Chapter 5 Trigonometry

Ch 5  Trigonometric Identities

Ch 5.1  Trigonometric Identities

Fundamental Trigonometric Identities: Reciprocal, Quotient, and Pythagorean Identities
http://www.youtube.com/watch?v=OmJ5fxyXrfg

Verifying sin2(θ)+cos2(θ)=1 for 150º
Verifying tan2(θ)+1=sec2(θ) for 45º
http://www.youtube.com/watch?v=KSmaRyDsNDs

Negative Angle identities : sin(–θ), cos(–θ), tan(–θ), csc(–θ), sec(–θ), and cot(–θ)
http://www.youtube.com/watch?v=WEOCLWiDF8Q

Simplifying tan3(x)·sec3(x) using identities

http://www.youtube.com/watch?feature=player_embedded&v=SZu_EVV4jjY

Simplifying sec(x)·cos(x)–cos2(x) using identities
http://www.youtube.com/watch?feature=player_embedded&v=FS6iQX7jY-s

Simplifying cotx·tanx–cos2http://www.youtube.com/watch?v=IaSzgJnc5wg&list=PL86281C72D802CE05&index=67&feature=plpp_video

Simplifying the following Trigonometric Expressions using Identities:

sin2(x)·cot(x)·csc(x)

cos2(t)-1 / cos2(t)tan2(t)

http://www.youtube.com/watch?v=Hf0AciRDDaE

Simplifying [1–cos2(x)][1+cos2(x)]
http://www.youtube.com/watch?v=C1nJIaxkWPs

Simplifying Trigonometric Expressions Involving Fractions:

sin(x)sec(x)/tan(x) and csc(θ)cot(θ)/tan(θ)sec(θ)

http://www.youtube.com/watch?feature=player_embedded&v=HGIHD7zee0Q

cos(x)-1/sin(x) - sin(x)/cos(x)-1

http://www.youtube.com/watch?v=Jx6zJUbWJKY&list=PL86281C72D802CE05&index=55&feature=plpp_video

sin2(x)-tan2(x) / tan2(x) sin2(x)

http://www.youtube.com/watch?v=3cZDPvJ7wMs

[1- tan2(x)/1+ tan2(x)]+1

http://www.youtube.com/watch?v=yRFzinqxt9o

sin(θ)/1+cos(θ) + 1+cos(θ)/sin(θ)
http://www.youtube.com/watch?v=VLBUtX5tuJE

Ch 5.2  Verifying Trigonometric Identities

Simplifying [(cos(x)–1][cos(x)+1]

[sec(x)+tan(x)][sec(x)–tan(x)]
http://www.youtube.com/watch?v=e0K-gWF9Y5g&list=PL86281C72D802CE05&index=56&feature=plpp_video

Simplifying [tan(θ)+cos2(θ)+sin2(θ)][tan(θ)–cos2(θ)–sin2(θ)]
http://www.youtube.com/watch?v=jbxM8CSOOLg&list=PL86281C72D802CE05&index=57&feature=plpp_video

Factoring sin2(θ)+cot2(θ) sin2(θ); 2- [cos2(x)/1-sin(x)]

http://www.youtube.com/watch?v=2tiv14iok0k&list=PL86281C72D802CE05&index=58&feature=plpp_video

2cos2(x)+2cos (x)–24  http://www.youtube.com/watch?v=IHieGivL8Cw&list=PL86281C72D802CE05&index=59&feature=plpp_video

Verifying that each Trigonometric Equation is An Identity :

cos(θ)tan(θ)=sin(θ);

cos2(θ)/1+sin(θ)=1-sin(θ);

[sin(θ)/1+cos(θ)]+[1+cos(θ)/sin(θ)]=2csc(θ);

sin2(θ)/cos(θ)=sec(θ)-cos(θ);

sin4(θ)–cos4(θ)=2sin2(θ)–1;
http://www.youtube.com/watch?v=Zktxkfr9zJE

tan(x)+cot (x)=sec(x)csc(x);

sec2(x)-1/sec2(x)=sin(x);

cos(x)/1-sin(x)= sec(x)+tan(x)

http://www.youtube.com/watch?v=zJTpoogPoJk

cos(θ)tan(θ)=sin(θ);
cos(x)[csc(x)+tan(x)]=cot(x)+sin(x);

http://www.youtube.com/watch?v=9uoKutwuCio

Verifying   [1-sin(t)/cos(t)]+[cos(t)/ 1-sin(t)]=2sec(t)     is an Identity

http://www.youtube.com/watch?feature=player_embedded&v=IE8q4WRubC4

Verifying    sec(x)/ 1+sec(x) = 1-cos(x)/ sin2(x)    is an Identity

http://www.youtube.com/watch?v=4QmzqwQSt8A&list=PL86281C72D802CE05&index=65&feature=plpp_video

Verifying    1+2cot2(x)+cot4(x)/ 1-cot2(x)= csc4(x)/ 1-cot2(x)  is an Identity

http://www.youtube.com/watch?v=dpFhEqdS3bU&list=PL86281C72D802CE05&index=66&feature=plpp_video

Ch 5.3  Sum and Difference Identities for Cosine

Sum and Difference Identities for Cosine :

Finding cos(A+B) if A=12/13  in Quadrant II and B= 4/5 in Quadrant I;
Determining the Exact Value of cos15°;
Determining the Exact Value of cos( 7π/12);
Determining the Exact Value of cos(40°)cos(50°)–sin(40°)sin(50°);
http://www.youtube.com/watch?v=H-0jQTzfkWQ

Verifying cos(x+ π/4)=√2/2[ cos(x) – sin(x)]

http://www.youtube.com/watch?v=FRNiVsS5bVc

Finding the function values and the Quadrant of A–B :

Finding cos(x–y) from tan(x)= -7/12 and cos(y)= 2/5 Where x and y Are in Quadrant IV
http://www.youtube.com/watch?v=yklLtxBOb4s&feature=player_embedded

Discussing Cofunction Identities :

Writing sin(18º); tan(65º), and csc(84º) in terms of Cofunction;
Writing cos(π/4) ); cot( π/3); sec( π/6) in terms of Cofunction;
http://www.youtube.com/watch?v=_gkuml–4_Q&feature=youtu.be

Solving cos (2θ+16°)=sin(θ+11°);

Solving cot(θ)=tan(θ+  π/6)
http://www.youtube.com/watch?v=S4D-0ykqL_8&feature=youtu.be

Ch 5.4  Sum and Difference Identities for Sine and Tangent

Sum and Difference Identities for Sine :

Finding sin(A–B) from sin(A)= 4/5 in Quadrant II and cos(B)=-5/13  Quadrant III
Determining the Exact Value of sin(105°);
Determining the Exact Value of sin(- π/12);
http://www.youtube.com/watch?v=hiNDDQyee2E

Determining the Exact Value of sin(75°);
http://www.youtube.com/watch?v=NZ2Y5_XxzTc&feature=player_embedded

Determining the Exact Value of sin(20°)cos(40°)–cos(20°)sin(40°)
http://www.youtube.com/watch?v=TNcNvSCBD30&feature=player_embedded

Determining the Exact Value of tan(–105°);

Determining the Exact Value of tan( 5π/12);
Using an Identity to Write tan(π–θ) as a Single Function of θ
http://www.youtube.com/watch?v=OQP78bwYcWw

Determining the Exact Value of tan( π/12)
http://www.youtube.com/watch?v=BuIhkzkJseM

Simplify tan(x+4π) using Sum and Difference Identities

http://www.youtube.com/watch?feature=player_embedded&v=7fy2U0Sm1Vc

Simplify tan(4π–x) using Sum and Difference Identities
http://www.youtube.com/watch?v=75PmzAEO89A&feature=player_embedded

Finding tan(2x) if sin(x)=12/13  and x is in Quadrant I

http://www.youtube.com/watch?v=CvKaolqn2-Q&feature=player_embedded

Finding sin(2x) if tan(x)= 5/3 and x is in Quadrant I
http://www.youtube.com/watch?v=G4s7ui6HmRY&feature=player_embedded

Ch 5.5  Double-Angle Identities

Double Angle Identities :

Finding the Exact Value of cos(2A), sin(2A) and tan(2A) and Quadrant of 2A  if  sin(A)=5/3  is in Quadrant II
Finding cos(A) given cos(2A)=-3/4  where 2A is in Quadrant III
http://www.youtube.com/watch?v=-zhCYiHcVIE

Using Double Angle Identities to Simplify and then Evaluate :
cos2( π/12)–sin2(π/12 )
2sin(π/4 )–cos(π/4 )
2cos2(π/2 )–1
http://www.youtube.com/watch?v=Ukq-9RzR5-M

Example : Determining Double Angle Trigonometric Function Values with Given Quadrant :

Finding sin(2θ), sin(2θ) and tan(2θ) from cos(θ)=-5/13 and θ is in Quadrant II
http://www.youtube.com/watch?v=Mkglhc1wYYo

Finding sin(2θ), sin(2θ) and tan(2θ) from tanθ=( 2/3) and sinθ<0
http://www.youtube.com/watch?v=uI_vx1qhruM

Verifying [sin(A)+cos(A)]2= sin(2A)+1

http://www.youtube.com/watch?v=FRNiVsS5bVc

Simplifying 1–16sin2(x)cos2(x) using Double Angle Identity

http://www.youtube.com/watch?v=eQW9a24flZA&feature=player_embedded

 

Product to Sum and Sum to Product Identities :

Product to Sum: sin(–4θ) sin(8θ);
Product to Sum: 2cos(7t/2)–cos(3t/2);
Product to Sum : sin(7π/8)–cos(π/8);
Sum to Product :cos(9x)+cos(4x);
Sum to Product : sin(17π/12)–sin(13π/12)
http://www.youtube.com/watch?v=ps4Z01gFOpM

Ch 5.6   Half-Angle Identities

Half Angle Identities  :

Finding the Exact Value of sin(π/8);
Determining the Exact Value of (cos105°);
Finding cos(A/2), sin(A/2) and tan(A/2) from cosA=(-2/3) in Quadrant II
http://www.youtube.com/watch?v=Rp61qiglwfg

Rewriting [sin(5x)]2 using Half Angle Identity;

Rewriting [cos(2x)]4 using Half Angle Identity;
http://www.youtube.com/watch?v=x6pfqRQ89fA

Determine cos(5π/12) using Half Angle Identity

http://www.youtube.com/watch?v=zaywBux2dv4

Determining sin(112.5°) using Half Angle Identity

http://www.youtube.com/watch?v=q4XvvKKGhTc

Determining sin(22.5°) using Half Angle Identity
http://www.youtube.com/watch?feature=player_embedded&v=uFbbF-IYFjM

Finding sin(a/2) if cos(a)=(3/5) for 0°≤a≤90°
http://www.youtube.com/watch?feature=player_embedded&v=FFXaeJYaGVY

Determining tan(π/8) using Half Angle Identity

http://www.youtube.com/watch?v=Nks0B4XB4QA

Determining tan(105°) using Half Angle Identity
http://www.youtube.com/watch?feature=player_embedded&v=Q48HHoLauyg

Verifying sin(2A)/2sin(A)=cos2(A/2)-sin2(A/2) using Half Angle and Double Angle Identities

http://www.youtube.com/watch?v=FRNiVsS5bVc (Last Part of the Video)