) can be written using the formula: cos(A+B)=cos‌A‌cos‌B−sin‌Asin‌B. cos‌
Ï€
3
=‌
1
2
and sin‌‌
Ï€
3
=‌
√3
2
. So, 3‌cos(θ+‌
Ï€
3
)=3(cos‌θ⋅‌
1
2
−sin‌θ⋅‌
√3
2
)=‌
3
2
‌cos‌θ−‌
3√3
2
sin‌θ Step 2: Combine all terms Now put this back into the main expression: 5sin‌θ+‌
3
2
‌cos‌θ−‌
3√3
2
sin‌θ+3 Combine the sin‌θ terms: 5sin‌θ−‌
3√3
2
sin‌θ=(5−‌
3√3
2
)sin‌θ So the expression becomes: (5−‌
3√3
2
)sin‌θ+‌
3
2
‌cos‌θ+3 Step 3: Find the maximum and minimum values This is now in the form asin‌θ+b‌cos‌θ+c, where a=5−‌
3√3
2
,b=‌
3
2
,c=3. The largest and smallest values of asin‌θ+b‌cos‌θ are R and −R, where R=√a2+b2.Calculate R:R=√(5−‌
3√3
2
)2+(‌
3
2
)2 Expanding: =√25+‌
27
4
−15√3+‌
9
4
=√25−15√3+9=√34−15√3 Step 4: Substitute the extreme values The minimum value is −R+3=α. The maximum value is R+3=β. So, α=−R+3 and β=R+3. Step 5: Calculate the final result Find α+β : α+β=(−R+3)+(R+3)=6 Find α−β : α−β=(−R+3)−(R+3)=−2R Now, (α−β)(α+β−6)=(−2R)×(6−6)=(−2R)×0=0