MTH5P1: Trigonometric Identities

This unit explains Trigonometric Identities

Extending the identities

for a and b positive acute angles. Is this identity still true for any angle at all?

Using the definition from Chapter 4, we can see that this formula works, for example, when a = 150° and b = 300°. And in fact, it will always work for angles of any size: Why is this true?

The Principle of Analytic Continuation: Higher mathematics to the rescue

Checking the formula for sin (a+ b) for general angles becomes very tedious. You can try it for other angles, reducing each sine or cosine to a function of a positive acute angle. But pack a lunch, because such a procedure takes a long time.
For this situation, a theorem from higher mathematics comes to our rescue. Called the Principle of Analytic Continuation, it says, roughly, that most of our identities will be preserved under the new definitions of the
trigonometric functions.
More precisely, the Principle of Analytic Continuation says that any identity involving rational trigonometric functions that is true for positive acute angles is true for any angle at all.
Since a proof of this statement will involve results from a course in calculus and another in complex analysis, we will only state this principle here. But to understand the statement above, we must explore some terminology.
A rational trigonometric function is a function you can get by taking the sine and cosine of various angles, together with all the constant functions, and adding, subtracting, multiplying, or dividing them.

Some examples of rational trigonometric functions are:

Some of our examples should seem familiar to you. In fact, you can check that most of our identities so far have involved rational trigonometric functions.
The Principle of Analytic Continuation tells us that if two such trigonometric rational functions are equal for numbers in any one interval (all the numbers between two real numbers) then they are equal for any numbers.
For example, in our list above of rational trigonometric functions, we have the examples sin (a + b) and sin a cos b + cos a sin b. Using geometry, we have already proved (three times!) that these two functions are equal for o < a, b < 45° (so that a, b, and a+ b are all acute angles).
The Principle of Analytic Continuation says that these two functions must

In the same way, if you start with integers, you can get all the rational numbers by adding, subtracting, multiplying, and dividing.

Then be equal for any values of a and b, and not just for the ones in the interval between 0 and 45°.

Back to our identities
You may imagine that a general statement such as the Principle of Analytic Continuation (and the full statement of this principle is even more  general!) must have its roots in rather deep properties of functions. And in fact it does. This is why one needs to follow two advanced courses of mathematics before understanding it fully.

So we can continue to work with our identities, with the assurance of  the mathematicians, who have proved the Principle of Analytic Continuation, that our work is valid for angles of any measure, and not just for positive acute angles.
Here, once again, are our formulas. We repeat them to emphasize their added meaning. Because of the Principle of Analytic Continuation, they are true for angles of any measure, and not just acute angles:

A formula for tan (a+ b).

In a way, this is nicer than the formula for sin (a + b) and cos (a+ b), since it uses only the tangents of a and b. The formula for sin (a+ b), on the other hand, uses cos a and cos {3 as well as sin a and sin b.

Double the angle
If we know sin a and cos a, we can find the value of sin 2a and cos 2a. We know that sin (a + b) = sin a cos b +cos a sin b

(The reader should check that the other two formulas for cos 2a lead to the same answer.)

The reader should check that in fact there are values of a for which each of our two answers is correct. If we are given the value of cos a, then the value of cos 2a is determined, but the value of sin 2a is not. (And certainly the value of a itself is not determined.)
The “double angle” formulas are often used in the following form. If we write a = 2b, then b = a/2, and we have:

Triple the angle
Let us now find formulas for sin 3a and cos 3a. We can write

(The reader should check the details.) In the same way, we can show that 

Derivation of the formulas for sin a /2 and cos a /2
Let us now derive formulas for sin a /2 and cos a /2 in terms of trigonometric functions of a.
We being with the formula for cos a in terms of cos a/2.

In the next section, we will see two formulas for tan(a/2) that are more convenient.

Another formula for tan a /2

So we have another formula for tan(a/2), without radicals, but with an ambiguous sign. But in fact there is a small miracle here: we don’t need the ambiguous sign! This miracle can easily be understood by looking at analytic continuation.

If the angle is positive and acute, that is, between 0″ and 90″, we must select the positive sign. In other words, in this case we have

(without the ambiguous sign). Unlike the formula we started with, this new formula is a rational trigonometric expression, so the Principle of Analytic Continuation guarantees that in fact the equation is true for any angle.

Products to sums

We can get some further useful results by working with the formulas for sin (a+ b) and cos (a+ b). For example, we can write cos (a + b) +cos (a – b) = 2 cos a cos b.
This simple yet remarkable formula says that the sum of the cosines of two angles can be written as the product of the cosines of two other angles.
Perhaps this is clearer if we write it as follows:

So the cosine function, in a rather complicated way, “converts” products to. sums. You may know that the logarithm function also “converts” products to sums, although in a much simpler fashion. In fact, people used to use cosine tables, like logarithm tables, to perform tedious multiplications by turning them into addition. If you study complex analysis you will learn of the rather deep relationship between the trigonometric functions and the exponential or logarithmic functions. In the same way, we can write

Sums to products
It is sometimes useful to convert sums of sines and cosines to products.
The following series of examples shows how this can be done.

