Download Precalc Notes Ch.5

Notes #3-___
Date:______
5.1 Fundamental Identities (423)
Identity: statements that are true for all values for which
both sides are defined (domain of validity).
x2 4
Example:
 x2
x2
Reciprocal Identities
1
1
sin 
cos 
csc
sec
csc 
1
sin 
sec 
Quotient Identities
sin 
tan 
cos
1
cos
cot  
tan 
1
cot 
cot  
1
tan 
cos
sin 
Pythagorean Identities
(sin θ)2 = sin2 θ
Why?
1
8
y
30°
30°
x
x2 + y2 = 1
1) sin2 θ + cos2 θ = 1
÷ by sin2 θ
2)
÷ by cos2 θ
3)
When simplifying:
#1 Look for identities.
#2 Reduce everything to sine and cosine.
Ex.1 Use basic identities to simplify the expressions:
a) cot θ (1 – cos2 θ)
b) tan θ ( csc θ)
Ex.2 Simplify the expressions:
a) (sin x – 1)(sin x + 1)
b)
(csc x 1)(csc x 1)
cos2 x
Most of the trig equations we will be solving have special
angles for solutions. Often answers are restricted to the
interval [0, 2π). You may get no solutions for part or all of
an equation.
Ex.3 Isolate the trig function:
Ex.4 Extract square roots:
Summary:
2tanx – 2 = 0.
4sin2x – 4 = 0.
Notes #3-___
Date:______
How do the 2 non-rt
angles of a rt triangle
compare?
cos θ is positive in
quadrants I & IV.
5.1 Fundamental Identities Day - 2
Cofunction Identities
sin (90° – θ) = cos θ
cos (90° – θ) = sin θ
sec (90° – θ) = csc θ
csc (90° – θ) = sec θ
tan (90° – θ) = cot θ
cot (90° – θ) = tan θ
Odd-Even Identities
sin (-x) = -sin x
cos (-x) = cos x
tan (-x) = -tan x
csc (-x) = -csc x
sec (-x) = sec x
cot (-x) = -cot x
1
30°
sin 30° =
cos 60° =
sin (90° – 60°) = cos 60°
(
,
)
(
,
)
1
45°
-45°
1
sin 45° =
sin -45° =
Ex.1 Simplify the expressions:
a) cos (θ – 90°)
b) sin (-x) csc (-x)
Ex.2 Simplify with factoring:
cos3x + cos x(sin2 x)
Ex.3 Simplify by combining fractions:
sin x cos x

1cos x sin x
Most of the trig equations we will be solving have special
angles for solutions. Often answers are restricted to the
interval [0, 2π). You may get no solutions for part or all of
an equation.
Ex.4 Isolate the trig function:
Ex.5 Extract square roots:
2cosx – 1 = 0.
4sin2x – 3 = 0.
Ex.6 Factor: 2cos2 x + cos x = 1
Try to get different trig f(x)s into multiplication & set equal
to zero.
Ex.7 Factor: 2secx·sinx – secx = 0.
Or use identities to change to a single trig function.
sin3 x
Ex.8 a) 2sin x + 3cosx = 3 b)
 tan x
cos x
2
Occasionally we need to use a calculator to find solutions.
Ex.9 cosx = .8
Ex.10 Make the suggested trigonometric substitution and
then use the Pythagorean Identities to write the
resulting function as a multiple of a basic trig f(x).
4 x 2 , x = 2cos θ
Summary:
Notes #3-___
Date:______
5.2 Proving Trigonometric Identities (432)
1)
2)
3)
4)
5)
We are not solving! Do not do anything to both sides!
Start with the more complicated side.
Look for identities.
Change everything to sin or cos.
Use algebra (common denominators, factoring etc)
Ex.1 cos3 x = (1 – sin2 x) cos x
Ex.2
1
1

 2sec2 x
1sin x 1sin x
Ex.3 (tan2 x + 1)(cos2 x – 1) = -tan2 x
Ex.4 (cos x – sin x)2 + (cos x + sin x)2 = 2
Ex.5
sec x  tan x 
cos x
1sin x
Ex.6 tan x + cot x = sec x(csc x)
Ex.7
Summary:
sec2  1
 sin 2 
2
sec 
Notes #3-___
Date:______
5.3 Sum & Difference Identities (439)
Show that:
36 64  36  64 so in general
a b 
(2 + 3)2 ≠ 22 + 32 so in general (x + y)2 ≠
sin(u + v) = sin u cos v + cos u sin v
–
–
Ex.1 Write the expression as the sine of an angle.
a) sin(5x) cos(2x) + cos(5x) sin(2x)
b) sin

