Ch 5 Trigonometric Identities

Find the exact value of cos(6π/12)+cos(2π/12)

cos(6π/12)+cos(2π/12)=(√3)/2 (cosA+cosB=2cos((A+B)/2)cos((A-B)/2))

Find the exact value of cos(6π/12)-cos(2π/12)

cos(6π/12)-cos(2π/12)=-(√3)/2 (cosA-cosB=-2sin((A+B)/2)sin((A-B)/2))

If sinθ=1/2, find cos (θ-(π/2))

cos(θ-(π/2))=1/2 (cos(-θ)=cosθ)

If sinθ=1/3, find cos(θ-(π/2))

cos(θ-(π/2))=1/3 (sinθ=cos((π/2)-θ))

Find the exact value of cos112.5°

cos112.5°=(√(2-√2))/2 (cos(θ/2)=±√((1+cosθ)/2))

Find the exact value of cos15°

cos15°=((√6)+(√2))/4 (cos(A-B)=cosAcosB+sinAsinB)

Find the exact value of cos15°

cos15°=((√6)-(√2))/4 (cos(A+B)=cosAcosB-sinAsinB)

If sinθ=2/5 on the interval (0, (π/2)), find cos2θ

cos2θ=17/25 (cos2θ=1-2sin^2θ)

If cosθ=6/7 on the interval (0, (π/2)), find cos2θ

cos2θ=23/49 (cos2θ=2cos^2θ-1)

If cosθ=4/5 on the interval (0, (π/2)), find cos2θ

cos2θ=7/25 (cos2θ=cos^2θ-sin^2θ)

Rewrite cos6xcos3x as a sum

cos6xcos3x=(1/2)cos3x+(1/2)cos9x (cosAcosB=(1/2)(cos(A-B)+cos(A+B)))

Rewrite cos6xsin4x as a difference

cos6xsin4x=(1/2)sin10x-(1/2)sin2x (cosAsinB=(1/2)(sin(A+B)-sin(A-B)))

Rewrite cos^4θ in terms with no power greater than 1

cos^4θ=(1/8) (3+4cos2θ+cos4θ) (cos^2θ=(1+cos2θ)/2)

If sinθ=1/2 and cosθ>0, find cosθ

cosθ= (√3)/2 (sin^2θ+cos^2θ=1)

If secθ=4, find cosθ

cosθ=1/4 (cosθ=1/secθ)

If tanθ=1, find cot(θ-(π/2))

cot(θ-(π/2))=-1 (tanθ=cot((π/2)-θ))

If tanθ=4/5, find cot(θ-(π/2))

cot(θ-(π/2))=-4/5 (cot(-θ)=-cotθ

If sinθ=1/4 and cos θ=3/4, find cotθ

cotθ=3 (cotθ=cosθ/sinθ)

If tanθ=3/8, find cotθ

cotθ=8/3 (cotθ=1/tanθ)

If secθ=10, find csc(θ-(π/2))

csc(θ-(π/2))=-10 (csc(-θ)=-cscθ)

If secθ=2, find csc(θ-(π/2))

csc(θ-(π/2))=-2 (secθ=csc((π/2)-θ))

If sinθ=3/7, find cscθ

cscθ=7/3 (cscθ=1/sinθ)

If cscθ=11, find sec(θ-(π/2))

sec(θ-(π/2))=11 (cscθ=sec((π/2)-θ))

If cscθ=4, find sec(θ-(π/2))

sec(θ-(π/2))=4 (sec(-θ)=secθ)

If cosθ=2/5, find secθ

secθ=5/2 (secθ=1/cosθ)

Find the exact value of sin(6π/12) + sin(2π/12)

sin(6π/12) + sin(2π/12)=3/2 (sinA+sinB=2sin((A+B)/2)cos((A-B)/2))

Find the exact value of sin(6π/12)-sin(2π/12)

sin(6π/12)-sin(2π/12)=1/2 (sinA-sinB=2cos((A+B)/2)sin((A-B)/2))

If cosθ=1/2, find sin(θ-(π/2))

sin(θ-(π/2))=-1/2 (sin(-θ)=-sinθ)

If cosθ=1/4, find sin(θ-(π/2))

sin(θ-(π/2))=-1/4 (cosθ=sin((π/2)-θ))

Find the exact value of sin105°

sin105°=((√2)+(√6))/4 (sin(A+B)=sinAcosB+cosAsinB)

Find the exact value of sin112.5°

sin112.5°=(√(2+√2))/2 (sin(θ/2)=±√(1+cosθ)/2)

Find the exact value of sin15°

sin15°=((√6)-(√2))/4 (sin(A-B)=sinAcosB-cosAsinB)

If sinθ=4/7 on the interval ((π/2),π), find sin2θ

sin2θ=(8√33)/49 (sin2θ=2sinθcosθ)

Rewrite sin4xcos3x as a sum

sin4xcos3x=(1/2)sin7x+(1/2)sinx (sinAcosB=(1/2)(sin(A+B)+sin(A-B)))

Rewrite sin5xsin3x as a difference

sin5xsin3x=(1/2)cos2x-(1/2)cos8x (sinAsinB=(1/2)(cos(A-B)-cos(A+B)))

Rewrite sin^2(x+1) in terms with no power greater than 1

sin^2(x+1)=(3-cos2x)/2 (sin^2θ=(1-cos2θ)/2)

If tanθ=-2 and sinθ>0, find sinθ and cosθ

sinθ=(2√5)/5, cosθ=-(√5)/5 (tan^2θ+1=sec^2θ)

If cotθ=2 and sinθ>0, find sinθ and cosθ

sinθ=(√5)/5, cosθ=(2√5)/5 (cot^2θ+1=csc^2θ)

If cscθ=5/3, find sinθ

sinθ=3/5 (sinθ=1/cscθ)

If cotθ=2/3, find tan(θ-(π/2))

tan(θ-(π/2))=-2/3 (tan(-θ)=-tanθ)

If cotθ=0, find tan(θ-(π/2))

tan(θ-(π/2))=0 (cotθ=tan((π/2)-θ))

Find the exact value of tan112.5°

tan112.5°=(-2-√2)/√2 (tan(θ/2)=(1-cosθ)/sinθ)

Find the exact value of tan112.5°

tan112.5°=-√((2+√2)/(2-√2)) (tan(θ/2)=±√((1-cosθ)/(1+cosθ)))

Find the exact value of tan112.5°

tan112.5°=-√2/(2-√2) (tan(θ/2)=sinθ/(1+cosθ))

Find the exact value of tan15°

tan15°=(3-√3)/(3+√3) (tan(A-B)=(tanA-tanB)/(1+tanAtanB)

If tanθ=3/5, find tan2θ

tan2θ=15/8 (tan2θ=(2tanθ)/(1-tan^2θ)

Find the exact value of tan75°

tan75°=((√3)+3)/(3-(√3)) (tan(A+B)=(tanA+tanB)/(1-tanAtanB))

Rewrite tan^4θ in terms with no power greater than 1

tan^4θ=(3-4cos2θ+cos4θ)/(3+4cos2θ+cos4θ) (tan^2θ=(1-cos2θ)/(1+cos2θ)

If sinθ=1/2 and cosθ=1/4, find tanθ

tanθ=2 (tanθ=sinθ/cosθ)

If cotθ=7/3, find tanθ

tanθ=3/7 (tanθ=1/cotθ)