trigonometry in a nutshell

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Trigonometr in a Nutshell

T .

I , L M, P . O L M

L M.

T : ? T , ?

T , .

T : , P ,

L M M N, L N L M

M N.

( )

I , . A

.

N : , L L , 360 .

Why?

I .

A L M P P

. T . S 2 ,

360 2 .

T " " , L M , 180 , . A

, 90 . T - " "

( = ) 45 .

T , ,

. I , . I , , 0

0 1 .

S M L P. I P

L M A B . T A

M. T . Y A M

M .

T : , M L

, .

(A .)

T , ,

, .

Why did they do that?

T .


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How so?

Well, he de a line egmen from the point A tangent to the unit circle around P to the line M. The length of this

segment the called the tangentof he angle fom M o L. (When he line ha a poiie lope he angen i aken o be negaie.)

The length of the line segment from P along M to the intersection of M with that tangent line at A the called the secant

of the angle from M to L. (The ecan i negaie hen hi line n aa fom M a P.)

The alo defined he complemen of an angle ha i le han a igh angle o be he diffeence beeen a igh angle and i. Thi go

hem o define he coine, coangen and coecan a he ine, angen and ecan of he complemen of he oiginal angle.

Fonael fo , all of hee i fncion ae eail elaed o he ine fncion, hich mean ha e need onl eall become

familia ih he ine, and e can hen fige o ha he ohe ill do. Noice ha he ine, b hee definiion change ign

hen e inechange L and M, hile coine a he ame.

Hee ae he elaion beeen hee fncion, all of hich follo fom he definiion fom he fac ha corresponding angles of

similar triangles are equal


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Exercises:

1. Draw a relevant picture for an angle that is less than showing all of these entities. Include the line perpendicular

to M at P, and its intersection with the tangent to the unit circle at A.

2. Identify which triangles are similar to one another. Remember that the two angles other than a right angle in a

triangle with a right angle are complementary.

3. Deduce all the claims above by using the similar triangles argument mentioned above.

T . T " "

, .

OK what are the basic theorems?

1.The Pythagorean Theorem: T the square of the hypotenuse of a right triangle is the sum of the

squares of its other two sides . T ,

,

2. The Law of Sines: T ABC A B

A B. I (BC) (AC) ,

T C AB

B BC A AC.

3. The Law of Cosines: T BC AB AC

A

l(BC)2= l(AB)2+ l(AC)2 2 l(AB)*l(AC)*cos A

T . O B AC Q. W

(AQ) = (AB)* A, AC = AQ + QC, (AQ) 2+ (BQ)2= (AB)2

(CQ)2+ (BQ)2= (BC)2

A .

What in the world is an addition theorem?

W , :

. T . T

not . T

.

And what are they?


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Addiion Theoem: Suppose we have two adjacent angles, of sizes q and j. Then the sine and cosine of their sum, q+ j are given

by

And hee do hee claim come fom?

Perhaps the easiest proof comes from the next discussion: the relation between sines and cosines and the exponential function. They

follow quickly from that relation, on substitution, given the fundamental properties of the exponential function.

Eecie:

4. Wie he coeponding el fo he ine and coine of and alo fo ine and coine of 2 .

5. Combine he la of hee ih he Phagoean heoem o ge epeion fo and fo in em of

co .

Sines Cosines and Eponents

Given any function, f, we can define two others that are symmetric and antisymmetric under reflection about the origin, as follows

In words, g(x) is the average f(x) and f(-x), while h(x) is the average of f(x) and f(-x). We can deduce from these definitions that

g is symmetric and h antisymmetric (which means it changes sign under this reflection). We may also notice, f = g + h.

In light of these facts, we can call g the symmetric part of f and h the antisymmetric part of f, under reflection about the origin.

The symmetric part of the exponential function, exis called cosh x, while the antisymmetric part is called sinh x. These are called

the hyperbolic cosine and sine respectively, and you may have noticed corresponding buttons on your calculator.

If we consider the exponential function of an imaginary argument, eixwe find that the symmetric part is real, while the

antisymmetric part is imaginary. In facthe mmeic pa of eii co , and he animmeic pa of epii i in .

These facts, and the fundamental property of exponents: eAeB= eA+B, hich i an addiion heoem fo eponen , provide for

straightforward deduction of the addition theorem for sines and cosines.

The formal statements of the relation between sines and cosines and exponentials are as follows

Eecie 6 The eponenial fncion being i on deiaie, can be ien a an infinie eie a follo


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Write out the first three terms of the series for sin and for cos implied b the relations just above and this statement.

Here is an applet to help ou get used to these concepts.

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trigonometry in a nutshell

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