1. trig curve

I have 7 problems here....
i need the complete solution please...

1. sketch the graph of the function defined by f(x)=-2tan(1/2(5pi-x))sin(1/2(x-4pi))+3. determine the domain, range, amplitude (if it has), period, asymptotes (if any).

2. if z = tan(x/2), express tanx/1-2sinx+3cosx in terms of z.

3. determine the fundamental solution set of the equation sinx=sin5x

4. prove or disprove:
sin a + sin b = 2sin((a+b)/2)cos((a-b)/2)

5. a picture x feet high is placed on a wall with its base y feet above the level of the eye of an observer. if the observer is z feet from the wall and Q is the radian measure of the angle subtended at the observer's eye by the picture, show that

Q= Arctan((xz)/z*2+xy+y*2)

6. prove heron's formula using cosine law

7. find the exact yalue of 100(cos1+cos2+...+cos44)/(sin1+sin2+...+sin44)

thanks...

2. 4. prove or disprove:
sin a + sin b = 2sin((a+b)/2)cos((a-b)/2)

sin(a+b)=cosaxsinb+cosbxsina
sin(a-b)=sinaxcosb- cosaxsinb
sin(a+b)+sin(a-b)=sinaxcosb+cosbxsina
=2sinaxcosb
(a+b)=A ---a=(A+B)/2
(a-b)=B ---b=(A-B)/2

So, sinA+sinB=2xsin(A+B)/2cos(A-B)/2

3. 3. determine the fundamental solution set of the equation sinx=sin5x

Since 5x and x has the same sinuse, they have to do with the formulas:
5x=k.360+x
5x=k.360+(180-x)
All the solutions of this equation are given by the formulas:
4x=k.360 --- x=k.90
6x=k.360+180--- x= k.60+30

4. Heron's formula - Wikipedia, the free encyclopedia

This has the proof you need for 6.

My CAS couldn't come up with an answer for 7, I have a feeling it can't be done.

You can make it a bit neater though using sigma notation.

$\frac{100 \sum_{i=1}^{44} \cos{i}}{\sum_{j=1}^{44} \sin{j}}$