4.3 Trigonometrische Eigenschaften
Aus Online Mathematik Brückenkurs 1
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Version vom 13:40, 12. Sep. 2008
Contents:
- The Pythagorean identity
- The double-angle and half-angle formulas
- Addition and subtraction formulas
Learning outcome:
After this section, you will have learned how to:
- Derive trigonometric relationships from symmetries in the unit circle.
- Simplify trigonometric expressions with the help of trigonometric formulas.
Introduction
There is a variety of trigonometric formulas to use if one wishes to transform between the sine, cosine or tangent for an angle or multiples of an angle. They are usually known as the trigonometric identities, since they only lead to different ways to describe a single expression using a variety of trigonometric functions. Here we will give some of these trigonometric relationships. There are many more than we can deal with in this course. Most can be derived from the so-called Pythagorean identity and the addition and subtraction formulas or identities (see below), which are important to know by heart.
The Pythagorean identity
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This identity is the most basic, but is in fact nothing more than Pythagorean theorem, applied to the unit circle. The right-angled triangle on the right shows that which is usually written as \displaystyle \sin^2\!v + \cos^2\!v = 1. |
Symmetries
With the help of the unit circle and reflection, and exploiting the symmetries of the trigonometric functions one obtains a large amount of relationships between the cosine and sine functions.
Instead of trying to learn all of these relationships by heart, it might be better to learn how to derive them from the unit circle.
Reflection in the x-axis
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Reflection in the y-axis
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Reflection in the line y = x
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Rotation by an angle of \displaystyle \mathbf{\pi/2}
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Alternatively, one can get these relationships by reflecting and / or displacing graphs. For instance, if we want to have a formula in which \displaystyle \cos v is expressed in terms of a sine one can displace the graph for cosine to fit the sine curve. This can be done in several ways, but the most natural is to write \displaystyle \cos v = \sin (v + \pi / 2). To avoid mistakes, one can check that this is true for several different values of \displaystyle v.
Check: \displaystyle \ \cos 0 = \sin (0 + \pi / 2)=1.
The addition and subtraction formulas and double-angle and half-angle formulas
One often needs to deal with expressions in which two or more angles are involved, such as \displaystyle \sin(u+v). One will then need the so-called "addition formulas" . For sine and cosine the formulas are
If one wants to know the sine or cosine of a double angle, that is \displaystyle \sin 2v or \displaystyle \cos 2v, one can write these expressions as \displaystyle \sin(v + v) or \displaystyle \cos(v + v) and use the addition formulas above and get the double-angle formulas
From these relationships, one can then get the formulas for half angles. By replacing \displaystyle 2v by \displaystyle v, and consequently \displaystyle v by \displaystyle v/2, in the formula for \displaystyle \cos 2v one gets that Vorlage:Displayed math
If we want a formula for \displaystyle \sin(v/2) we use the Pythagorean identity to get rid of \displaystyle \cos^2(v/2) Vorlage:Displayed math i.e.
Similarly, we can use the Pythagorean identity to get rid of \displaystyle \sin^2(v/2). Then we will have instead
Study advice
The basic and final tests
After you have read the text and worked through the exercises, you should do the basic and final tests to pass this section. You can find the link to the tests in your student lounge.
Keep in mind that:
The unit circle is an invaluable tool for finding trigonometric relationships. They are a multitude and there is no point in trying to learn all of them by heart. It is also time-consuming to have to look them up all the time. Therefore, it is much better that you learn how to use the unit circle.
The most famous trigonometric formula is the so-called Pythagorean identity. It applies to all angles, not just for acute angles. It is based on the Pythagoras theorem.
Useful web sites
