In Section FundamentalTrigonometricIdentities, we saw the utility of identities in finding the values of the circular functions of a given angle as well as simplifying expressions involving the circular functions. In this section, we introduce several collections of identities which have uses in this course and beyond.

Our first set of identities is the ‘Even / Odd’ identities. We observed the even and odd properties of the circular functions graphically in Sections GraphsofSineandCosine and GraphsofOtherCircularFunctions. Here, we take the time to prove these properties from first principles. We state the theorem below for reference.

We start by proving \(\cos (-\theta ) = \cos (\theta )\) and \(\sin (-\theta ) = -\sin (\theta )\).

Consider an angle \(\theta \) plotted in standard position. Let \(\theta _0\) be the angle coterminal with \(\theta \) with \(0 \leq \theta _0 < 2\pi \). (We can construct the angle \(\theta _0\) by rotating counter-clockwise from the positive \(x\)-axis to the terminal side of \(\theta \) as pictured below.) Since \(\theta \) and \(\theta _0\) are coterminal, \(\cos (\theta ) = \cos (\theta _0)\) and \(\sin (\theta ) = \sin (\theta _0)\).

We now consider the angles \(-\theta \) and \(-\theta _0\). Since \(\theta \) is coterminal with \(\theta _0\), there is some integer \(k\) so that \(\theta = \theta _0 + 2\pi \cdot k\). Hence, \(-\theta = -\theta _0 - 2\pi \cdot k = -\theta _0 + 2\pi \cdot (-k)\). Since \(k\) is an integer, so is \((-k)\), which means \(-\theta \) is coterminal with \(-\theta _0\). Therefore, \(\cos (-\theta ) = \cos (-\theta _0)\) and \(\sin (-\theta ) = \sin (-\theta _0)\).

Let \(P\) and \(Q\) denote the points on the terminal sides of \(\theta _0\) and \(-\theta _0\), respectively, which lie on the Unit Circle. By definition, the coordinates of \(P\) are \((\cos (\theta _0),\sin (\theta _0))\) and the coordinates of \(Q\) are \((\cos (-\theta _0),\sin (-\theta _0))\).

Since \(\theta _0\) and \(-\theta _0\) sweep out congruent central sectors of the Unit Circle, it follows that the points \(P\) and \(Q\) are symmetric about the \(x\)-axis. Thus, \(\cos (-\theta _0) = \cos (\theta _0)\) and \(\sin (-\theta _0) = -\sin (\theta _0)\).

Since the cosines and sines of \(\theta _0\) and \(-\theta _0\) are the same as those for \(\theta \) and \(-\theta \), respectively, we get \(\cos (-\theta ) = \cos (\theta )\) and \(\sin (-\theta ) = -\sin (\theta )\), as required.

As we saw in Section GraphsofOtherCircularFunctions, the remaining four circular functions ‘inherit’ their even/odd nature from sine and cosine courtesy of the Reciprocal and Quotient Identities, Theorem recipquotidfull.

Our next set of identities establish how the cosine function handles sums and differences of angles.

We first prove the result for differences. As in the proof of the Even / Odd Identities, we can reduce the proof for general angles \(\alpha \) and \(\beta \) to angles \(\alpha _0\) and \(\beta _0\), coterminal with \(\alpha \) and \(\beta \), respectively, each of which measure between \(0\) and \(2\pi \) radians. Since \(\alpha \) and \(\alpha _0\) are coterminal, as are \(\beta \) and \(\beta _0\), it follows that \((\alpha - \beta )\) is coterminal with \((\alpha _0 - \beta _0)\). Consider the case below where \(\alpha _0 \geq \beta _0\).

Since the angles \(POQ\) and \(AOB\) are congruent, the distance between \(P\) and \(Q\) is equal to the distance between \(A\) and \(B\). The distance formula, Equation distanceformula, yields

\[ \begin{array}{rcl} \sqrt {(\cos (\alpha _0) - \cos (\beta _0))^2 + (\sin (\alpha _0) - \sin (\beta _0))^2 } & = & \sqrt {(\cos (\alpha _0 - \beta _0) - 1)^2 + (\sin (\alpha _0 - \beta _0) - 0)^2} \\ \end{array} \]

