Trigonometric Identities (Revision : 1.4)
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Trigonometric Identities you must remember
The “big three” trigonometric identities are
\(\large \begin{equation} \sin^{2} t + cos^{2} t = 1 \tag{1} \end{equation}\)
\(\large \begin{equation} \sin(A + B) = \sin A \cos B + \cos A \sin B \tag{2} \end{equation}\)
\(\large \begin{equation} \cos(A + B) = \cos A \cos B − \sin A \sin B \tag{3} \end{equation}\)
Using these we can derive many other identities. Even if we commit the other useful identities to memory, these three will help be sure that our signs are correct, etc.
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Two more easy identities
From equation (1) we can generate two more identities. First, divide each term in (1) by \(cos^{2} t\) (assuming it is not zero) to obtain
\(\large \begin{equation} \tan^{2} t + 1 = \sec^{2} t \tag{4} \end{equation}\)
When we divide by \(sin^2 t\) (again assuming it is not zero) we get
\(\large \begin{equation} 1 + cot^{2} t = csc^{2} t \tag{5} \end{equation}\)
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Identities involving the difference of two angles
From equations (2) and (3) we can get several useful identities. First, recall that
cos(−t) = cos t, sin(−t) = − sin t. (6)
From (2) we see that
sin(A − B) = sin(A + (−B))
= sin A cos(−B) + cos A sin(−B)
which, using the relationships in (6), reduces to
sin(A − B) = sin A cos B − cos A sin B. (7)
In a similar way, we can use equation (3) to find
which simplifies to
cos(A − B) = cos(A + (−B))
= cos A cos(−B) − sin A sin(−B)
cos(A − B) = cos A cos B + sin A sin B. (8)
Notice that by remembering the identities (2) and (3) you can easily work out the signs in these last two identities.
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4 Identities involving products of sines and cosines
If we now add equation (2) to equation (7)
sin(A−B) = sinAcosB−cosAsinB +(sin(A+B) = sinAcosB+cosAsinB)
we find
and dividing both sides by 2 we obtain the identity
sin A cos B = 21 sin(A − B) + 12 sin(A + B). (9) In the same way we can add equations (3) and (8)
cos(A−B) = cosAcosB+sinAsinB +(cos(A+B) = cosAcosB−sinAsinB)
sin(A − B) + sin(A + B) = 2 sin A cos B
to get
which can be rearranged to yield the identity
cos A cos B = 21 cos(A − B) + 12 cos(A + B). (10)
Suppose we wanted an identity involving sin A sin B. We can find one by slightly modi- fying the last thing we did. Rather than adding equations (3) and (8), all we need to do is subtract equation (3) from equation (8):
cos(A−B) = cosAcosB+sinAsinB −(cos(A+B) = cosAcosB−sinAsinB)
This gives
or, in the form we prefer,
cos(A − B) − cos(A + B) = 2 sin A sin B
cos(A − B) + cos(A + B) = 2 cos A cos B
sin A sin B = 12 cos(A − B) − 12 cos(A + B). (11) 5 Double angle identities
Now a couple of easy ones. If we let A = B in equations (2) and (3) we get the two identities
sin2A = 2sinAcosA, (12) cos2A = cos2 A−sin2 A. (13)
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6 Identities for sine squared and cosine squared
If we have A = B in equation (10) then we find
cosAcosB = 12cos(A−A)+12cos(A+A) cos2A = 12cos0+12cos2A.
Simplifying this and doing the same with equation (11) we find the two identities
cos2A = 12(1+cos2A), (14)
sin2A = 12(1−cos2A). (15) 7 Identities involving tangent
Finally, from equations (2) and (3) we can obtain an identity for tan(A + B): tan(A+B)= sin(A+B) = sinAcosB+cosAsinB.
cos(A + B) cos A cos B − sin A sin B
Now divide numerator and denominator by cos A cos B to obtain the identity we wanted:
tan(A+B)= tanA+tanB . (16) 1−tanAtanB
We can get the identity for tan(A − B) by replacing B in (16) by −B and noting that tangent is an odd function:
tan(A−B)= tanA−tanB . (17) 1+tanAtanB
8 Summary
There are many other identities that can be generated this way. In fact, the derivations above are not unique — many trigonometric identities can be obtained many different ways. The idea here is to be very familiar with a small number of identities so that you are comfortable manipulating and combining them to obtain whatever identity you need to.