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

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

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

negative 20 over 29.

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

tangent of theta, and cosecant of theta.

Now we define the trigonometric functions in terms of

coordinates of points on the unit circle as follows.

Suppose that theta is an angle in standard position,

whose terminal side intersects the unit circle at X Y.

Then the six trigonometric functions are defined as follows.

Sine of theta is equal to the Y coordinate,

cosine of theta is equal to the X coordinate,

tangent of theta is equal to Y divided by X,

cosecant of theta is equal to 1 divided by Y,

secant of theta is equal to 1 divided by X,

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

Now these definitions are consistent with

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

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

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

Now this length then is Y,

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

Using our trig ratios,

we have that sine of theta is equal to opposite,

over hypotenuse which is equal to Y divided by 1,

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

that the sine of theta is equal to Y?

And cosine of theta is equal to adjacent,

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

which looking over here is exactly what this says.

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

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

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

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

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

tangent of theta, and cosecant of theta.

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

which is negative 20 divided by 29,

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

which is equal to negative 20 divided by 29,

divided by negative 21 divided by 29,

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

as well as the negatives,

and we get 20 divided by 21.

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

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

which is equal to negative 29 divided by 20.

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

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

All right, let's look at another example.

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

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

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

cotangent of theta and secant of theta.

Again we are going to use the following definition,

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

cotangent of theta and secant of theta.

So were given that the terminal side of theta intersects

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

as shown in the figure here.

Therefore the X coordinate at this point is three fourths and

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

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

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

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

and the 4's will cancel,

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

which we then can rationalize,

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

And finally, the secant of theta,

is equal to 1 divided by the X coordinate or 1

divided by three fourths which is equal to four thirds.

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

cosine of theta is 3 divided by 4,

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

and secant of theta is 4 divided by 3.

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

Thank you and we'll see you next time.