Squaring both sides, we expand the left hand side of this equation as

\[ \begin{array}{rcl} (\cos (\alpha _0) - \cos (\beta _0))^2 + (\sin (\alpha _0) - \sin (\beta _0))^2 & = & \cos ^2(\alpha _0) - 2\cos (\alpha _0)\cos (\beta _0) + \cos ^2(\beta _0) \\ & & + \sin ^2(\alpha _0) - 2\sin (\alpha _0)\sin (\beta _0) + \sin ^2(\beta _0) \\ & = & \cos ^2(\alpha _0) + \sin ^2(\alpha _0) + \cos ^2(\beta _0) + \sin ^2(\beta _0) \\ & & - 2\cos (\alpha _0)\cos (\beta _0) - 2\sin (\alpha _0)\sin (\beta _0) \end{array}\]

From the Pythagorean Identities, \(\cos ^2(\alpha _0) + \sin ^2(\alpha _0) = 1\) and \(\cos ^2(\beta _0) + \sin ^2(\beta _0) = 1\), so

\[ \begin{array}{rcl} (\cos (\alpha _0) - \cos (\beta _0))^2 + (\sin (\alpha _0) - \sin (\beta _0))^2 & = & 2 - 2\cos (\alpha _0)\cos (\beta _0) - 2\sin (\alpha _0)\sin (\beta _0) \end{array}\]

Turning our attention to the right hand side of our equation, we find

\[ \begin{array}{rcl} (\cos (\alpha _0 - \beta _0) - 1)^2 + (\sin (\alpha _0 - \beta _0) - 0)^2 & = & \cos ^2(\alpha _0 - \beta _0) - 2\cos (\alpha _0 - \beta _0) + 1 + \sin ^2(\alpha _0 - \beta _0) \\ & = & 1 + \cos ^2(\alpha _0 - \beta _0) + \sin ^2(\alpha _0 - \beta _0) - 2\cos (\alpha _0 - \beta _0) \\ \end{array} \]

Once again, we simplify \(\cos ^2(\alpha _0 - \beta _0) + \sin ^2(\alpha _0 - \beta _0)= 1\), so that

\[ \begin{array}{rcl} (\cos (\alpha _0 - \beta _0) - 1)^2 + (\sin (\alpha _0 - \beta _0) - 0)^2 & = & 2 - 2\cos (\alpha _0 - \beta _0) \\ \end{array} \]

Putting it all together, we get \(2 - 2\cos (\alpha _0)\cos (\beta _0) - 2\sin (\alpha _0)\sin (\beta _0) = 2 - 2\cos (\alpha _0 - \beta _0)\), which simplifies to: \(\cos (\alpha _0 - \beta _0) = \cos (\alpha _0)\cos (\beta _0) + \sin (\alpha _0)\sin (\beta _0)\).

Since \(\alpha \) and \(\alpha _0\), \(\beta \) and \(\beta _0\), and \((\alpha - \beta )\) and \((\alpha _0- \beta _0)\) are all coterminal pairs of angles, we have established the identity: \(\cos (\alpha - \beta ) = \cos (\alpha ) \cos (\beta ) + \sin (\alpha ) \sin (\beta )\).

For the case where \(\alpha _0 \leq \beta _0\), we can apply the above argument to the angle \(\beta _0 - \alpha _0\) to obtain the identity \(\cos (\beta _0 - \alpha _0) = \cos (\beta _0)\cos (\alpha _0) + \sin (\beta _0)\sin (\alpha _0)\). Using this formula in conjunction with the Even Identity of cosine gives us the result in this case, too:

\[ \begin{array}{rcl} \cos (\alpha _0 - \beta _0) = \cos ( - (\alpha _0 - \beta _0)) = \cos (\beta _0 - \alpha _0) & = & \cos (\beta _0)\cos (\alpha _0) + \sin (\beta _0)\sin (\alpha _0) \\ & = & \cos (\alpha _0)\cos (\beta _0) + \sin (\alpha _0)\sin (\beta _0). \end{array} \]

To get the sum identity for cosine, we use the difference formula along with the Even/Odd Identities

\[ \cos (\alpha + \beta ) = \cos (\alpha - (-\beta )) = \cos (\alpha ) \cos (-\beta ) + \sin (\alpha ) \sin (-\beta ) = \cos (\alpha ) \cos (\beta ) - \sin (\alpha ) \sin (\beta ). \]

We put these newfound identities to good use in the following example.

