(5) Trigonometric functions

Please find the answers below:

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cos(θ+α)=p
cosθcosα-sinθsinα=p
sinθ=(cosθcosα-p)/sinα----(1) or cosθ=(p+sinθsinα)/cosα------(3)

sin(θ+x)=q
sinθcosx+cosθsinx=q
sinθ=(q-cosθsinx)/cosx-----(2) or cosθ=(q-sinθcosx)/sinx----------(4)

sub (1) into (2)
(cosθcosα-p)/sinα=(q-cosθsinx)/cosx
cosx(cosθcosα-p)=sinα(q-cosθsinx)
cosxcosθcosα-pcosx=qsinα-sinαcosθsinx
cosθ(cosαcosx+sinαsinx)=pcosx+qsinα
cosθcos(α-x)=pcosx+qsinα
cosθ=pcosx+qsinα)/(cos(α-x))

sub(3) into (4)
(p+sinθsinα)/cosα=(q-sinθcosx)/sinx
sinx(p+sinθsinα)=cosα(q-sinθcosx)
psinx+sinxsinθsinα=qcosα-cosαsinθcosx
sinxsinθsinα+cosαsinθcosx=qcosα-psinx
sinθ(cosαcosx-sinαsinx)=qcosα-psinx
sinθcos(α-x)=qcosα-psinx
sinθ=(qcosα-psinx)/(cos(α-x))

Hence show that p²+q²+2pqsin(α-x)=cos²(α-x).

cosθ=(pcosx+qsinα)/(cos(α-x))-----(1)
sinθ=(qcosα-psinx)/(cos(α-x)) -----(2)

(1)²+(2)²
1=(pcosx+qsinα)²/cos²(α-x) +(qcosα-psinx)²/cos²(α-x)
cos²(α-x)=(pcosx+qsinα)² +(qcosα-psinx)²
cos²(α-x)=p²cos²x+2(pcosx)(qsinα)+q²sin²α +q²cos²α-2(qcosα)(psinx)+p²sin²x
cos²(α-x)=p²(cos²x+sin²x)+2pq(sinαcosx-sinxcosα)+q²(sin²α+q²cos²α)
p²+q²+2pqsin(α-x)=cos²(α-x).

Let cos(θ+α)=p and sin(θ+x)=q
a) Express cosθ and sinθ in erms of α,x,p and q.
b) Hence show that p²+q²+2pqsin(α-x)=cos²(α-x).

a)...............cos(θ+α)=p
cosθcosα – sinθsinα = p
.......................cosθ = ( p + sinθsinα ) / cosα ------(1)

...................sin(θ+x)=q
sinθcosx + cosθsinx = q
.........................sinθ= (q – cosθsinx ) / cosx --------(2)

Sub (2) in (1),
...................................cosθ = { p + [(q – cosθsinx ) / cosx )]sinα } / cosα
...................................cosθ = ( p cosx + q sinα– cosθsinx sinα ) / cosαcosx
.....................cosθcosαcosx = p cosx + q sinα– cosθsinx sinα
cosθ (cosαcosx + sinx sinα ) = p cosx + q sinα
..................cosθ cos (α+ x ) = p cosx + q sinα
.....................................cosθ= ( p cosx + q sinα ) / cos (α+ x )

# 好不容易會做了. follow the steps do it again for finding sinθ.
# i do not know how to do the second part. Sorry!

2007-03-01 20:48:05 補充：
Something wrong! last 2 line of Subit should be ..................cosθ cos (α - x ) = p cosx + q sinα.....................................cosθ= ( p cosx + q sinα ) / cos (α - x )there should be ' - ' in cos (α - x )