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Study Notes 031:
Trigonometry 5:
By William E. Steinman:
Cofunctions are pretty much what the name implies.
For example, the side opposite of 300 is the side adjacent of 600.
Therefore, the sin of 300 is the same as the cos of 600.
Both are equal to 0.5
Sin = side opposite/hypotenuse and cos = side adjacent/hypotenuse.
Likewise the sin 0f 290 is equal to the cos of 610.
Both are equal to 0.48481.
Check the table in Study Notes 27 to verify this.
What does this mean.
It means that the sin and cos are cofunctions as are the secant and cosecant. Also the tangent and cotangent are cofunctions.
If we wanted to know the cot of 270 we could subtract 270 from 900 and find the tangent of the resultant angle.
In equation form, cot Ө = tan (900 - Ө)
We said previously that the secant, cosecant, and cotangent are implicit in our table even though they are not listed directly.
Here is how.
We have some basic relationships for the trigonometric functions that allow us to find the missing values.
First we have some reciprocal relationships.
csc Ө = 1/sin Ө
sec Ө = 1/cos Ө
cot Ө = 1/tan Ө
How do we know this?
Let’s go back to the basic trigonometric functions.
We can take our first relationship as an example.
csc Ө = 1/sin Ө
We know that the csc is defined as the hypotenuse/side opposite.
Also sin = side opposite/hypotenuse.
Substituting we can get
hypotenuse/side opposite = 1/side opposite/hypotenuse
Remember your algebra. To divide we can invert and multiply.
So hypotenuse/side opposite = hypotenuse/side opposite.
Ain’t science wonderful?
If we wanted to find the csc of 300 we can look up the sin of 300 and take the reciprocal to get the csc.
In this case the csc 300 = 1/sin 300 = 1/0.5 = 2
You can work out the other proofs without my help.
We also have quotient relationships.
These are
tan Ө = sin Ө /cos Ө
cot Ө = cos Ө /sin Ө
Again, the proofs of these are implicit in the basic trigonometric functions.
You can work it out if you wish.
Finally we have the Pythagorean relationships.
These are
sin2 Ө + cos2 Ө = 1
1 + tan2 Ө = sec2 Ө
1 + cot2 Ө = csc2 Ө
Let’s work out the first one.
sin Ө = side opposite/ hypotenuse, we can use shorthand and make this (so/h)
cos Ө = side adjacent/ hypotenuse or (sa/h)
Substituting we get (so/h)2 + (sa/h)2 = 1
Or so2/h2 + sa2/h2 = 1
(so2 + sa2)/h2 = 1
so2 + sa2 = h2
But, this is the original Pythagorean’s theorem that we know and love.
The square of the hypotenuse of a right triangle is equal to the sum of the squares of the two sides.
It proves out.
These basic relationships are called identities.
The identities are valid for all values of the angle for which the functions are defined.
In addition to the eight identities already defined we have two more.
These are:
cos Ө csc Ө = cot Ө
cos Ө tan Ө= sin Ө
Lets check out the easy one for 300.
cos 300 = 0.86603
tan 300 = 0.57735
0.57735 x 0.86603 = 0.5000, which is the sin of 300.
We should notice that these calculations will only be valid to the accuracy of the table we are using. They are valid for most practical purposes.
What is all this crap good for?
Later, if we get far enough into science we may end up with some pretty hairy equations. If they involve the trigonometric functions, we can validate them through manipulation of the equations whereby we can reduce them using our basic identities.
For example, you might get an equation like this:
Tan Ө + 2 cot Ө = (sin2 Ө + 2 cos2 Ө)/ sin Ө cos Ө
Is it true?
Lets reduce it. For this I will leave out the symbol for the angle and just use the functions.
First we can manipulate the fractions in the right side of the equation.
(sin2 + 2 cos2)/sin cos = sin2/(sin cos) + 2cos2/(sin cos)
this reduces to become sin/cos + 2 cos/sin
so tan + 2cot = sin/cos + 2cos/sin
But these are the basic identities that you just proved above.
tan = sin/cos
cot = cos/sin
This requires a large fanfare!
Enough for today!