Professor Dave again, let’s learn trig identities.
We now know about the six trigonometric functions, so let’s learn about some of the identities
that apply to these functions.
First, let’s list the functions themselves.
We have sine, cosine, tangent, cosecant, secant, and cotangent.
We should recall that these three are reciprocals of the other three, so we can express these
as reciprocals of the others, or vice versa.
Another identity we should be aware of is that tangent is equal to sine over cosine.
This is true because if sine is opposite over hypotenuse, and cosine is adjacent over hypotenuse,
then the hypotenuses cancel, and we are left with opposite over adjacent, which is the
definition of tangent.
Likewise, since cotangent is the reciprocal of tangent, that means that cotangent equals
cosine over sine.
There is another identity we must know, the Pythagorean identity.
To derive this, let’s start with the Pythagorean theorem that we already know.
Now let’s divide both sides by C squared.
A squared over C squared is the same thing as A over C quantity squared, since exponents
distribute over products and quotients, so we can rewrite this like this.
From here, we can rationalize that A over C is the same as sine theta, because sine
is opposite over hypotenuse.
Similarly, B over C equals cosine theta.
So we can rewrite this as sin theta squared plus cosine theta squared equals one.
When we raise a trig function to an exponent, so that we don’t risk ambiguity as to whether
the exponent is operating on the term that the function is operating on, or if the exponent
is operating on the whole function, sine theta quantity squared can be written as sine squared
theta, with the exponent right here.
This identity will be used a lot, so it’s worth memorizing.
We can also use this to derive other useful identities.
If we divide everything by cosine squared theta, this term becomes tangent squared,
this term becomes one, and this term becomes one over cosine squared, or secant squared.
If instead, we were to divide by sin squared theta, we get one plus cotangent squared equals
cosecant squared.
So these are the three Pythagorean identities.
Next we want to understand the cofunction identities.
Let’s go back to the unit circle.
Looking at the first quadrant, we see that the same values are used for the X and Y coordinates,
just in reverse order.
In other words, the sin of zero is the cosine of ninety.
The sin of thirty is the cosine of sixty.
This continues all the way through the quadrant.
For this reason, we can say that sine of theta is equal to the cosine of ninety minus theta.
For example, the sine of thirty is one half, and that’s also equal to the cosine of ninety
minus thirty, or the cosine of sixty.
So this is a relationship that occurs between any pair of complementary angles.
Likewise, the cosine of theta is equal to the sine of ninety minus theta.
And because of these identities, we can also write similar ones relating tangent and cotangent,
as well as secant and cosecant.
Let’s just quickly list the basic identities we have highlighted in this clip for ease
of memorization.
Once you’ve properly absorbed these, let’s check comprehension.