**POINT OF INTEREST 2
VALUES OF TRIGONOMETRIC RATIOS OF ANGLES OF ANY MAGNITUDE
Copyright by Ingrid Stewart, Ph.D. Please Send Questions and Comments to ingrid.stewart@csn.edu.**

Let's place a right triangle into the *Rectangular Coordinate
System s*o that the right angle lies opposite the origin.

Observe
that the length of leg * a* and the length of leg

For the subsequent discussion, we will use a **30-60-90** triangle. Specifically, we will draw right triangles containing a * 30^{o}* and

NOTE: We could have also used a **45-45-90** triangle.

Specific to our discussion, we will let the length of the *side adjacent* to angle * 60^{o}* equal

As we can see, these triangles produce the points , , , and .

Now, instead of using the sides of the triangles to define the values of the cosine, sine, and tangent ratios, we will use the four points stated above.

Their their* x-*coordinates will be equivalent to the *sides adjacent*, and their *y*-coordinates will be equivalent to the *sides opposite* the * 60^{o}* angle.

Angle

60^{o}Its initial side is the positive

x-axis, and we define its terminal side to be the line through the point in Quadrant I.Using

andadj 1, we findhyp 2=cos 60^{o}.Using

andopp, we findhyp 2=sin 60^{o}.Using

andopp, we findadj 1=tan 60^{o}.The secant, cosecant, and cotangent will all be positive.

Angle

120^{o}Its initial side is the positive

x-axis, and we define its terminal side to be the line through the point in Quadrant II.Using

andadj 1, we findhyp 2=cos 120^{o}.Using

andopp, we findhyp 2=sin 120^{o}.Using

andopp, we findadj 1=tan 120^{o}.The secant and cotangent will be negative and the cosecant will be positive.

Angle

.240^{o}Its initial side is the positive

x-axis, and we define its terminal side to be the line through the point in Quadrant III.Using

andadj 1, we findhyp 2=cos 240^{o}.Using

andopp, we findhyp 2=sin 240^{o}.Using

andopp, we findadj 1=tan 240^{o}.The secant and cosecant will be negative and the cotangent will be positive.

Angle

.300^{o}Its initial side is the positive

x-axis, and we define its terminal side to be the line through the point in Quadrant IV.Using

andadj 1, we findhyp 2=cos 300^{o}.Using

andopp, we findhyp 2=sin 300^{o}.Using

andopp, we findadj 1=tan 300^{o}.The cosecant and cotangent will be negative and the secant will be positive.

Do you notice in the above that some values are

negative? As a matter of fact, we can actually establish the following pattern for any angle.Do you further notice in the above that the multiples of angle

produce values equal to the60^{o}absolute value of the values in QI? This finding allows us to understand how some values of trigonometric ratios are connected.