1 Trigonometric Functions

Transcription

1 Tigonometic Functions. Geomet: Cicles and Radians cicumf. = π θ Aea = π An angle of adian is defined to be the angle which makes an ac on the cicle of length. Thus, thee ae π adians in a cicle, so π ad = 360. Angles ae nomall measued anti clockwise fom the -ais as indicated.. Cicula Functions Given a ight angled tiangle as in the diagam: θ The side labelled is called the hpotenuse, the side labelled the adjacent and the side labelled the opposite. The following functions ae defined fo the vaiable θ: sin θ = cos θ = tan θ = = sin θ cos θ cosec θ = = sin θ sec θ = = cos θ

2 cot θ = = tan θ You ae familia with the following esult. Theoem. Pthagoas Theoem In a ight angled tiangle, the sum of the squae on the hpotenuse is equal to the sum of the squaes on the othe two sides, i.e. = + whee is the length of the hpotenuse,,, the lengths of the othe two sided. Poof: Stat with one tiangle: and place thee moe identical ones aound it The aea of the oute squae can be epessed as the aea of the inne squae plus the aeas of the fou tiangles: ( + ) = + 4.

3 fom which we obtain + + = + which is the statement of Pthagoas Theoem. + =.3 Tigonometic Identities We can use the definitions of the tigonometic functions, togethe with Pthagoas Theoem to obtain the following identities. Theoem. Fo all values of θ fo which the functions ae defined: cos θ + sin θ = + tan θ = sec θ cot θ + = cosec θ Poof: Fo the fist fomula, we have ( ) ( ) = + = cos θ + sin θ + = + = = The othe two fomulas ae obtained b dividing thoughout b cos θ and sin θ espectivel. Eample.3 Thee common ight angled tiangles ae: = π 6 45 = π 4 60 = π 3 sin π 6 = sin π 4 = sin π 3 = 3 cos π 6 = 3 cos π 4 = cos π 3 = tan π 6 = 3 tan π 4 = tan π 3 = 3

4 .4 Gaphs of Cicula Functions ove θ [0, π] = sin θ 0 π = cos θ 0 π 4 = tan θ 0 π 4

5 .5 Sum and Diffeence Identities We have alead met some identities in section.3, now we conside some othe identities which will be useful late. Theoem.4 Sum and Diffeence identities Fo an angles A and B, sin(a + B) = sina cos B + cosa sinb sin(a B) = sina cos B cosa sinb cos(a + B) = cosa cos B sina sinb cos(a B) = cosa cosb + sin A sinb Poof of: sin (A + B) = sin A cos B + cos A sin B cos B sin A cos B A cos A cos B B cos A sin B cos A B cos B p q A+B 90 In both diagams the lowe angle is 90 B, which poves that the angle at the top of the ight hand p diagam eall is B. Now we have cos B = sin (A + B) and p = cos B. Togethe this gives q sin (A + B) = and so: sin (A + B) = sin A cos B + cos A sin B as equied. Poof of: cos (A + B) = cos A cos B sin A sin B p cos B = q cos B cos B = q Fom the gaphs of cos and sin, we have that cos θ = sin (θ + π ). So now cos (A + B) = sin (A + B + π ) = sin A cos (B + π ) + cos A sin (B + π ) We also note fom the gaphs that cos (θ + π ) = sin θ. Thus giving the esult. cos (A + B) = sin (A + B + π ) = sin A sin B + cos A cos B

6 .6 Futhe Tigonometic Identities Othe identities ma be obtained fom the fomulas sin(a + B) and cos(a + B). Theoem.5 Double Angle Fomulae sinθ = sinθ cosθ cosθ = cos θ sin θ The latte identit ma also be witten cos θ = cos θ cos θ = sin θ Poof: Let θ = A = B in the sum identities sin θ = sin (θ + θ) = sin θ cos θ + cos θ sin θ = sin θcos θ cos θ = cos (θ + θ) = cos θ cos θ sin θ sin θ = cos θ sin θ Using the identit cos θ + sin θ = to eliminate eithe cos θ o sin θ fom the identit fo cos θ completes the poof. Theoem.6 Facto fomulae sinx + siny = sin cos sinx siny = cos sin cosx + cosy = cos cos cosx cosy = sin sin Poof: sin(a + B) = sina cos B + cosa sinb sin(a B) = sina cos B cosa sinb Adding, sin(a + B) + sin(a B) = sina cosb Let X = A + B, Y = A B then A = (X + Y ), B = (X Y ) and sinx + siny = sin cos Subtacting, i.e. sin(a + B) sin(a B) = cosa sinb sinx siny = cos sin cos(a + B) = cosa cos B sina sinb

7 cos(a B) = cosa cosb + sin A sinb Adding, cos(a + B) + cos(a B) = cosa cosb i.e. cosx + cosy = cos cos Subtacting, cos(a + B) cos(a B) = sina sinb i.e. cosx cosy = sin sin Theoem.7 tangent identities tan(a + B) = tana + tanb tana tan B tan(a B) = tana tanb + tana tan B.7 Invese Tigonometic Functions Fistl, ecall the idea of an invese function. Eample.8 The functions = and = ae mutuall invese functions. Thus let = and = then = = =. Similal let = and = then = =. Thus, the effect of one function is undone b the othe. Note that = gives two possible values fo, namel = + and =. It is usual to take the positive value and to call this the pincipal value. But it is impotant to emembe thee ae two possible values, and sometimes it ma be the negative one that is needed. Eample.9 The functions = ep and = ln ae also mutuall invese functions. Thus let = ep and = ln then = ep = ep (ln ) =. Similal let = ep and = ln then = ep (ln ) =. Again, the effect of one function is undone b its invese. Now define the invese functions of the tigonometic functions. Definition.0 Let = sin then wite = Sin and sa that is an angle whose sine is. Let = cos then wite = Cos and sa that is an angle whose cosine. Let = tan then wite = T an and sa that is an angle whose tangent is.

8 π π π 0 π = sin θ Pincipal Values = cos θ 0 π = Sin θ = Cos θ Definition. = sin is the value of = Sin which is such that π π. = cos is the value of = Cos which is such that 0 π = tan is the value of = T an which is such that π π. Eample. Let cos =. Find all values of. Solution: The tiangle above shows that the pincipal value is = π 3. Now, fom the gaph anothe value is 5π 3. We can also add o subtact an multiple of π. Hence, the values ae = π 3 + nπ o = 5π 3 + nπ whee n can be an intege (positive o negative) Eample.3 Find all values of tan = 5 in the ange 0 < π. Solution: = tan 5 =.3734 and the gaph indicates that this is the onl value in this ange. Finall, BEWARE! = sin, = cos, = tan denote the INVERSE functions of = sin, = cos, = tan, espectivel. Do NOT confuse them with the RECIPROCALS of = sin, = cos, = tan, which ae, of couse, = cosec, = sec, = cot, espectivel. An altenative notation, which avoids this poblem is to wite = acsin, = accos, = actan in place of = sin, = cos, = tan, espectivel.