Presentation on theme: "JMerrill, 2010 IIt will be imperative that you know the identities from Section 5.1. Concentrate on the reciprocal, quotient, and Pythagorean identities."— Presentation transcript:
IIt will be imperative that you know the identities from Section 5.1. Concentrate on the reciprocal, quotient, and Pythagorean identities. TThe Pythagorean identities are crucial!
SSolve sin x = ½ WWhere on the circle does the sin x = ½ ? Particular Solutions General solutions Solve for [0,2π] Find all solutions
Find all solutions to: sin x + = -sin x
Round to nearest hundredth
Solve Verify graphically These 2 solutions are true because of the interval specified. If we did not specify and interval, you answer would be based on the period of tan x which is π and your only answer would be the first answer.
Quick review of Identities
Reciprocal Identities Also true:
Pythagorean Identities These are crucial! You MUST know them.
sin 2 cos 2 tan 2 cot 2 sec 2 csc 2 (Add the top of the triangle to = the bottom) 1
SStrategies CChange all functions to sine and cosine (or at least into the same function) SSubstitute using Pythagorean Identities CCombine terms into a single fraction with a common denominator SSplit up one term into 2 fractions MMultiply by a trig expression equal to 1 FFactor out a common factor
Hint: Make the words match so use a Pythagorean identity Quadratic: Set = 0 Combine like terms Factor—(same as 2x 2 -x-1)
You cannot divide both sides by a common factor, if the factor cancels out. You will lose a root…
Common factor— lost a root No common factor = OK
Sometimes, you must square both sides of an equation to obtain a quadratic. However, you must check your solutions. This method will sometimes result in extraneous solutions.
SSolve cos x + 1 = sin x in [0, 2π) TThere is nothing you can do. So, square both sides ((cos x + 1) 2 = sin 2 x ccos 2 x + 2cosx + 1 = 1 – cos 2 x 22cos 2 x + 2cosx = 0 NNow what? Remember—you want the words to match so use a Pythagorean substitution!
SSolve 2cos3x – 1 = 0 for [0,2π) 22cos3x = 1 ccos3x = ½ HHint: pretend the 3 is not there and solve cosx = ½. AAnswer: But….
In our problem 2cos3x – 1 = 0 What is the 2? What is the 3? This graph is happening 3 times as often as the original graph. Therefore, how many answers should you have? amplitude frequency 6
Add a whole circle to each of these And add the circle once again.
Final step: Remember we pretended the 3 wasn’t there, but since it is there, x is really 3x:
WWork the problems by yourself. Then compare answers with someone sitting next to you. RRound answers: 11. csc x = -5 (degrees) 22. 2 tanx + 3 = 0 (radians) 33. 2sec 2 x + tanx = 5 (radians)
44. 3sinx – 2 = 5sinx – 1 55. cos x tan x = cos x 66. cos 2 - 3 sin = 3