Trigonometric Functions

Transcription

1 Trigonometric Functions Here is a review the basic definitions and properties of the trigonometric functions. We provide a list of trig identities at the end.. Definitions The trigonometric functions are defined as ratios of the lengths of the sides of a right angle triangle as shown below. In the figure, H stands for Hpotenuse, stands for djacent side and P stands for opposite side. (,P) H P csc = H P Here are the graphs of the trigonometric functions. sin sin = P H cos = H tan = P sec = H cot = P cos (a) (b) π π π π π π tan cot π π π π π π csc sec π π π π π π. Radians For use in calculus, angles are best measured in units called radians. B definition, an arc of length on a circle of radius one subtends an angle of radians at the center of the circle. c Joel Feldman. 04. ll rights reserved. March 0, 04

2 Because the circumference of a circle of radius one is π, we have π radians = 60 π radians = 80 π π radians = 60 π 4 radians = 45 π 6 radians = 90 radians = 0. rclength and rea onsider a circle of radius r. Denote b L() the length of an arc of that circle that subtends an angle of radians. The angle subtended,, is the fraction of the full circle. So L() is π the fraction of the circumference of the full circle, which is πr. Similarl, denote b () π the area of a wedge of that circle that subtends an angle of radians. The area () is the fraction of the area of the full circle, which is π πr. So () L() L() = π πr = r () = π πr = r 4. Special Triangles From the triangles 0 45 we can read off the values of all of the trigonometric functions for the angles = π 6 = 0, = π 4 = 45 and = π = sin cos tan csc sec cot 0 = 0 rad = π 6 rad 45 = π 4 rad 60 = π rad 90 = π rad = π rad The empt boxes mean that the trig function is undefined (i.e. ± ) for that angle. c Joel Feldman. 04. ll rights reserved. March 0, 04

3 5. Trig Identities Basic The following identities are all immediate consequences of the definitions in (). csc = sin sec = cos tan = sin cot = = cos cos tan sin Because π radians is 60, the angles and +π are reall the same, so sin(+π) = sin cos(+π) = cos The following trig identities are consequences of the figure to their right and Pthagoras theorem, +P = H. (cos,sin) sin( ) = sin() sin +cos = cos( ) = cos() H = = cos P = sin The following trig identities are consequences of the figure to their left. (cos, sin) cos π sin sin ( π ) = cos cos ( π ) = sin 6. Trig Identities ddition and Double ngle Formulae The following trig identities are derived below in 9. Setting = x gives sin(x+) = sinxcos +cosxsin sin(x ) = sinxcos cosxsin cos(x+) = cosxcos sinxsin cos(x ) = cosxcos +sinxsin sin(x) = sinxcosx cos(x) = cos x sin x = cos x since sin x = cos x = sin x since cos x = sin x Solving cos(x) = cos x for cos x and cos(x) = sin x for sin x gives cos x = +cos(x) sin x = cos(x) () c Joel Feldman. 04. ll rights reserved. March 0, 04

4 7. Trig Identities the Sine Law The sine law sas that, if a triangle has sides of length, B and and the angles opposite those sides are a, b and c, then sina = sinb B = sinc To derive sina = sinc, drop a perpendicular to the side of length B from the vertex opposite it, as in the figure below. c The perpedicular has length P = sinc = sina. Dividing b gives sina = sinc. 8. Trig Identities the osine Law B The cosine law sas that, if a triangle has sides of length, B and and the angle opposite the side of length is, then P = +B Bcos Observe that, when = π, this reduces to, (surpise!) Pthagoras theorem = + B. To derive the osine law, appl Pthagoras to the shaded triangle in the right hand figure of B sin a cos That triangle is a right triangle whose hpotenuse has length and whose other two sides have lengths ( B cos ) and sin. So Pthagoras gives as desired. = ( B cos ) + ( sin ) = B B cos+ cos + sin = B B cos+ (since sin +cos = ) B c Joel Feldman. 04. ll rights reserved. 4 March 0, 04

5 9. Trig Identities Derivation of the ddition Formulae We now derive the addition formulae (). The first step is to prove the fourth addition formula, cos(x ) = cosxcos+sinxsin. The angle, of the upper triangle in the figure, (cosx,sinx) x (cos,sin) that is opposite the side of length is x. So, b the cosine law, = + cos(x ) = cos(x ) But the side of length joins the points (cos,sin) and (cosx,sinx) and so we also have, b Pthagoras, = (cos cosx) +(sin sinx) = cos cosxcos +cos x+sin sinxsin +sin x = cosxcos sinxsin (using cos z +sin z = for z = x and z = ) Setting the two formulae for equal to each other gives cos(x ) = cosxcos sinxsin = cos(x ) = cosxcos sinxsin = cos(x ) = cosxcos +sinxsin which is the fourth addition formula. Replacing b gives cos(x+) = cosxcos( )+sinxsin( ) = cosxcos sinxsin which is the third addition formula. Now, replacing x b π x gives cos ( π x+) = cos ( π x) cos sin ( π x) sin Recalling that sin ( π z) = cosz and cos ( π z) = sinz, sin ( x ) = sinxcos cosxsin which is the second addition formula. Finall, replacing b gives the first addition formula. c Joel Feldman. 04. ll rights reserved. 5 March 0, 04

6 0. Trig Identities Summar The basic trig identities are tan = sin csc = sec = cot = = cos (T) cos sin cos tan sin sin( ) = sin cos( ) = cos (T) sin( +π) = sin cos(+π) = cos (Ta) sin( +π) = sin cos(+π) = cos (Tb) sin( π ) = cos cos(π ) = sin (Tc) sin +cos = (T4) sin() = sincos cos() = cos sin sin( +ϕ) = sincosϕ+cossinϕ cos(+ϕ) = coscosϕ sinsinϕ (T5) (T6) (T7a) (T7b) The following trig identities are easil derived from the basic identities above. We use the code that the identities in (T4 ) are easil derived from the identit (T4), that the identit in(t5,t6 ) is easil derived b dividing the identities in(t5) and(t6) and that the identities in (T7 ) are easil derived from the identities in (T7) and (T7 ). tan + = sec +cot = csc (T4 ) tan() = tan tan cos() = cos = sin cos = +cos() sin = cos() tan = cos() +cos() sin( ϕ) = sincosϕ cossinϕ cos( ϕ) = coscosϕ+sinsinϕ tan(+ϕ) = tan+tanϕ tan( ϕ) = tan tanϕ tantanϕ +tantanϕ { } sincosϕ = sin(+ϕ)+sin( ϕ) { } sinsinϕ = cos( ϕ) cos(+ϕ) { } coscosϕ = cos(+ϕ)+cos( ϕ) sinα+sinβ = sin α+β cos α β sinα sinβ = cos α+β sin α β cosα+cosβ = cos α+β cos α β cosα cosβ = sin α+β sin α β (T5,T6 ) (T6 ) (T7 ) (T7 ) c Joel Feldman. 04. ll rights reserved. 6 March 0, 04