the sum of the square of the other two sides. The three important Pythagorean identities of trigonometric ratios can be easily understood and proved with the unit circle. At 90º and at 270º the cosθ value is equal to 0 and hence the tan values at these angles are undefined.Įxample: Find the value of tan 45º using sin and cos values from the unit circle. The entire circle represents a complete angle of 360º and the four quadrant lines of the circle make angles of 90º, 180º, 270º, 360º(0º). Applying this further we have tanθ = sinθ/cosθ or tanθ = y/x.Īnother important point to be understood is that the sinθ and cosθ values always lie between 1 and -1, and the radius value is 1, and it has a value of -1 on the negative x-axis. Here we have cosθ = x, and sinθ = y, and these values are helpful to compute the other trigonometric ratio values. For any values of θ made by the radius line with the positive x-axis, the coordinates of the endpoint of the radius represent the cosine and the sine of the θ values. Also by changing the θ values we can obtain the principal values of these trigonometric ratios.Īny point on the unit circle has coordinates(x, y), which are equal to the trigonometric identities of (cosθ, sinθ). Similarly, we can obtain the values of the other trigonometric ratios using the right-angled triangle within the unit circle. We now have sinθ = y, cosθ = x, and using this we now have tanθ = y/x.
Applying this in trigonometry, we can find the values of the trigonometric ratio, as follows: Now we have a right angle triangle with the sides 1, x, y. Here the values of x and y are the lengths of the base and the altitude of the right triangle. The radius vector makes an angle θ with the positive x-axis and the coordinates of the endpoint of the radius vector is (x, y). The radius of the circle represents the hypotenuse of the right triangle. Consider a right triangle placed in a unit circle in the cartesian coordinate plane. Let us apply the Pythagoras 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. The above equation satisfies all the points lying on the circle across the four quadrants.įinding Trigonometric Functions Using a Unit Circle Here for the unit circle, the center lies at (0,0) and the radius is 1 unit.
This is simplified to obtain the equation of a unit circle. Hence the equation of the unit circle is (x - 0) 2 + (y - 0) 2 = 1 2. A unit circle is formed with its center at the point(0, 0), which is the origin of the coordinate axes. This equation of a circle is simplified to represent the equation of a unit circle. The general equation of a circle is (x - a) 2 + (y - b) 2 = r 2, which represents a circle having the center (a, b) and the radius r. The locus of a point which is at a distance of one unit from a fixed point is called a unit circle. The unit circle has applications in trigonometry and is helpful to find the values of the trigonometric ratios sine, cosine, tangent. The unit circle is algebraically represented using the second-degree equation with two variables x and y. The unit circle is generally represented in the cartesian coordinate plane. A unit circle is a circle with a radius measuring 1 unit.
0 kommentar(er)
