Addition and Subtraction Formulas
We shall turn our attention to some useful formulas for the addition and subtraction of trigonometric functions. It would be nice if we could just add trig functions like we add numbers. Unfortunately, adding and subtracting trig functions is not as simple as just adding the individual pieces. It can be verified that cos(45°) + cos(45°) ≠ cos(90°). (Recall, and cos(90°) = 0.) As a result, we need to develop a more sophisticated formula. For that, we turn to Geometry.
We begin with two right triangles (BAC and CAD) stacked on top of each other, the first with angle x and the second with angle y. The vertices have been labeled.
From our earlier discussion on the properties of right triangles, we know that the length AB is equal to cos(x) and the length CB is equal to sin(x).
Also, we can see that the length AC is cos(y) and the length CD is sin(y). We shall return to this later on.
Next, we draw a line from D down to the segment AB so that it meets at a right angle. (The right triangle is not drawn.)
Then we draw a line from C to the line that we just made. These two lines meet at right angles as well.
Also, we shall label the new vertices created. The lines just created appear in red.
Recall from Geometry, we know that angle FCE is also equal to x. (Given two parallel line segments, CE and AB and a transversal AC, then the alternate interior angles will be the same.)
Since ACD is a right triangle, that means that angle ECD is equal to 90º - x, which in turn makes angle CDE equal to x (because triangle DEC is a right triangle.) These are labeled in the picture to the left.
Now, notice that will be sin(x + y). And likewise, will be cos(x + y). To determine sin(x + y), notice that .
(We have used the fact that the length of EG is equal to the length of CB.)
The next step is the trickiest. Let us multiply the first term by by and the second term by . Doing this does not change the value of , but it does allow us to relate the lengths CB and DE to known quantities, namely sin(x), sin(y), cos(x) and cos(y).
Doing this multiplication, we have . Rewriting, we get , which is equal to sin(x) cos(y) + cos(x) sin(y).
And so, we see that sin(x + y) = sin(x) cos(y) + cos(x) sin(y).
For cos(x + y), notice that .
(Similar to above, we used the fact that BG is the same thing as EC.)
Just as before, we multiply the first term by and the second term by . This gives us the following: . Rewriting, we get , which is equal to cos(x) cos(y) − sin(x) sin(y).
And so, cos(x + y) = cos(x) cos(y) − sin(x) sin(y).
Now, we want to derive the subtraction formulas for sine and cosine. We can use the above formulas if we recall that sine is an odd function and cosine is an even function. That is, sin(-x) = -sin(x) and cos(-x) = cos(x).
We see that . Since sine is odd, we have that .
We see that . Since cosine is even, we have that .
Let us summarize the results of the previous two pages in the following table. These formulas will be used quite extensively, so it is worthwhile to spend a few minutes looking them over before continuing.
Addition and Subtraction Formulas
Pay careful attention to the signs of the above formulas. It is rather easy to change a sign somewhere and get the wrong answer.
The best way to remember them is to remember that sine equals sine-cosine, cosine-sine. And since we are dealing with the sine function, the “sign” inside goes between two quantities. For cosine, we have cosine-cosine, sine-sine and the sign is the opposite.
Before we go too far, let us verify that this formula actually works. Let us consider the following, simple example.
Verify that sin(90°) = 1 by finding sin(x + y) if x = 30° and y = 60°.
Using the formula for sin(x + y), we have that sin(90°) = sin(30°)cos(60°) + cos(30°)sin(60°). Recall that .
Plugging those values in, we have , as expected.
We can use the addition and subtraction formulas to find exact values for trig functions of many angles. In particular an exact (real) expression for any angle that is a multiple of 3° can be found. The reason the others cannot be written in such a way is due to difficulties solving cubic equations. However, this goes beyond the scope of this book.
Let us move on to an example that is relevant, though. We shall return to this example again in the next section when we examine half angles.
Find an exact expression for cos(15°) and sin(15°).
Notice that we need to find someway to express 15°. After thinking for a moment, we see that we can use 45° 30° = 15°. That means that we should use the subtraction formulas.
So far, we have limited our attention to the addition and subtraction formulas for sine and cosine. But there are also formulas for tangent. They can be developed from the technique at the beginning of the section, but it might be easier to remember if they are derived from the definition of tangent.
Recall that tan(θ) = sin(θ) / cos(θ). That means that tan(x + y) = sin(x + y) / cos(x + y). We can simplify the expression into a more compact formula by dividing all terms by cos(x)cos(y) and then simplifying. Something similar is done for tan(x y).
Addition and Subtraction Formulas
When using the addition and subtraction formulas, it is wise to pay attention to what quadrant our angle should be in. Recall that when we first introduced sine, cosine and tangent, we used a table to show in which quadrants they are positive and negative. The table is included here again.
The table can be useful in checking that you have a plausible answer. If, for example, two angles add to an angle which is in the fourth quadrant, we expect sine to be negative.
But if our calculations reveal that it is positive, we should go back through and check our work for mistakes. Or, perhaps, we copied the formula down incorrectly.
Let us move on to a slightly more complicated example. This time we shall find expressions for sine, cosine and tangent.
Determine the exact value of (a) sin(105°), (b) cos(105°), and (c) tan(105°).
First off, recognize that 105° = 60° + 45°. We shall use this fact in all three parts. Also, we will need the values for sine, cosine and tangent at 45° and 60°. Those are included below.
A useful consequence of the angle addition and subtraction formulas is a proof for the relationship between sine and cosine. Recall when we discussed the graphs of sine and cosine, we pointed out that the graph of cosine was just a phase shift of of π/2 (or 90°) from the graph of sine, and vice versa. At that time, we wrote down formulas relating the two functions. Now, using the addition and subtraction formulas, we can justify those formulas algebraically.
We had observed that cos(x) = sin(x + 90°) and sin(x) = cos(x 90°). But then using the fact that sine is an odd function and cosine is an even function, we rewrote the above as cos(x) = sin(90° x) and sin(x) = cos(90° x).
Using the subtraction formulas for sine and cosine, verify that cos(x) = sin(90° x) and sin(x) = cos(90° x).
We use the subtraction formulas, replacing x with 90° and y with x.
sin(90° x) = sin(90°)cos(x) − cos(90°)sin(x) = 1cos(x) 0sin(x) = cos(x).
cos(90° x) = cos(90°)cos(x) + sin(90°)sin(x) = 0cos(x) + 1sin(x) = sin(x).