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Study Notes 030:
Trigonometry 4:
By William E. Steinman:
In our previous study notes we began looking at the trigonometric functions. We defined them as functions of a right triangle.
We can also define the same functions in terms of the coordinates on a point on a circle. Let’s look.
P is a point on a circle of radius r.
Ө is the angle of interest.
-Ө is the same angle in the negative direction.
r is the radius of the circle
x then becomes the abscissa
y becomes the ordinate
Well, this thing looks a lot like a right triangle because it is.
Therefore:
Sin Ө = ordinate/radius = y/r
Cos Ө = abscissa/radius = x/r
Tan Ө = y/x
Csc Ө = r/y
Sec Ө= r/x
Cot Ө = x/y
Now, since it is clear that the negative angle is a mirror of our angle, these definitions hold true for the negative angle and for any other angle on the circle.
Great!
In scientific applications, certain angles tend to occur frequently. Hence, it is handy to be able to calculate the trigonometric functions simply and quickly.
The angles are 0o, 30o, 45o, 60o, 90o, 180o, 270o, and 360o.
We can easily calculate the functions for 45o using an isosceles right triangle. Remember an isosceles triangle is one having two equal sides.
If we consider a right triangle with legs equal to 1 and 1, we have an isosceles right triangle. We can notice that the other angles of this triangle must be equal and must equal 45o. That is 180o – 90o/2 = 45o.
What about 180o and 360o?
It’s kind of a trick question.
A quadrantal angle is defined as an angle where the terminal side coincides with one of the axes. This means either the ordinate or the abscissa will be 0, so it will not matter how long the other one is.
All this means is some of the functions just as in 0o and 90o will be zero or undefined.
For example, for 90o the sin will be 1, the cos will 0, and the tan will be infinite or undetermined.
For180o the sin will be 0, the cos will be -1, the tan will be 0, the cot will be undefined, the sec will be -1 and the csc will be undefined.
If you think about it, you realize that 270o is the same as – 90o. Since it is negative the ordinate will have a negative value and some of the calculations will produce negative results. So what?
Work it out and you will see.
Okay?
Enough already!