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Trig Functions of angles Note that by what we know of 'similar triangles' from geometry, that the ratios of the sides in the blue and red trangles must be the same. there are six possible ratios, hence six trig functions! Sine, Cosine, Tangent, Cosecant, Secant, and Cotangent are the names of the ratios and the definitions (and abbreviations!) are given below: (pause video to note them!) sin( θ ) = cos( θ ) = tan( θ ) = opposite hypotenuse adjacent hypotenuse opposite adjacent csc ( θ ) = hypotenuse sec ( θ ) = hypotenuse cot( θ ) = adjacent opposite adjacent opposite Two things to note for memory's sake: Always write them in this order as each is opposite the 'upside down version' of itself Easy to memorize LHS by SOH-CAH-TOA Some simple trigonometric identities are apparent from the 'upside down' functions opposite each other: sin( θ ) = cos( θ ) = tan( θ ) = 1 csc( θ ) 1 sec ( θ ) 1 cot( θ ) Note that the letters sin, cos,tan etc have no meaning without an angle. Sine is not defined at all, in fact but sin of theta is defined. For today's lesson, assume that the argument (angle, if you want) for each function is acute. 0o <θ<90o Ex given that tan( θ ) = 3 4 , find the other five trig functions of theta. (Pause and think about how to do this!) sin( θ ) = cos( θ ) = tan( θ ) = opposite hypotenuse adjacent hypotenuse 3 4 csc ( θ ) = hypotenuse sec ( θ ) = hypotenuse cot( θ ) = 4 opposite adjacent 3 (Pause here again and think of a way to find the third side.) Pythagorean theorem!! a2+b2=h2 Hypotenuse can be found if we know opp and adj sides! 2 2 2 h = 3 + 4 = 25 sin( θ ) = cos( θ ) = tan( θ ) = 3 5 4 5 3 4 h =5 csc ( θ ) = 5 sec ( θ ) = 5 cot( θ ) = 4 3 4 3 Ex if sin( θ ) = 1 2 find the other five trig functions of θ. (pause and try it first!) sin( θ ) = 2 cos( θ ) = tan( θ ) = csc ( θ ) = 1 3 2 2 1 sec ( θ ) = 3 cot( θ ) = 1 3 2 3 1 During the semester, some angles are quite commonly used. We do NOT have to actually memorize the functions of the angles, but can get all from memory by knwing the definitions of the six trig functions and memorizing the following picture of two special right triangles π 6 o = 30 π = 1 6 2 csc π = 3 6 2 sec π = 1 6 3 cot sin cos tan π = 2 6 1 π = 2 6 3 π = 3 6 1 π 3 o = 60 π = 3 3 2 csc π = 2 3 3 π = 1 3 2 sec π = 3 3 1 cot sin π = 2 3 1 cos π = 1 3 3 tan π 4 o = 45 π = 1 4 2 csc π = 2 4 1 π = 1 4 2 sec π = 1 4 1 cot sin π = 2 4 1 cos π =1 4 tan Ex Suppose I am 90 feet from the base of the ACM clock tower, the angle of elevation to the top of the tower is 30o. How high is the tower? ( o) = 90x tan 30 1 3 = x 90 x= 90 3 = 30 3 feet tall Calculator has 'mode' button allowing you to change from degrees to radian mode.Older models also has 'grads'. It will give you decimal approximations of the values of all six trig functions of a given angle, but you must be cautious about the mode your calcluator is in. Ex set your calculator to degrees and find sin(60) π The set it to radians and compare to sin , they should give the same result. 3 Check it against 3 . 2 (pause here and do this!) Note that you merely have a decimal approximation of an irrational number. Use exact numbers whenever possible to lessen roundoff errors. So how would you find sec(60o) with your calculator? (Pause and think!) Notation for powers of trig functions: sin(θ)*sin(θ) usually written sin2(θ) This software doesn't understand that properly, so sin( θ ) if I'm typing using this. 2 ( 2) sin θ Problem is with this pair of different things sin(θ2) and sin2(θ). One squares the angle, the other squares the sine of the angle. So do not write sin θ2 ! (Even though I must!) 8 Fundamental Identities csc ( θ ) = sec ( θ ) = 1 sin( θ ) opp tan( θ ) = opp adj = hyp = adj 1 cos( θ ) sin( θ ) cot( θ ) = cot( θ ) = cos( θ ) 1 tan( θ ) 1 tan( θ ) = cos( θ ) sin( θ ) hyp Pythagorean identities: sin2 θ +cos 2 θ = 1 1 + cot 2θ = csc2θ tan2 θ+1=sec 2 θ All 8 are here: csc ( θ ) = 1 sin( θ ) tan( θ ) = sin( θ ) cos( θ ) sin ( θ ) + cos ( θ ) = 1 2 2 sec ( θ ) = cot( θ ) = 1 cos( θ ) 1 tan( θ ) cos( θ ) sin( θ ) sec ( θ ) − tan ( θ ) = 1 2 cot( θ ) = 2 csc ( θ ) − cot ( θ ) = 1 2 2 Ex prove the identities sin( θ ) ⋅ cot( θ ) = cos( θ ) 1 sin( θ ) ⋅ cos( θ ) sin( θ ) cos( θ ) cos ( 2θ ) − sin ( 2θ ) = 2cos ( 2θ ) − 1 2 2 2 2 cos ⋅ ( 2 ⋅ θ ) − sin ⋅ ( 2 ⋅ θ ) + cos ( 2θ ) − cos ( 2θ ) 2 2 2 2 2cos ⋅ ( 2 ⋅ θ ) − sin ⋅ ( 2 ⋅ θ ) − cos ( 2θ ) 2 2 2 2 ⋅ cos ⋅ ( 2 ⋅ θ ) − sin ⋅ ( 2 ⋅ θ ) + cos ⋅ ( 2 ⋅ θ ) 2 2 2cos ( 2θ ) − 1 2 3 sin( θ ) + cos( θ ) cos( θ ) sin( θ ) cos( θ ) sin( θ ) cos( θ ) + = 1 + tan( θ ) cos( θ ) cos( θ ) +1 tan( θ ) + 1 1 + tan( θ ) 2 4 Write all six trig functions in terms of sin(θ) sin( θ ) = sin( θ ) cos( θ ) = ± ⋅ tan( θ ) = ± ⋅ csc ( θ ) = 2( 1 − sin θ) sin( θ ) 1 − sin ( θ ) 2 cos ( θ ) + sin ( θ ) = 1 1 2 sin( θ ) sec ( θ ) = ± ⋅ 2 1 1 − sin ( θ ) 2 cot( θ ) = ± ⋅ 1 − sin ( θ ) 2 sin( θ ) Trigonometric Functions of ANY Angles Ex Place the angle 2π/3 in standard position, and find values for all 6 trig functions 2π = 3 3 2 csc 2π = 2 3 3 2π = − 1 2 3 sec sin 2π = −2 3 cos 2π = − 3 3 tan 2π = − 1 3 3 cot Ex Find the exact values of all six trig functions if (2,-1) is on the terminal side of an angle in standard position. sin( θ ) = − 1 csc ( θ ) = − 5 5 cos( θ ) = sec ( θ ) = 2 5 tan( θ ) = − Ex 1. find all six trig functions of an angle of π 1 2 5 2 cot( θ ) = −2 sin( π ) = 0 cos( π ) = −1 tan( π ) = 0 csc ( π ) = ∞ sec ( π ) − 1 cot( π ) = ∞