Reduction principle and simple trigonometric

1) sin225*cos315* - tan120*/sin(-60*)tan210*
= - sin 45* cos45* - (-tan 60*) / -sin 60* tan30*
= - sin 45* cos45* - tan 60* / sin 60* tan30*
= - (√2 /2 )( √2 /2 ) - √3 / (√3 / 2 )( 1 / √3 )
= - 1/2 - 2√3

2) 5sin217*cos233*+5sin127*cos323*
= 5 ( sin37*cos53* + sin53*cos37* )
= 5 sin ( 37* + 53*)
= 5 sin 90*
= 5

3) 5sin^2θ-2sinθcosθ-3cosθ^2=0
Dividing by cosθ^2,
5tanθ^2 - 2 tanθ - 3 = 0
( 5 tanθ + 3 )( tan θ - 1 ) = 0
tan θ = -3/5 or tan θ = 1
θ = 149* , 329* or θ = 45*, 225* ( cor. to 3 s.f.)

4) 4sinA=3tanA
4 sin A = 3 sin A / cos A
4 sin A cos A = 3 sin A
sin A ( 4 cos A -3 ) = 0
sin A = 0 or cos A = 3/4
A = 180* or A = 41.4*, 319* (cor. to 3s.f.) for 0<θ<360

(5) Slove cos(3θ-45*)=1/2
3θ-45*= 60* , 300*, 420*, 660*, 780*, 1020*
3θ = 105*, 345*, 465*, 705*, 825*,1065*
θ = 35*, 115*, 155*, 235*, 275*, 355* for 0<θ<360

Sorry that I haven't come across the term "Reduction principle" but anyway, I work out all the questions as far as I know. Hope it helps.

1) sin225cos315- tan120/sin(-60)tan210
= - sin 45 cos45+ tan 60 / -sin 60 tan30
= - sin 45 cos45 + tan 60 /-sin 60 tan30
= - (√2 /2 )( √2 /2 ) - √3 / (√3 / 2 )( 1 / √3 )
= - 1/2 - 2√3

2) 5sin217cos233+5sin127cos323
= 5 ( (-sin37)(-cos53) + sin53cos37 )
= 10 sin ^2 53
=6.378186779...

3) 5sin^2θ-2sinθcosθ-3cosθ^2=0
(5 tanθ + 3)(tan θ - 1) = 0
tan θ = -3/5 or tan θ = 1
θ=149.036 or 329.036 or 45 or 225

4) 4sinA=3tanA
4 sin A=3 sin A/cos A
4cos A=3 or sinθ=0
θ=0(reject) or 41.4

5)cos(3θ-45*)=1/2where0<θ<360
3θ-45= 60 or 300 or 420 or 660 or 780 or 1020
3θ = 105 or 345 or 465 or 705 or 825 or 1065
θ = 35 or 115 or 155 or 235 or 275 or 355

2007-06-16 19:24:57 補充：
sor第二題係5 ( (-sin37)(-cos53) sin53cos37 ) =5(sin^2 37 sin^2 53) sin@=cos(90-@) & sin^2@ sin^2(90-@)=1=5

2007-06-16 19:26:54 補充：
4)θ=0(reject) or 41.4θ=360-41.4=315.6 or θ=180

Reduction principle 好似 outsee 左