Download 5.1 Fundamental Identities: Identities Review: tan θ cot θ

Math 170
Trigonometry Lecture Notes
Chapter 5
5.1 Fundamental Identities:
Identities Review:
tan θ
cot θ
Pythagorean Identities:
Using the Identities to Simplify Expressions: (pg 273)
♦ Change everything to sine and cosine functions – Start with basic reciprocal & ratio
identities.
♦ If there are squared terms, think: ____________________
♦ 1 can be changed to the sum or difference of functions
Matching: Rewrite the functions on the left to match the ones on the right.
a) sin (-x)
1) sec2 x – 1
2) –tan x cos x
3)
sec x
csc x
b) tan x
c)
sin 2 x
cos 2 x
CA: Review Exercises – 5.1 #7 & #11 + Algebra Review Activity (on last page of these notes)
Write each expression in terms of sine, cosine. Simplify. Then, if needed, replace appropriate
quotient with tan or cot so that no quotients appear in the final answer.
1) (1 – cos x) ( 1 + sec x)
2) sin2x (csc2x – 1)
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Math 120
Trigonometry Lecture Notes
cos 2 x − sin 2 x
3)
sin x cos x
4)
Chapter 5
cos x + sin x
sin x
5.1 Verifying Trigonometric Identities
Think of these as Puzzles! 
Work with the more complicated side first. (pg 273 for solving ideas)
If you can’t get the expression on the other side, try to simplify that side as well.
sin 2 x
+ sin x csc x = sec 2 x
1)
2
cos x
3)
1
1
+
= 2 sec 2 x
1 − sin x 1 + sin x
2)
cos x + 1 cos x
=
tan 2 x sec x −1
∗4)
1 − cos x
=
(cot x − csc x) 2
1 + cos x
2
Math 120
Trigonometry Lecture Notes
Chapter 5
5.2 Sum & Difference Identities for Cosine
Cos (A + B) = cos A cos B – sin A sin B
Cos (A – B) = cos A cos B + sin A sin B
Use these to find exact values for the cosine of certain angles:
Anything value we can get by adding or subtracting our special angles.
30, 45, 60, 90, etc.
Evaluate the following:
1) cos (105⁰) = cos ( ________ + _________)
2) cos(15⁰) = cos (________– _________)
3) cos
2π
7π
2π
7π
− sin
cos
sin
9
9
9
9
4) cos (270⁰ – θ)
5) Let sin s = ⅔ and sin t = -⅓, s is in Quad II, and t is in Quad IV. Find cos(s + t) and cos (s – t)
CA 5.2 #21, 53
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Math 120
Trigonometry Lecture Notes
Chapter 5
5.2 Sum & Difference Identities for Sine & Tangent
A) Sine
B) Tangent
sin (A + B) = sin A cos B + cos A sin B
tan( A + B ) =
tan A + tan B
1 − tan A tan B
sin (A – B) = sin A cos B – cos A sin B
tan( A − B) =
tan A − tan B
1 + tan A tan B
Use the sum or difference formula for sine to find the EXACT VALUES for the following:
1) sin -15⁰
2) sin (7π/12)
3) Evaluate: sin 43⁰ cos 137⁰ + cos 43⁰ sin 137⁰
Use the sum or difference formula for tangent to find the EXACT VALUE for the following:
1) tan 75⁰ =
Let sin s = -¾ and cos t = 5/13, with s and t in Quadrant IV.
1) Find sin (s + t)
in CA book: 2) Find tan (s + t) (for above info) also do 5.2 #55, 45
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Math 120
Trigonometry Lecture Notes
Chapter 5
5.3 Double Angle Identities
A) The Identities
cos (2θ) = cos (____ + ____)
sin (2θ) = sin (____ + ____)
NOTE: cos 2θ ≠ 2 cos θ
sin 2θ ≠ 2 sin θ
tan 2θ ≠ 2 tan θ
tan (2θ) = tan
B) Applications:
1) cos θ = -4/5 and sin θ < 0. Find:
a) sin 2θ
b) cos 2θ
c) tan 2θ
CA Book: 3) Given tan x = 2 and cos x >0. Find sin 2x and cos 2x.; 5.3 #19
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Math 120
Trigonometry Lecture Notes
Chapter 5
C) Identities:
1) Use an identity to write each expression as a single trigonometric function:
a) cos2 15⁰ – sin2 15⁰
b) 2 sin 25⁰ cos 25⁰
c)
2 tan 34°
(1 − tan 2 34°)
CA: Find sin 4θ
3) Verify the following Identities:
a) (sin x + cos x)2 = sin 2x + 1
b) sin 2 x =
2 tan x
1 + tan 2 x
5.5 Identities and Formulas Involving Inverse Functions
1. Re-name the inverse function as an angle using a variable (usually θ, α, or β) Substitute this
angle variable into your function.
2. Draw a triangle for the original inverse function(s) that you replaced with variables.
Determine the dimensions of the triangle according to that substituted value.
3. Re-write the function in simplest form, using trig identities when helpful
4. Use the triangle to find the values of the re-rewritten function.
Example:
3
sin(2 tan −1 )
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In CA book: Examples: 5.5 #16; practice: 5.5 #14, 18, 20
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Math 120
Trigonometry Lecture Notes
Chapter 5
Algebra Review Activity:
a) Add:
1 5
+
x x
Write as two separate fractions. Simplify if possible.
b)
7+d
x
c)
Factor
d) x2 – 25
e) x2 – 1
7+ x
x
f) x4 – y4
Expand
 x+7
g) 

 y 
2
Multiply numerator and denominator by the conjugate of the denominator:
h)
5
x−7
7