Trigonometric Functions of any Angle
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1 Trigonometric Functions of an Angle Wen evaluating an angle θ, in standard position, wose terminal side is given b te coordinates (,), a reference angle is alwas used. Notice ow a rigt triangle as been created. Tis will allow us to evaluate te si trigonometric functions of an angle. Notice te side opposite te angle θ as a lengt of te value of te given coordinates. Te adjacent side as a lengt of te value of te coordinates. Te lengt of te potenuse is given b +. Lets sa, for te sake of argument, te lengt of te potenuse is 1 unit. Tis would mean te following would be true. 1 cscθ 1 secθ cotθ You must tink of te sine function as giving ou te value, wereas te cosine function ields te value. Tis is ow we will determine weter te sine, cosine, tangent, cosecant, secant or cotangent of a given angle is a positive or negative value. If te angle to be evaluated is in quadrant IV, for instance, te sine of te angle θ will be negative. Te cosine of θ, in tis instance, will be positive, wile te tangent of te angle θ will be negative.
2 Eample Evaluate te si trigonometric functions of an angle θ, in standard position, wose terminal side as an endpoint of (-3,). Te angle wit terminal side is first drawn. Remember, in order to evaluate te si trigonometric ' functions for θ, use te reference angle θ. From te endpoint of te terminal side of te angle, a line is drawn to te ais. Tis is te reason reference angles are alwas drawn in relation to te ais. It will alwas create a rigt triangle wit wic to work. Now all tat is needed to solve te problem, is to find te lengt of te potenuse ten te values of te si trigonometric functions can be found. Using te Ptagorean Teorem, te lengt of te potenuse ma be found. + 3 c c 13 c 13 c cscθ secθ cotθ 3 Te first tree functions are evaluated using So-Ca-Toa. opp adj opp p p adj To find te second set of functions take te reciprocals of te first tree. Rationalize an denominators if needed. Note te terminal side to tis angle is in quadrant II. Tis means cosine, tangent, secant and cotangent are all negative values.
3 Eample Evaluate te si trigonometric functions of te angle θ, in standard position, wose terminal side as an endpoint of (-4,-3). Begin b drawing te angle θ in standard position wose terminal side as te endpoint of (-4,-3). A rigt triangle is formed b drawing a line segment to te ais. Now use te reference angle tat is drawn in relation to te ais to evaluate te si trigonometric functions. Since tis is obviousl a rigt triangle, tere is no need to use te Ptagorean Teorem in tis case. 3 5 cscθ secθ cotθ 4 3 Since tis angle resides in quadrant III, sine, cosine, cosecant and secant are negative values. Tangent is. Tis means bot tangent and cotangent is and cotangent will be positive values.
4 Once again, tink of te sine of an angle θ as ielding te value, wile te cosine ields te value wen te potenuse is 1. Since te tangent of an angle is over, as and ; it is a trigonometric identit tat. If te potenuse is an oter lengt, te following is true. cscθ secθ cotθ Tese are te actual equations used for evaluating te si trigonometric functions. Te reason we tink of sine being te value, cosine being te value, and tangent being sine divided b cosine is to determine weter te value of a trigonometric function is positive or negative. Tis of course all depends on were te terminal side of te angle lies. Te following questions will require evaluating te si trigonometric functions of an angle θ given different tpes of information. Understand tat tese are te same tpes of questions encountered on te previous pages, just asked in a different manner. Evaluate te si trigonometric functions of an angle θ, in standard position, were sin 3 θ and < 0. Tis question will be done sortl. Since te sine of an angle can be tougt of as te value tere are two quadrants in wic sine is positive. It is terefore necessar to ave one more piece of information to answer te question. Tere is some vital information tat is needed to answer tis tpe of question. Te following guidelines will elp determine in wic quadrant an angle lies. Sine is positive in quadrants I and II. Sine is negative in quadrants III and IV. Cosine is positive in quadrants I and IV. Cosine is negative in quadrants II and III. Tangent is positive in quadrants I and III. Tangent is negative in quadrants II and IV. For tis particular problem, > 0 and < 0, tis means te angle θ must reside in quadrant II. Tis information tells us were to construct our triangle.
5 Eample Evaluate te si trigonometric functions of an angle θ, in standard position, were and < 0. 3 According to te information given, te sine of te angle is positive and cosine is negative. Tis means te terminal side of te angle to be evaluated must be in quadrant II. From ere, we will use te reference angle drawn in relation to te ais. A rigt triangle is ten constructed. Since te sine of an angle is opposite over potenuse, te and te 3 can be placed on te appropriate sides of te triangle.
6 Using te Ptagorean Teorem, te adjacent side is found to be 5 units cscθ secθ cotθ 5 5 At te beginning of te problem, te value of sine was given. Terefore, we can fill in te values of sine and cosecant rigt awa. From tat point, te oter values can be found using: So-Ca-Toa. opp adj opp p p adj 3 cscθ secθ cotθ 5 Here are te values of te si trigonometric functions of te angle θ.
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