Understanding Trigonometric Equations and Graphs
Explore trigonometric equations, graphs, and laws to master solving trigonometric equations with symmetry concepts. Learn about the roots of trigonometric equations, angle laws, and practical problem-solving techniques.
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Presentation Transcript
Trigonometric Equations
Sine Graph -360 -270 -180 -90 90 180 270 360 Suppose we know that sin(30) = 0.5. By thinking about symmetry in the graph, how could we work out: ? ? ? sin(150) = 0.5 sin(-30) = -0.5 sin(210) = -0.5
Cosine Graph -360 -270 -180 -90 90 180 270 360 Suppose we know that cos(60) = 0.5. By thinking about symmetry in the graph, how could we work out: cos(120) = -0.5 cos(-60) = 0.5 cos(240) = -0.5 ? ? ?
Tangent Graph -360 -270 -180 -90 90 180 270 360 Suppose we know that tan(30) = 1/ 3. By thinking about symmetry in the graph, how could we work out: ? ? tan(-30) = -1/ 3 tan(150) = -1/ 3
Laws of Trigonometric Functions ? sin ? = sin 180 ? ? cos ? = cos 360 ? ? sin and cos repeat every 360 tan repeats every 180 ?
How many roots does a trigonometric equation have?
Solving Trig Equations Solve sin ? = 0.6 in the range 0 ? < 360 ? = ??? ??.? = ??.?? ,???.?? ? Angle Law #1: ??? ? = ???(??? ?) ? 0.6 ? ??.?? -360 -270 -180 -90 90 180 270 360
Solving Trig Equations Angle Law #2: ??? ? = ???(??? ?) Solve 3cos ? = 2in the range0 ? < 360 cos ? =2 3 ? ? ? ? = ??? ? = ??.?? ,???.?? ? ? ? ? ??.?? ? -360 -270 -180 -90 90 180 270 360
Solving Trig Equations Angle Law #3: Sin and cos repeat every 360 Solve sin ? = 0.3 in the range 0 ? < 360 ? = ??? ? ?.? = ??.?? ,???.?? ,???.?? ? ? ? ??.?? ? -360 -270 -180 -90 90 180 270 360 -0.3
Test Your Understanding Solve cos ? = 0.9 in the range 0 ? < 360 ? = cos 10.9 = ??.?? 360 25.84 = ???.?? ? Solve tan ? = 1 in the range 0 ? < 360 ? = tan 11 = ?? 45 + 180 = ??? ? Set 4 Paper 2 Q14 sin? = 0.68 180 42.84 = ???.?? 42.84 + 360 = ???.?? ? = 42.84 ?
Exercise 1 Angle Law #1: ??? ? = ???(??? ?) Solve the following in the range 0 ? 360 sin ? = 0.5 ? = ?? ,??? 3 2 1 Angle Law #2: ??? ? = ???(??? ?) ? ? a Angle Law #3: Sin and cos repeat every 360 b cos ? = ? = ?? ,??? ? ? ? ?? c tan ? = sin ? = 0.1 4cos ? = 3 6tan ? = 5 ? = ??.?? ,???.?? sin ? = 0.4 ? = ??.?? ,???.?? 3 ? = ?? ,??? ? = ?.?? ,???.?? ? = ??.?? ,???.?? d e f g Solve the following in the range 0 ? 360 sin ? = 0.5 ??? ,??? cos ? = 0.5 ??? ,??? tan ? = 1 ??? ,??? sin ? = 0.4 ???.?? ,???.?? cos ? = 0.7 ???.? ,???.? tan ? = 0.2 ???.?? ,???.?? 2 ?? ? a b c d e ? ?? f
Challenge Questions ? ? ?
