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Let's find trigonometric values given a point on the unit circle.

Â For example that suppose that theta is an angle in standard position

Â whose terminal side intersects the unit circle at negative 21 over 29,

Â negative 20 over 29.

Â Let's find the exact values of sine of theta,

Â tangent of theta, and cosecant of theta.

Â Now we define the trigonometric functions in terms of

Â coordinates of points on the unit circle as follows.

Â Suppose that theta is an angle in standard position,

Â whose terminal side intersects the unit circle at X Y.

Â Then the six trigonometric functions are defined as follows.

Â Sine of theta is equal to the Y coordinate,

Â cosine of theta is equal to the X coordinate,

Â tangent of theta is equal to Y divided by X,

Â cosecant of theta is equal to 1 divided by Y,

Â secant of theta is equal to 1 divided by X,

Â and cotangent of theta is equal to X divided by Y.

Â Now these definitions are consistent with

Â the right triangle trigonometric ratios that you might be used to.

Â To see this, suppose that theta is acute like the angle shown here,

Â let's drop a perpendicular from the point XY to the X axis.

Â Now this length then is Y,

Â and this is X and remember the radius of the circle is 1, so this length is 1.

Â Using our trig ratios,

Â we have that sine of theta is equal to opposite,

Â over hypotenuse which is equal to Y divided by 1,

Â or Y, which looking over here isn't that what this says,

Â that the sine of theta is equal to Y?

Â And cosine of theta is equal to adjacent,

Â divided by Hypotenuse, which is equal to X divided by 1,

Â which looking over here is exactly what this says.

Â And we can validate the other 4 equations in a similar way.

Â Okay, so let's apply this to our situation here.

Â Where given that X is equal to negative 21 divided by 29,

Â and Y is equal to negative 20 divided by 29 as shown in this figure.

Â So let's use this definition to find sine of theta,

Â tangent of theta, and cosecant of theta.

Â Namely sine of theta is equal to the Y coordinate of this point,

Â which is negative 20 divided by 29,

Â and tangent of theta is equal to the Y coordinate divided by the X coordinate,

Â which is equal to negative 20 divided by 29,

Â divided by negative 21 divided by 29,

Â which is equal to, the 29's will cancel,

Â as well as the negatives,

Â and we get 20 divided by 21.

Â And finally, cosecant of theta is equal to 1 divided by Y,

Â which is equal to 1 divided by negative 20 over 29,

Â which is equal to negative 29 divided by 20.

Â So these are the three values that we were looking for,

Â sine of theta, tangent of theta and cosecant of theta.

Â All right, let's look at another example.

Â Suppose that theta is an angle in standard position whose terminal side

Â intersects the unit circle at three fourths negative square root 7 divided by 4.

Â Let's find the exact values of cosine of theta,

Â cotangent of theta and secant of theta.

Â Again we are going to use the following definition,

Â but now we're going to find cosine of theta,

Â cotangent of theta and secant of theta.

Â So were given that the terminal side of theta intersects

Â the unit circle at three fourths negative square root 7 over 4,

Â as shown in the figure here.

Â Therefore the X coordinate at this point is three fourths and

Â the Y coordinate is negative square root 7 divided by 4.

Â Therefore cosine of theta is equal to the X coordinate or 3 divided by 4 and cotangent

Â of theta is equal to the X coordinate divided by the Y coordinate which is

Â equal to three fourths divided by negative square root of 7 divided by 4,

Â and the 4's will cancel,

Â which leaves us with negative 3 divided by the square root of 7,

Â which we then can rationalize,

Â which leaves us with our answer of negative 3 square root of 7 divided by 7.

Â And finally, the secant of theta,

Â is equal to 1 divided by the X coordinate or 1

Â divided by three fourths which is equal to four thirds.

Â So these are the three values we're looking for,

Â cosine of theta is 3 divided by 4,

Â cotangent of theta is negative 3 square root of 7 divided by 7,

Â and secant of theta is 4 divided by 3.

Â And this is how we find trigonometric values given a point on the unit circle.

Â Thank you and we'll see you next time.

Â