4
cos



– cos
sin
(Give the exact value)
12
4
12
cos(u + v) = cos u cos v – sin u sin v
–
+
Ex.2 Find the exact value of cos195º.
tan(u + v) =
tan u  tan v
1  tan u tan v
tan(u – v) =
tan u  tan v
1  tan u tan v
Ex.3 Write as one trig f(x) & find an exact value if possible:
a)
tan 80 tan 55
1 tan 80 tan 55
b)
tan 43 tan10
1 tan 43 tan10
Ex.4 Find the exact value of tan 345º.
Ex.5 Find cos(u – v) given cos u = 
sin v =

4
, 0v  .
2
5
Ex.6 Verify the identities:
F
I
a) sinG   J cos
H2 K
Summary:
b)
15
3
,  u
and
17
2
sin    
 tan   tan 
cos  cos 
Notes #3-___
Date:______
5.1 to 5.3 Select Review
Ex.1 Find cos(x + y) and tan(x + y) given cos y = 

2
 y   and sin x =
4

, 0 x .
2
5
12
,
13
1
Ex.2 Find sin(x – y) and tan(x – y) given cos y =  ,
3
3
3

and sin x = , 0  x  .
  y
2
2
4
Ex.3 Make the suggested trigonometric substitution and
then use the Pythagorean Identities to write the
resulting function as a multiple of a basic trig f(x).
25 x2 , x = 5cos θ
Ex.4 Use the Quadratic Formula to solve:
csc2t – 3csct – 2 = 0
Remember other answers in the interval.
Cofunctions
Sine and cosine
Tangent and cotangent
Secant and cosecant
Pythagorean Identities:
Ex.5 A ship leaves Miami with a heading 100° at 15 knots.
An hour later it changes heading to 190°. How far is
the ship 2 hrs later and what is the bearing from Miami?
163.4°
Summary:
Notes #3-___
Date:______
5.4 Multiple Angle Identities - Day 1 (446)
Double-Angle Identities:
Use the sum formulas
to derive the double
angle identities.
sin(2u) = sin(u + u) =
cos(2u) =
tan(2u) =
Use the double angle
identities for cos2u
and solve for sin2u.
Power Reducing Identities:
cos (2u) =
sin2u =
cos2u =
tan2u =
Ex.1 Find the exact values of sin 2u, cos 2u and tan 2u
3
 u  2 .
given cot u = -5,
2
Ex.2 Find the solutions to 2cos x + sin 2x = 0 in [0, 2π).
Ex.3 Find the solutions to 4sin x·cos x = 1 in [0, 2π).
Ex.4 Express sin 3x in terms of sin x.
Ex.5 Rewrite cos5x in terms of trigonometric functions with
no power greater than 1.
Ex.6 Verify: sin 2θ =
Ex.7 Verify:
Summary:
2 tan 
.
1 tan 2 
2 cos 2
= cot α – tan α.
sin 2
Notes #3-___
Date:______
5.5 Law of Sines (453)
Law of Sines (can also use the reciprocals)
a
b
c


sin A sin B sin C
 If a  h & b then there is no .
 If h  a < b then there are two s.
• h = bsinA
a
b
b
A
a
a
A
You can find the number of Δs during the problem.
Ex.1 How many triangles are there in each case?
a) A = 62°, a = 10 and b = 12
b) B = 98°, b = 10 and c = 3
c) C = 54°, a = 10 and c = 7
Ex.2-4 Solve the triangle(s) if possible.
Ex.2 A = 58°, a = 20 and c = 10
Ex.3 B = 78°, b = 207 and c = 210
Ex.4 C = 31°, b = 46 and c = 29
Ex.5 The angle of elevation to a mountain is 3.5°. After
driving 13 miles it is 9°. Approximate the height.
Summary:
Notes #3-___
Date:______
5.6 Law of Cosines & Area (461)
I. Law of Cosines
What happens if C is
a right angle?
c2 = a2 + b2 – 2abcosC
Ex.1-2 Solve the triangle:
Ex.1 A = 49°, b = 42 and c = 15
Ex.2 a = 31, b = 52 and c = 28
Ex.3 A plane takes off and travels 60 miles, then turns 15°
and travels for 80 miles. How far is the plane from
the airport?
Ex.4 Two ships leave port with a 19° angle between their
planned routes. If they are traveling at 23 mph and 31
mph, how far apart are they in 3 hours?
II. Area of a triangle =
1
absinC
2
Ex.5 Find the area of a Δ with B = 120°, a = 32 and c = 50.
III. Heron’s Area Formula
Semi-perimeter (s) = (a + b + c) / 2
Area of a triangle =
s( s  a)( s b)( s  c)
Ex.6 Find the area of a Δ if: a = 5 m, b = 6 m and c = 9 m.
Summary: