# Ch 5  Trigonometric Identities

## Ch 5.1  Trigonometric Identities

Fundamental Trigonometric Identities: Reciprocal, Quotient, and Pythagorean Identities

Verifying sin2(θ)+cos2(θ)=1 for 150º
Verifying tan2(θ)+1=sec2(θ) for 45º

Negative Angle identities : sin(–θ), cos(–θ), tan(–θ), csc(–θ), sec(–θ), and cot(–θ)

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

Simplifying sec(x)·cos(x)–cos2(x) using identities

Simplifying the following Trigonometric Expressions using Identities:

sin2(x)·cot(x)·csc(x)

cos2(t)-1 / cos2(t)tan2(t)

Simplifying [1–cos2(x)][1+cos2(x)]

Simplifying Trigonometric Expressions Involving Fractions:

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

cos(x)-1/sin(x) – sin(x)/cos(x)-1

sin2(x)-tan2(x) / tan2(x) sin2(x)

[1- tan2(x)/1+ tan2(x)]+1

sin(θ)/1+cos(θ) + 1+cos(θ)/sin(θ)

## Ch 5.2  Verifying Trigonometric Identities

Simplifying [(cos(x)–1][cos(x)+1]

[sec(x)+tan(x)][sec(x)–tan(x)]

Simplifying [tan(θ)+cos2(θ)+sin2(θ)][tan(θ)–cos2(θ)–sin2(θ)]

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

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;

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

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

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

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

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

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

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

## 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°);

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

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

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;

Solving cos (2θ+16°)=sin(θ+11°);

Solving cot(θ)=tan(θ+  π/6)

## 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);

Determining the Exact Value of sin(75°);

Determining the Exact Value of sin(20°)cos(40°)–cos(20°)sin(40°)

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

Determining the Exact Value of tan( π/12)

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

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

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

Finding sin(2x) if tan(x)= 5/3 and x is in Quadrant I

## 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

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

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

Finding sin(2θ), sin(2θ) and tan(2θ) from tanθ=( 2/3) and sinθ<0

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

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

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)

## 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

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

Rewriting [cos(2x)]4 using Half Angle Identity;

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

Determining sin(112.5°) using Half Angle Identity

Determining sin(22.5°) using Half Angle Identity

Finding sin(a/2) if cos(a)=(3/5) for 0°≤a≤90°