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Trigonometric Equations (II)

Trigonometric Equations (II). In this section we'll learn various techniques to manipulate trigonometric equations so we can solve them. We'll find solutions on the interval from 0 to 2 . The first tip is to try using identities to get in terms of the same trig function. .

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Trigonometric Equations (II)

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  1. Trigonometric Equations (II)

  2. In this section we'll learn various techniques to manipulate trigonometric equations so we can solve them. We'll find solutions on the interval from 0 to 2. The first tip is to try using identities to get in terms of the same trig function. Use the Pythagorean Identity to replace this with an equivalent expression using sine. Combine like terms, multiply by -1 and put in descending order Factor (think of sin  like x and this is quadratic) Set each factor = 0 and solve

  3. When we don't have squared trig functions, we can't use the Pythagorean identities. If you have two terms with different trig functions you can try squaring both sides. Square both sides. Must do whole side together NOT each term (so left side will need to be FOILed). 2 2 re-order terms Pythagorean Identity---this equals 1 Double angle Identity Get sine term alone Where is the sine -1? Remember to do another loop when you have 2

  4. HELPFUL HINTS FOR SOLVING TRIGONOMETRIC EQUATIONS • Try to get equations in terms of one trig function by using identities. • Be on the look-out for ways to substitute using identities • Try to get trig functions of the same angle. If one term is cos2 and another is cos for example, use the double angle formula to express first term in terms of just  instead of 2 • Get one side equals zero and factor out any common trig functions • See if equation is quadratic in form and will factor. (replace the trig function with x to see how it factors if that helps) • If the angle you are solving for is a multiple of , don't forget to add 2 to your answer for each multiple of  since  will still be less than 2 when solved for.

  5. There are some equations that can't be solved by hand and we must use a some kind of technology. Use a graphing utility to solve the equation. Express any solutions rounded to two decimal places. Graph this side as y1 in your calculator Graph this side as y2 in your calculator You want to know where they are equal. That would be where their graphs intersect. You can use the trace feature or the intersect feature to find this (or these) points (there could be more than one point of intersection).

  6. This was graphed on the computer with graphcalc, a free graphing utility you can download at www.graphcalc.com This is off a little due to the fact we approximated. If you carried it to more decimal places you'd have more accuracy. check: After seeing the initial graph, lets change the window to get a better view of the intersection point and then we'll do a trace. Rounded to 2 decimal places, the point of intersection is x = 0.53

  7. Acknowledgement I wish to thank Shawna Haider from Salt Lake Community College, Utah USA for her hard work in creating this PowerPoint. www.slcc.edu Shawna has kindly given permission for this resource to be downloaded from www.mathxtc.com and for it to be modified to suit the Western Australian Mathematics Curriculum. Stephen Corcoran Head of Mathematics St Stephen’s School – Carramar www.ststephens.wa.edu.au

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