The Unit Circle

Hello guys, it's Renz and today's blog would be all about the Unit Circle.

Unit Circle

The "Unit Circle" is a circle with a radius of 1. Being so simple, it is a great way to learn and talk about lengths and angles. The center is put on a graph where the x axis and y axis cross, so we get this neat arrangement he re.

Sine, Cosine and Tangent Because the radius is 1, you can directly measure sine, cosine and tangent. What happens when the angle, θ is 0°? cos=1, sin=0 and tan=0 What happens when θ is 90°? cos=0, sin=1 and tan is undefined Pythagoras Pythagoras' Theorem says that for a right angled triangle, the square of the long side equals the sum of the squares of the other two sides: x2 + y2 = 12 But 12 is just 1, so: x2 + y2 = 1  (the equation of the unit circle) Also, since x=cos and y=sin, we get: (cos(θ))2 + (sin(θ))2 = 1  (a useful "identity") Important Angles: 30°, 45° and 60° You should try to remember sin, cos and tan for the angles 30°, 45° and 60°. Yes, yes, it is a pain to have to remember things, but it will make life easier when you know them, not just in exams, but other times when you need to do quick estimates, etc. These are the values you should remember! Angle Sin Cos Tan=Sin/Cos 30° 1/√3 = √3/3 45° 1 60° √3 ow To Remember? To help you remember, think "1,2,3" : sin(30°) = √1/2 = 1/2 (because √1 = 1) sin(45°) = √2/2 sin(60°) = √3/2 And cos goes "3,2,1" cos(30°) = √3/2 cos(45°) = √2/2 cos(60°) = √1/2 = 1/2 (because √1 = 1) What about tan? tan = sin/cos, so you can calculate: tan(30°) = sin(30°)  =  1/2  =  1 cos(30°) √3/2 √3   tan(45°) = sin(45°)  =  √2/2  = 1  cos(45°) √2/2     tan(60°) = sin(60°)  =  √3/2  = √3  cos(60°) 1/2     Where did those values come from? We can use the equation x2 + y2 = 1 to find the lengths of x and y (which are equal to cos and sin when the radius is 1): 45 Degrees For 45 degrees, x and y are equal, so y=x: x2 + x2 = 1 2x2 = 1 x2 = ½ x = y = √½ 60 Degrees Take an equilateral triangle (all sides are equal and all angles are 60°) and split it down the middle. The "x" side is now ½, And the "y" side will be: (½)2 + y2 = 1 ¼ + y2 = 1 y2 = 1-¼ = ¾ y = √¾ 30 Degrees 30° is just 60° with x and y swapped, so x = √¾ and y = ½ Summary √½ is usually changed to this:   And √¾ is usually changed to this:   And here is the result (as before): Angle Sin Cos Tan=Sin/Cos 30° 1/√3 = √3/3 45° 1 60° √3 Putting it All Together And here they are for every quadrant. With the correct sign (plus or minus) as per Cartesian Coordinates. Note that cos is first and sin is second, so it goes (cos, sin): And this is the same Unit Circle in radians. Here's a website that may help you as well: https://www.khanacademy.org/math/trigonometry/basic-trigonometry/unit_circle_tut/e/unit_circle Source: http://www.mathsisfun.com/geometry/unit-circle.html