Sin [0,1]

Clear["Global`*"]

fsin[x_,a1_,a2_,a3_,a4_,a5_]=a1 Sin[1 π x]+a2 Sin[2 π x]+a3 Sin[3 π x]+a4 Sin[4 π x]+a5 Sin[5 π x];
fsin[x,1,2,3,4,5]

Plot[fsin[x,1,2,3,4,5],{x,0,1}]

Manipulate[
Plot[fsin[x,a1,a2,a3,a4,a5]//Re ,{x,0,2},PlotRange->All],
{a1,0,1},
{a2,0,1},
{a3,0,1},
{a4,0,1},
{a5,0,1}
]

Sin Part 2 [-1,1]

Clear["Global`*"]

fsin[x_,a1_,a2_,a3_,a4_,a5_]=a1 Sin[1 π x]+a2 Sin[2 π x]+a3 Sin[3 π x]+a4 Sin[4 π x]+a5 Sin[5 π x];
fsin[x,1,2,3,4,5]

Plot[fsin[x,1,2,3,4,5],{x,0,1}]

Manipulate[
Plot[fsin[x,a1,a2,a3,a4,a5]//Re ,{x,-1,1},PlotRange->All],
{a1,0,1},
{a2,0,1},
{a3,0,1},
{a4,0,1},
{a5,0,1}
]

Cos  [-1,1]

Clear["Global`*"]

fcos[x_,a1_,a2_,a3_,a4_,a5_]=a1 Cos[1 π x]+a2 Cos[2 π x]+a3 Cos[3 π x]+a4 Cos[4 π x]+a5 Cos[5 π x];
fcos[x,1,2,3,4,5]

Plot[fcos[x,1,2,3,4,5],{x,0,1}]

Manipulate[
Plot[fcos[x,a1,a2,a3,a4,a5]//Re ,{x,-1,1},PlotRange->All],
{{a1,0},-1,1},
{{a2,0},-1,1},
{{a3,0},-1,1},
{{a4,0},-1,1},
{{a5,0},-1,1}
]

Exp

Clear["Global`*"]

f1[x_,a0_,a1_,a2_,a3_,a4_,a5_]=a0+a1 (E^( I π x))+a2 (E^( I 2π x))+a3 (E^( I 3π x))+a4 (E^( I 4π x))+a5 (E^( I 5π x));
f1[x,0,1,2,3,4,5]

Plot[f1[x,0,1,2,3,4,5]//Re ,{x,0,1}]

Manipulate[
Plot[f1[x,a0,a1,a2,a3,a4,a5]//Re ,{x,0,2},PlotRange->All],
{a0,0,1},
{a1,0,1},
{a2,0,1},
{a3,0,1},
{a4,0,1},
{a5,0,1}
]