Addition formulas allow you to express sin(2*a), cos(2*a) and tan(a) through trigonometric functions of angle a.
1. cos(a+b) = cos(a)*cos(b) - sin(a)*sin(b).
2. sin(a+b) = sin(a)*cos(b) + cos(a)*sin(b).
3. tg(a+b) = (tg(a) +tg(b))/(1-tg(a)*tg(b)).
Let us put a = b in these formulas. As a result, we obtain the following identities:
1. sin(2*a) = 2*sin(a)*cos(a).
2. cos(2*a) = (cos(a)) 2 - (sin(a)) 2 .
3. tg(2*a) = (2*tg(a))/(1-(tg(a)) 2).
These identities are called double angle formulas. Let's look at several examples of using double angle formulas.
Example 1. Find the value of sin(2*a), knowing that cos(a) = -0.8 and a is the 3rd quarter angle. Solution:
First let's calculate sin(a). Since angle a is the third quarter, the sine in the third quarter will be negative:
sin(a) = -v(1-(cos(a)) 2) = -v(1-0.64) = -v0.36 = -0.6.
Using the double angle sine formula we have:
sin(2*a) = 2*sin(a)*cos(a) = 2*sin(a)*cos(a) = 2*(-0.6)*(-0.8) = 0.96 .
Answer: sin(2*a) = 0.96.
Example 2. Simplify the expression sin(a)*(cos(a)) 3 - (sin(a)) 3 *cos(a). Solution:
Let's take sin(a)*cos(a) out of brackets. We get:
sin(a)*(cos(a)) 3 - (sin(a)) 3 *cos(a) = sin(a)*cos(a)*(cos(a)) 2 - (sin(a)) 2).
Now let's use the double angle formulas:
= (1/2)*(2*sin(a)*cos(a))*cos(2*a) = (1/2)*sin(2*a)*sin(2*a) = (1 /4)*sin(4*a).
Answer: sin(a)*(cos(a)) 3 - (sin(a)) 3 *cos(a) = (1/4)*sin(4*a).
Using the double angle formulas you can obtain the following expressions
1 - cos(2*a) = 2*(sin(a)) 2 ,
1 + cos(2*a) = 2*(cos(a)) 2 .
Sometimes when solving examples it is very convenient to use these formulas. Consider the following example:
Example 3. Simplify the expression (1-cos(a))/(1+cos(a)). Solution:
Let's apply the formulas written above for the expressions (1-cos(a)) and (1+cos(a)). To do this, we first represent angle a in the form of the following product 2*(a/2).
As a result of the transformations we get:
(1-cos(a))/(1+cos(a)) = (2*(sin(a/2)) 2)/(2*(cos(a/2)) 2),
Using the definition of tangent we have:
(2*(sin(a/2)) 2)/(2*(cos(a/2)) 2)= (tg(a/2)) 2 .
Answer: (1-cos(a))/(1+cos(a))= (tg(a/2)) 2 .
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In trigonometry, many formulas are easier to derive than to memorize. Cosine of double angle is a wonderful formula! It allows you to obtain formulas for reducing degrees and formulas for half angles.
So, we need the cosine of the double angle and the trigonometric unit:
They are even similar: in the double angle cosine formula it is the difference between the squares of the cosine and sine, and in the trigonometric unit it is their sum. If we express the cosine from the trigonometric unit:
and substitute it into the cosine of the double angle, we get:
This is another double angle cosine formula:
This formula is the key to obtaining the reduction formula:
So, the formula for reducing the degree of sine is:
If in it the alpha angle is replaced by a half angle alpha in half, and the double angle two alpha is replaced by an alpha angle, then we obtain the half angle formula for sine:
Now we can express the sine from the trigonometric unit:
Let's substitute this expression into the double angle cosine formula:
We got another formula for the cosine of a double angle:
This formula is the key to finding the formula for reducing the power of cosine and the half angle for cosine.
Thus, the formula for reducing the degree of cosine is:
If we replace α with α/2, and 2α with α, we obtain the formula for the half argument for the cosine:
Since tangent is the ratio of sine to cosine, the formula for tangent is:
Cotangent is the ratio of cosine to sine. Therefore, the formula for cotangent is:
Of course, in the process of simplifying trigonometric expressions, there is no point in deriving the formula for half an angle or reducing a degree every time. It is much easier to put a sheet of paper with formulas in front of you. And simplification will move faster, and visual memory will turn on memorization.
But it’s still worth deriving these formulas several times. Then you will be absolutely sure that during the exam, when it is not possible to use a cheat sheet, you will easily get them if the need arises.