The identity verified in Example cosinesumdiffex, namely, \(\cos \left (\frac {\pi }{2} - \theta \right ) = \sin (\theta )\), is the first of the celebrated ‘cofunction’ identities. These identities were first hinted at in Exercise cofunctionforeshadowing in Section AppRightTrig.

From \( \sin (\theta ) = \cos \left (\frac {\pi }{2} - \theta \right ) \), we get: \(\sin \left (\frac {\pi }{2} - \theta \right ) = \cos \left (\frac {\pi }{2} -\left [\frac {\pi }{2} - \theta \right ]\right ) = \cos (\theta )\), which says, in words, that the ‘co’sine of an angle is the sine of its ‘co’mplement. Now that these identities have been established for cosine and sine, the remaining circular functions follow suit. The remaining proofs are left as exercises.

The Cofunction Identities enable us to derive the sum and difference formulas for sine. We first convert to sine to cosine and expand:

\[ \begin{array}{rcl} \sin (\alpha + \beta ) & = & \cos \left ( \frac {\pi }{2} - (\alpha + \beta ) \right ) \\ & = & \cos \left ( \left [ \frac {\pi }{2} - \alpha \right ] - \beta \right ) \\ & = & \cos \left ( \frac {\pi }{2} - \alpha \right ) \cos (\beta ) + \sin \left ( \frac {\pi }{2} - \alpha \right )\sin (\beta ) \\ & = & \sin (\alpha ) \cos (\beta ) + \cos (\alpha ) \sin (\beta ) \\ \end{array} \]

We can derive the difference formula for sine by rewriting \(\sin (\alpha - \beta )\) as \(\sin (\alpha + (-\beta ))\) and using the sum formula and the Even / Odd Identities. Again, we leave the details to the reader.

We try out these new identities in the next example.

The formula developed in Exercise sinesumanddiffex for \(\tan (\alpha + \beta )\) can be used to find a formula for \(\tan (\alpha - \beta )\) by rewriting the difference as a sum, \(\tan (\alpha + (-\beta ))\) and using the odd property of tangent. (The reader is encouraged to fill in the details.) Below we summarize all of the sum and difference formulas.

In the statement of Theorem circularsumdifference, we have combined the cases for the sum ‘\(+\)’ and difference ‘\(-\)’ of angles into one formula. The convention here is that if you want the formula for the sum ‘\(+\)’ of two angles, you use the top sign in the formula; for the difference, ‘\(-\)’, use the bottom sign. For example,

\[\tan (\alpha - \beta ) = \frac {\tan (\alpha ) - \tan (\beta )}{1 + \tan (\alpha ) \tan (\beta )}\]

If we set \(\alpha = \beta \) in the sum formulas in Theorem circularsumdifference, we obtain the following ‘Double Angle’ Identities:

The three different forms for \(\cos (2\theta )\) can be explained by our ability to ‘exchange’ squares of cosine and sine via the Pythagorean Identity. For instance, if we substitute \(\sin ^{2}(\theta ) = 1 - \cos ^{2}(\theta )\) into the first formula for \(\cos (2\theta )\), we get \(\cos (2\theta ) = \cos ^{2}(\theta ) - \sin ^{2}(\theta ) = \cos ^{2}(\theta ) - (1 - \cos ^{2}(\theta )) = 2 \cos ^{2}(\theta ) - 1\).

It is interesting to note that to determine the value of \(\cos (2\theta )\), only one piece of information is required: either \(\cos (\theta )\) or \(\sin (\theta )\). To determine \(\sin (2\theta )\), however, it appears that we must know both \(\sin (\theta )\) and \(\cos (\theta )\). In the next example, we show how we can find \(\sin (2\theta )\) knowing just one piece of information, namely \(\tan (\theta )\).

In the last problem in Example doubleangleex, we saw how we could rewrite \(\cos (3\theta )\) as sums of powers of \(\cos (\theta )\). In Calculus, we have occasion to do the reverse; that is, reduce the power of cosine and sine.

Solving the identity \(\cos (2\theta ) = 2\cos ^{2}(\theta ) -1\) for \(\cos ^{2}(\theta )\) and the identity \(\cos (2\theta ) = 1 - 2\sin ^{2}(\theta )\) for \(\sin ^{2}(\theta )\) results in the aptly-named ‘Power Reduction’ formulas below.