Adding, we find that sin (y + &) + sin (y – &) = 2 sin y cos &, which represents a factored form of the given expression.

Example 52 A bottle and a cork together cost $1.10. The bottle costs $1 more than the cork. How much does the cork cost?
Solution. It is tempting to say immediately that the bottle costs $1 and the cork costs 10 cents, but this is incorrect. With those prices, the bottle would cost only 90 cents more than the cork. ·
Algebra will quickly supply the correct answer. If the price of the bottle is b, and the price of the cork is c, then we have

We may solve for b and c by adding these two equations. We find that 2b = 2.1, sob = 1.05. Using this result, we know how to calculate the value of c from either equation. For example, using the first equation, we obtain c = 1.1- b = 1.1- 1.05 = 0.05 .
Thus, the bottle costs $1.05 and the cork costs 5 cents.

Example 53 If x + y = a and x – y = b, express x and y separately in terms of a and b.
Solution. Proceeding as in the problem with the bottle and the cork,

Please remember this result. It will be useful in many applications of algebra and trigonometry, and not just in problems about bottles (of undetermined contents).

Example 54 Write the expression sin a + sin b as a product of sines and cosines.
Solution. With the experience of the previous examples, this is not difficult to do. We may use

Example 51 if we can find angles y and 8 such that y + & =a andy- & = b. Example 53 shows us how to do this. We
just need to choose

Substituting into the result of Example 51, we obtain the useful formula

We may also express the difference sin a – sin b as a product of sines and cosines. We use the angles y and 8 found before, such that y + & = a and y – & = b, and write
sin a – sin f3 = sin (y + &) -sin (y – &) = 2 cosy sin &.
We now express this result in terms of the original variables a and b, and find that

Expressions for sin b, cos b, and tan b in terms of tan b /2.
We can use our results in trigonometry to obtain some results in number theory. Let us begin by reviewing some results obtained earlier.

Uniformization of sin a, cos a, and tan a, We can rewrite our new identities by letting a = 2b:

These formulas provide a uniformization of the trigonometric: functions.
That is, they allow us to represent all these functions using rational expressions of a single function, tan a /2. So, for instance, if we have a trigonometric identity, or an equation involving trigonometric functions, we can rewrite these functions as rational functions of this single variable.
Then the trigonometric equation or identity becomes an algebraic equation or identity.
While this may be important theoretically, it rarely makes things easier when we have an actual problem to solve. However, this uniformization yields some very interesting results in a most unexpected area. We can use it to find Pythagorean triples: solutions in natural numbers to the equation;

We know that if the numbers a, b, and c form a Pythagorean triple, then there is a right triangle with legs a and band hypotenuse c. Then each acute angle of this triangle has a rational sine, cosine, and tangent. For example, we are familiar with the fact that the numbers 3, 4, and 5 satisfy

We can use our uniformization to find other triangles with angles whose sine, cosine, and tangent are rational by following this process backwards.
If we let tan a /2 be some rational number, then our uniformization tells us that sina, cosa, and tana will also be rational. We can then form a right triangle with rational sides and, by scaling it up, we can form a right triangle with integer sides. The sides of this triangle will be a Pythagorean triple.
For example, let

Then we have

In this case, a right triangle with an acute angle a can have sides 12/ 13, 5/13, and 1. Multiplying each side by 13, we form a similar right triangle with sides 12, 5, and 13. Because they are the sides of a right triangle, these

Themes and variations
We return to a theme that we introduced in Chapter 1, and develop it more fully.
Theme: The maximum value of sin x cos x Variation 1: Find the largest possible value of the expression sin x cos x. Certainly sin x cos x < 1, since both sin x and cos x are at most 1 (and cannot be equal to 1 for the same angle). But is this the best estimate?

which, as we already know, is correct (note that we choose the positive sign for the radical).
Similarly, we have

Note that the expressions we get contain “nested radicals.”

Trigonometric series
In this section we use the identities we have learned to the find the sum of series whose terms involve trigonometric expressions. This topic turns out to be of great importance in later work.
We introduce some of the techniques used by first looking at some purely algebraic problems.

Of course, if you already know the general formula for the sum of a geometric progression, this result is not unexpected. But if you don’t already know the general formula for the sum of a geometric progression, you have essentially learned it above: the general case will work in just the same way.
The key to this trick is forming a “telescoping” sum: a sum of terms in which many pairs add up to zero.

Summing a trigonometric series

We would like to find the sum of the series S = sin x + sin 2x + sin 3x + · · · + sin nx .
We can form a telescoping sum, as in Example 58 above. The trick is to multiply by 2 sin (x/2):

This technique is quite general, and can be used to sum the sines or cosines of angles which are in arithmetic progression. We can find a general formula for the sum

Similarly, we can find a general formula for the sum
S = cos x + cos (x + a) + cos (x + 2a) + · · · + cos (x + na) , again by multiplying by 2 sin a /2, and using the identity
2 cos A sin B = sin (A + B) – sin (A – B) .

In the following exercises, we recommend using the hints provided, then checking the results by applying the formulas directly.

ASSIGNMENT : Trigonometry Assignment MARKS : 50  DURATION : 1 week, 3 days

 

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