Trig Graphs and Equations for Advanced Students
Explore trigonometric graphs and equations, understand key features, and solve equations using radians. Learn about graph transformations, double transformations, and symmetry of trig graphs. Practice with starters and check your answers with a calculator.
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Presentation Transcript
Trig Graphs And equations
Trigonometry: Graphs to equations BAT know the key features of trig graphs using radians BAT solve trig equations using radians KUS objectives Starter: Simplify (without a calculator) sin30 1 + 3 + 3cos60 2sin45 + cos45 1 + 2 tan150 1 + 3 3tan120 Then check with a calculator
Notes These are the Trigonometric graphs, but with radians instead y y = sin 1 0 - -180 - 270 3 2 -90 2 -360 -2 -270 -3 2 360 2 90 2 180 -1 y y = cos 1 0 - -180 - 270 3 2 -90 2 -360 -2 -270 -3 2 360 2 90 2 180 -1 y = tan 1 0 - -180 - 270 3 2 -90 2 -360 -2 -270 -3 2 360 2 90 2 180 -1
WB11 Transformations of sin graph y Sketch the graph of ??? ? +? ? 1 a) ??? ? +? ????? ? 0 ? 2 ? 2? b) ???? ? 3? 2 y = sin -1 c) ??? ?? ? ???? ? ?
WB12b double transformations (ax+b) ? = 2 cos ? +3? 2 y ? = cos ? +3? Sketch the graph of 2 y = cos 1 a) 2cos ? +3? 2 - 2 -3 2 0 - 3 2 -2 2 2 -1 Transformations in order are i) shift 3? 2in the x direction ii) Stretch 2 in the y direction
WB12b double transformations (ax+b) y ??? ? ? ? Sketch the graph of b) sin 2? ? ??? ? ? ? ? 1 y = sin 0 ? 2 ? 2? 3? 2 = sin 2 ? ? 2 -1 Transformations in order are i) shift +? 2in the x direction ii) Stretch 1 2in the y direction
WB12c double transformations (ax+b) 1 2 ? +? y = tan Sketch the graph of 2 y = tan ? +? y = tan 2 ? 2+? c) tan 4 1 1 2? +? - 2 - -3 2 -2 = tan 3 2 2 2 2 -1 Transformations in order are i) shift ? 2in the x direction ii) Stretch 2 in the x direction
Reminder Symmetry of trig graphs y y = sin 1 For any angle , 0 ? 2 -? -? 2? 3? 2 -3? 2? ? 2 2 -1 sin(- ) = - sin y cos(- ) = cos y = cos 1 tan(- ) = - tan 0 -? ? 2 -? -3? 3? 2 2? 2? ? 2 2 -1 Complete: y = tan sin ? ???? = ?= ? ? 1 ???? ? cos ? = ?= ? 0 ? 2 -? -? ? 2? -3? 3? 2 2? ? 2 2 -1 ???? tan ? = ?= ? ?
Trig equations Solve trig equations in radians the same way as in degrees i) Rearrange to sin / cos / tan = ii) Use arcsin / arccos / arctan on calculator to get a first answer in a list iii) Use a graph to list all the answers in the given range iv) Sometimes complete the solution with more inverses ( e.g. when its cos 3? +? 2 you do the operations ? 2 and 3 to your list
WB13 Solve sin ? +? 3 2in the range 0 ? 2? = 2 y ? +? 3 1 = arcsin 2 2 y = sin 0 =? 2? 3, . 3 , ? 2 ? 2? 3? 2 ? 3 2? 3 7? 3 -1 ? 3 ? 2? 3 ? , . ? = 2, 2 ? = ? ? 6, ..What do you notice? 6, 7? 3 ? =11? The other answer in range is 2 6 ? 6,11? The final answers are ? = 6 check they work with your calculator
WB14 Solve each equation for x in the interval 0 ? 2? giving your answers in terms of ? 2= ? 4, 3? 4, .. ?) ? 2 2 ?) sin = 2 ? 2, 3? 2, ? = 2 ?) cos ? + ? = 2 ? 4, 7? 4,9? 4 .. ?) ? + = 3? 4, 5? 4, ? = ? 2 ? ?) tan = 3 4 ? 2 ? 3, 4? 3, .. c) 4= 7? 6, only ? =
WB15 Solve each equation for x in the interval 0 ? 2? giving your answers to 2 decimal places ?) 2? = 0.644,2.498, 6.927,8.781 .. ?) sin2? = 0.6 ? = 0.32, 1.25, 3.46, 4.39 ?) cos ? 0.3 = 0.8 ? 0.3 = 0.644, 0.644, 5.640, 6.927 .. ?) 2cos ? 0.3 = 1.6 ? =, 0.94, 5.94 ?) 3 2 tan ? ? c)tan ? ? =3 = 0 2 3 3 ? ? 3 = 0.9828, 4.1244 , .. ? = 2.03, 5.17
WB16 Solve each equation for in the interval ? ? ? giving your answers in terms of ?) 2 ???2? = 1 2 gives ? =? 2 4,3? ???2? =1 2 sin? = + 4 2 2 gives ? = ? 2 4, 3? sin? = sin? = 4 2 4???2? + 4cos? + 1 = 0 ?) cos? = 1 gives ? = ?, ? cos? + 1 2cos? 1 = 0 sin? =1 2 gives ? = ? 3,? cos? = 1 ,1 3 2
WB17a Solve each equation for in the interval ? ? ? giving your answers to 3 significant figures cos? = 0.4343 gives ? = 2.02,2.02 ?) 2cos? 1 = sin?tan? 2cos? 1 = sin x .sin x cos x 2???2? cos? = ???2? 2 2 sin? = 2???2? cos? = 1 ???2? gives ? = 0.696,0.696 3???2? cos? 1 = 0 ( 1)2 4(3)( 1) 2(3) ( 1) cos? = cos? = 0.4343, 0.7676
WB17b Solve each equation for in the interval ? ? ? giving your answers to 3 significant figures ?) ???2? 3sin? ???2? = 2 sin? = 1 2 (1 ???2?) 3 sin x ???2? = 2 gives ? = 5? 6, ? 6 0 = 2???2? + 3sin? + 1 sin? = 1 2sin? + 1 sin? + 1 = 0 gives ? = ? 2 sin? 1 2 , 1
WB18 Exam Q Sketch the curve ? = 2cos? for x in the interval 0 ? 2?. Sketch on the same diagram the curve ? = cos(? ? 3) use your graph to find the values of x in the interval 0 ? 2? for which 2cos? = cos ? ? 3 From the graph 2cos? = cos ? ? 3 Has two intersections When y = 1 and y=-1 ? 3,1 ? ?? ????? 4? 3, 1 B ?? ?????
KUS objectives BAT know the key features of trig graphs using radians BAT solve trig equations using radians self-assess One thing learned is One thing to improve is