Our next example is a typical application of Theorem powerreduction that you’ll likely see in Calculus.

Another application of the Power Reduction Formulas is the Half Angle Formulas. To start, we apply the Power Reduction Formula to \(\cos ^{2}\left (\frac {\theta }{2}\right )\)

\[ \cos ^{2}\left ( \frac {\theta }{2}\right ) = \frac {1 + \cos \left (2 \left (\frac {\theta }{2}\right )\right )}{2} = \frac {1 + \cos (\theta )}{2}.\]

We can obtain a formula for \(\cos \left (\frac {\theta }{2}\right )\) by extracting square roots. In a similar fashion, we may obtain a half angle formula for sine, and by using a quotient formula, obtain a half angle formula for tangent.

We summarize these formulas below.

Our next batch of identities, the Product to Sum Formulas, are easily verified by expanding each of the right hand sides in accordance with Theorem circularsumdifference and as you should expect by now we leave the details as exercises. They are of particular use in Calculus, and we list them here for reference.

Related to the Product to Sum Formulas are the Sum to Product Formulas, which we will have need of in Section TrigonometricEquationsandInequalities. These are essentially restatements of the Product to Sum Formulas (by re-labeling the arguments of the sine and cosine functions) and as such, their proofs are left as exercises.

The reader is reminded that all of the identities presented in this section which regard the circular functions as functions of angles (in radian measure) apply equally well to the circular (trigonometric) functions regarded as functions of real numbers.

1 Sinusoids, Revisited

We first studied sinusoids in Section ??. Using the sum formulas for sine and cosine, we can expand the forms given to us in Theorem ??:

\[ S(t) = A \sin (\omega t + \phi ) + B = A\sin (\omega t) \cos (\phi ) + A \cos (\omega t)\sin (\phi ) + B,\]

and

\[C(t) = A \cos (\omega t + \phi ) + B = A\cos (\omega t) \cos (\phi ) - A \sin (\omega t) \sin (\phi ) + B.\]

As we’ll see in the next example, recognizing these ‘expanded’ forms of sinusoids allows us to graph functions as sinusoids which, at first glance, don’t appear to fit the forms of either \(C(t)\) or \(S(t)\).

A couple of remarks about Example 7 are in order. First, had we chosen \(A = -2\) instead of \(A = 2\) as we worked through Example 7, our final answers would have looked different. The reader is encouraged to rework Example 7 using \(A = -2\) to see what these differences are, and then for a challenging exercise, use identities to show that the formulas are all equivalent.

It is important to note that in order for the technique presented in Example 7 to fit a function into one of the forms in Theorem ??, the frequencies of the sine and cosine terms much match. For example, in the Exercises, you’ll be asked to write \(f(t) = 3\sqrt {3}\sin (3t) - 3\cos (3t)\) in the form of \(S(t)\) and \(C(t)\) above, and since both the sine and cosine terms have frequency \(3\), this is possible.

However, a function such as \(f(t) = \sin (t) - \sin (3t)\) cannot be written in the form of \(S(t)\) or \(C(t)\). The quickest way to see this is to examine its graph below which is decidedly not a sinusoid. That being said, we can still analyze this curve using identities.

Using our result from number 2 Example 6, we may rewrite \(f(t) = \sin (t) - \sin (3t) = -2 \sin (t) \cos (2t)\). Grouping factors, we can view \(f(t) = [ -2 \sin (t) ] \cos (2t) = A(t) \cos (2t)\) as the curve \(y = \cos (2t)\) with a variable amplitude, \(A(t) = -2 \sin (t)\).

Overlaying the graphs of \(f(t)\) with the (dashed) graphs of \(A_{1}(t) = 2 \sin (t)\) and \(A_{2}(t) = -2 \sin (t)\), we can see the role these two curves play in the graph of \(y = f(t)\). They create a kind of ‘wave envelope’ for the graph of \(y = f(t)\). This is an example of the beats phenomenon. Note that when written as a product of sinusoids, it is always the lower frequency factor which creates the ‘wave-envelope’ of the curve.

Note that in order to rewrite a sum or difference of sine and cosine functions with different frequencies into a product using the sum to product identities, Theorem 10, we need the amplitudes of each term to be the same. We explore more examples of these functions and this behavior in the Exercises.