\left[\frac{5}{\sqrt{2}}cos\:30+\frac{6}{\sqrt{2}}cos60\right]+\left[\frac{5}{\sqrt{2}}sin\:30+\frac{6}{\sqrt{2}}sin60\right]

asked by guest
on Nov 17, 2024 at 5:27 pm



You asked:

Evaluate the expression: 52cos⁡(30)+62cos⁡(60)+52sin⁡(30)+62sin⁡(60)\frac{5}{\sqrt{2}} \cos\left( 30 \right) + \frac{6}{\sqrt{2}} \cos\left( 60 \right) + \frac{5}{\sqrt{2}} \sin\left( 30 \right) + \frac{6}{\sqrt{2}} \sin\left( 60 \right)

MathBot Answer:

52cos⁡(30)+62cos⁡(60)+52sin⁡(30)+62sin⁡(60)=32cos⁡(60)+52sin⁡(30)2+32sin⁡(60)+52cos⁡(30)2≈−8.28180608060151373548635646962202\frac{5}{\sqrt{2}} \cos\left( 30 \right) + \frac{6}{\sqrt{2}} \cos\left( 60 \right) + \frac{5}{\sqrt{2}} \sin\left( 30 \right) + \frac{6}{\sqrt{2}} \sin\left( 60 \right) = 3 \sqrt{2} \cos{\left(60 \right)} + \frac{5 \sqrt{2} \sin{\left(30 \right)}}{2} + 3 \sqrt{2} \sin{\left(60 \right)} + \frac{5 \sqrt{2} \cos{\left(30 \right)}}{2} \approx -8.28180608060151373548635646962202

