Trigonometric Equations of Quadratic Type

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 factoring. You can replace the trig function with u to make the factoring easier to see. Move everything to the left side and factor. DO NOT divide both sides by sin x. You don’t know if sin x = 0 in which case you’d be dividing by 0. You will loose solutions if you do this. Factor (If it makes it easier for you to factor, substitute u for sin x) Set each factor = 0 and solve

The next 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 x like u and this is quadratic) Set each factor = 0 and solve

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. re-order terms Get sine term alone 2 2 Square both sides. Must do whole side together NOT each term (so left side will need to be FOILed). Pythagorean Identity---this equals 1Double angle Identity Remember to do another loop when you have 2x Where is the sine -1?

Try to get equations in terms of one trig function by using identities. Try to get trig functions of the same angle. If one term is cos2x and another is cosx for example, use the double angle formula to express first term in terms of just instead of 2x 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 u to see how it factors if that helps) If the angle you are solving for is a multiple of x, don't forget to add 2  to your answer for each multiple of x since x will still be less than 2  when solved for. HELPFUL HINTS FOR SOLVING TRIGONOMETRIC EQUATIONS Be on the look-out for ways to substitute using identities You can square each side when necessary to enable you to use the Pythagorean identities