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Topic 2: Trigonometry Website: Click here or use Achievement Standard 91575: Apply trigonometric methods in solving problems. This standard is an Internal worth 4 credits. Week 8: Trigonometry definitions and transformations of trig graphs. This needs to be completed by Wednesday 30 March. Learning Objectives 1. To understand the formal definition of the 3 trig functions from the unit circle. 2. To know the basic features of the graphs of the 3 trig functions. 3. To understand transformations of trig graphs and how these transformations are linked to the equation. 4. To be able to find the amplitude and period from a trig equation. Topic Notes Definitions from the unit circle. The unit circle is a circle radius 1 centre (0,0). As a point rotates around the circumference of the circle, the horizontal component is defined as cosθ (the pink line), the vertical component is defined as sinθ ( the blue line) and if the horizontal distance is fixed at 1 the distance to the intersection with the tangent to the circle is defined as tanθ (the red line). The basic graph of any function can be transformed using reflections, changes of scale and translations. This work was covered in the Level 2 graphs achievement standard. We are now looking at the effect of these transformations on the trig graphs. Graphs of y=sinθ, y=cosθ and y=tanθ and their basic features. Transformations of trig graphs If we put all the possible transformations together in one equation we get equations like 𝑦 = 𝐴𝑠𝑖𝑛𝐵(𝑥 + 𝐶) + 𝐷 A changes the amplitude B changes the period C translates horizontally D translates vertically Finding period and amplitude from transformed equations. Before we start using these trig equations to model real life situations we need to be able to find the period and amplitude from the equation. From the equation 𝑦 = 𝐴𝑠𝑖𝑛𝐵(𝑥 + 𝐶) + 𝐷 The amplitude is equal to A and 2𝜋 The period= 𝐵 Delta Mathematics 2nd Edition Old Book The basic graphs and their features are summarised on Page 304-305. Also go to the websites listed on the next page to watch the construction of the 3 graphs from the unit circle. Delta Maths NCEA Level 3 New Book The basic graphs and their features are summarised on Page 72. Read through pages 305 to 308 and the summary of these transformations on page 308. Read through pages 73 to 76 and the summary of these transformations on page 77. Use the link on the next page to an online graphing package to have a play with transforming the trig graphs. The graphs are easier to see than when using your graphical calculators. Use the link on the next page to an online graphing package to have a play with transforming the trig graphs. The graphs are easier to see than when using your graphical calculators. Page 309-310 Ex 33.2 Q 2, 3, 9, 10, 11 and 12 Page 79-80 Ex 5.02 Q 2, 3, 10, 11, 12 and 13 Also go to the websites listed on the next page to watch the construction of the 3 graphs from the unit circle. Useful web links for trigonometry definitions and graphs. Description Interactive graphing of sine and cosine from the unit circle Web link https://tube.geogebra.org/student/m1525 Click on either the sine or cosine curve and click the trace on. Then use the mouse to rotate the point around the unit circle. A trace of dots is left to show the curve being drawn. The sine curve is red and the cosine is blue. You can erase the trace or have both curves drawn on the same axes. Graphing of sine and cosine from the unit circle https://tube.geogebra.org/student/m220433 Select sine, cosine or both and click start animation to see the curves drawn as the point rotates around the unit circle. Video explaining the 3 trig graphs. https://www.youtube.com/watch?v=OjHgoZOdRKM Video explaining special triangles https://www.youtube.com/watch?v=ZT65zTqMjXQ The special triangles are given on the formula sheet and this video explains how to use them to find exact values for the 3 trig fucnctions and multiples of 30, 45 and 60 degrees. Online graphing program https://www.desmos.com/calculator Use this graphing package to enter equations of the following form and play with changing the values of A, B, C and D. Make sure you understand how each value changes the standard trig graph. 𝑦 = 𝐴𝑠𝑖𝑛𝐵(𝑥 + 𝐶) + 𝐷 A changes the amplitude B changes the period C translates horizontally D translates vertically QR Code