Download 5.1 Notes - Madison County Schools

5.1 Notes
AMP Pre-Cal
Learning Targets
1. Know the fundamental identities: reciprocal, quotient, Pythagorean, co-function and even-odd.
2. Rewrite trigonometric expression using the following techniques:
a. Rewrite with sine and/or cosine
b. Use the fundamental identities listed below
c. Factor with the GCF
3. Solving trigonometric equations
WHY? Read the chapter overview on page 404.
 This chapter is particularly important for those continuing on into college mathematics
o (not JUST math majors….but DEFINITELY them)
 We are now shifting our focus from problem solving to theory and proof.
 We are studying the connections among the trigonometric functions themselves.
Identities: Mathematical sentences that are true for all values of the variable for which both sides of the equation are defined.
 Reciprocal Identities:
1
csc 𝜃 =
sin 𝜃
sin 𝜃 =
1
csc 𝜃
sec 𝜃 =
1
cos 𝜃
cot 𝜃 =
1
tan 𝜃
cos 𝜃 =
1
sec 𝜃
tan 𝜃 =
1
cot 𝜃
 Quotient Identities:
tan 𝜃 =
 Pythagorean Identities:
sin 𝜃
cos 𝜃
cot 𝜃 =
cos 𝜃
sin 𝜃
 Cofunction Identities (aka Complementary Identities)
 Odd-Even Identities:
sin(−𝜃) =
𝑐𝑜𝑠(−𝜃) =
tan(−𝜃) =
csc(−𝜃) =
𝑠𝑒𝑐(−𝜃) =
cot(−𝜃) =
Now we are going to put all those identities to work…
To simplify or rewrite trigonometric expressions:
 The final answer should be simpler than the original expression.
 Prefer one trig. function.
 Prefer NO fractions.
 If you are stuck try the following strategies (more strategies in the next lesson)
By changing to sines and cosines. . .
Example1: Simplify the expression
cot 𝜃 sec 𝜃 sin 𝜃
By using Pythagorean and/or Co-Function identities. . .
Example2: Simplify the expression
𝜋
sin2 𝜃 + cos( − 𝜃) − 1 + cos 2 𝜃
2
By factoring the GCF…
Example3: Simplify the expression
cos 𝜃 − cos 𝜃 sin2 𝜃
By using Even-Odd and Reciprocal identities…
Example4: Simplify the expression
sin(−𝑥) csc(𝑥)
Solving Trigonometric Equations
We will use the identities, simplifying process, and our knowledge of the unit circle to solve more complex trig equations.
To Solve Trig Equations:
1. Use identities to get one trigonometric function, if possible.
2. Identify your variable.
a. If 𝑥 only OR 𝑥 2 only … isolate 𝑥.
b. If 𝑥 and 𝑥 2 … factor so you can use the Zero Product Property to isolate 𝑥.
3. Use the inverse to find the missing angle.
4. Give the correct solution(s) based on the directions and context of the problems.
Example5: Solve each equation for [0, 2𝜋).
a. tan2 𝑥 + tan 𝑥 = 0
b. 2 sin2 𝑥 − 5 sin 𝑥 + 2 = 0
Example6: Find all real solutions.
a. 2 sin2 𝑥 − 1 = 0
b. 4 sin2 𝑥 − 4 sin 𝑥 + 1 = 0
HW: page 410: 1-21 every other odd, 23-55 odd