52cos⁡(30)+62cos⁡(60)+52sin⁡(30)+62sin⁡(60)=5⋅12⋅cos⁡(30)+6⋅12⋅cos⁡(60)+5⋅12⋅sin⁡(30)+6⋅12⋅sin⁡(60)=5⋅12⋅2⋅cos⁡(30)+6⋅12⋅cos⁡(60)+5⋅12⋅sin⁡(30)+6⋅12⋅sin⁡(60)=52⋅2⋅cos⁡(30)+6⋅12⋅cos⁡(60)+5⋅12⋅sin⁡(30)+6⋅12⋅sin⁡(60)=52⋅2⋅cos⁡(30)+6⋅12⋅2⋅cos⁡(60)+5⋅12⋅sin⁡(30)+6⋅12⋅sin⁡(60)=52⋅2⋅cos⁡(30)+3⋅2⋅cos⁡(60)+5⋅12⋅sin⁡(30)+6⋅12⋅sin⁡(60)=52⋅2⋅cos⁡(30)+3⋅2⋅cos⁡(60)+5⋅12⋅2⋅sin⁡(30)+6⋅12⋅sin⁡(60)=52⋅2⋅cos⁡(30)+3⋅2⋅cos⁡(60)+52⋅2⋅sin⁡(30)+6⋅12⋅sin⁡(60)=52⋅2⋅cos⁡(30)+3⋅2⋅cos⁡(60)+52⋅2⋅sin⁡(30)+6⋅12⋅2⋅sin⁡(60)=52⋅2⋅cos⁡(30)+3⋅2⋅cos⁡(60)+52⋅2⋅sin⁡(30)+3⋅2⋅sin⁡(60)=(3⋅2⋅cos⁡(60)+52⋅2⋅cos⁡(30))+52⋅2⋅sin⁡(30)+3⋅2⋅sin⁡(60)=(3⋅2⋅cos⁡(60)+52⋅2⋅cos⁡(30)+52⋅2⋅sin⁡(30))+3⋅2⋅sin⁡(60)=3⋅2⋅cos⁡(60)+3⋅2⋅sin⁡(60)+52⋅2⋅cos⁡(30)+52⋅2⋅sin⁡(30)\begin{aligned}\frac{5}{\sqrt{2}} \cos\left( 30 \right) + \frac{6}{\sqrt{2}} \cos\left( 60 \right) + \frac{5}{\sqrt{2}} \sin\left( 30 \right) + \frac{6}{\sqrt{2}} \sin\left( 60 \right)&=5 \cdot \frac{1}{\sqrt{2}} \cdot \cos{\left(30 \right)} + 6 \cdot \frac{1}{\sqrt{2}} \cdot \cos{\left(60 \right)} + 5 \cdot \frac{1}{\sqrt{2}} \cdot \sin{\left(30 \right)} + 6 \cdot \frac{1}{\sqrt{2}} \cdot \sin{\left(60 \right)}\\&=5 \cdot \frac{1}{2} \cdot \sqrt{2} \cdot \cos{\left(30 \right)} + 6 \cdot \frac{1}{\sqrt{2}} \cdot \cos{\left(60 \right)} + 5 \cdot \frac{1}{\sqrt{2}} \cdot \sin{\left(30 \right)} + 6 \cdot \frac{1}{\sqrt{2}} \cdot \sin{\left(60 \right)}\\&=\frac{5}{2} \cdot \sqrt{2} \cdot \cos{\left(30 \right)} + 6 \cdot \frac{1}{\sqrt{2}} \cdot \cos{\left(60 \right)} + 5 \cdot \frac{1}{\sqrt{2}} \cdot \sin{\left(30 \right)} + 6 \cdot \frac{1}{\sqrt{2}} \cdot \sin{\left(60 \right)}\\&=\frac{5}{2} \cdot \sqrt{2} \cdot \cos{\left(30 \right)} + 6 \cdot \frac{1}{2} \cdot \sqrt{2} \cdot \cos{\left(60 \right)} + 5 \cdot \frac{1}{\sqrt{2}} \cdot \sin{\left(30 \right)} + 6 \cdot \frac{1}{\sqrt{2}} \cdot \sin{\left(60 \right)}\\&=\frac{5}{2} \cdot \sqrt{2} \cdot \cos{\left(30 \right)} + 3 \cdot \sqrt{2} \cdot \cos{\left(60 \right)} + 5 \cdot \frac{1}{\sqrt{2}} \cdot \sin{\left(30 \right)} + 6 \cdot \frac{1}{\sqrt{2}} \cdot \sin{\left(60 \right)}\\&=\frac{5}{2} \cdot \sqrt{2} \cdot \cos{\left(30 \right)} + 3 \cdot \sqrt{2} \cdot \cos{\left(60 \right)} + 5 \cdot \frac{1}{2} \cdot \sqrt{2} \cdot \sin{\left(30 \right)} + 6 \cdot \frac{1}{\sqrt{2}} \cdot \sin{\left(60 \right)}\\&=\frac{5}{2} \cdot \sqrt{2} \cdot \cos{\left(30 \right)} + 3 \cdot \sqrt{2} \cdot \cos{\left(60 \right)} + \frac{5}{2} \cdot \sqrt{2} \cdot \sin{\left(30 \right)} + 6 \cdot \frac{1}{\sqrt{2}} \cdot \sin{\left(60 \right)}\\&=\frac{5}{2} \cdot \sqrt{2} \cdot \cos{\left(30 \right)} + 3 \cdot \sqrt{2} \cdot \cos{\left(60 \right)} + \frac{5}{2} \cdot \sqrt{2} \cdot \sin{\left(30 \right)} + 6 \cdot \frac{1}{2} \cdot \sqrt{2} \cdot \sin{\left(60 \right)}\\&=\frac{5}{2} \cdot \sqrt{2} \cdot \cos{\left(30 \right)} + 3 \cdot \sqrt{2} \cdot \cos{\left(60 \right)} + \frac{5}{2} \cdot \sqrt{2} \cdot \sin{\left(30 \right)} + 3 \cdot \sqrt{2} \cdot \sin{\left(60 \right)}\\&=\left(3 \cdot \sqrt{2} \cdot \cos{\left(60 \right)} + \frac{5}{2} \cdot \sqrt{2} \cdot \cos{\left(30 \right)}\right) + \frac{5}{2} \cdot \sqrt{2} \cdot \sin{\left(30 \right)} + 3 \cdot \sqrt{2} \cdot \sin{\left(60 \right)}\\&=\left(3 \cdot \sqrt{2} \cdot \cos{\left(60 \right)} + \frac{5}{2} \cdot \sqrt{2} \cdot \cos{\left(30 \right)} + \frac{5}{2} \cdot \sqrt{2} \cdot \sin{\left(30 \right)}\right) + 3 \cdot \sqrt{2} \cdot \sin{\left(60 \right)}\\&=3 \cdot \sqrt{2} \cdot \cos{\left(60 \right)} + 3 \cdot \sqrt{2} \cdot \sin{\left(60 \right)} + \frac{5}{2} \cdot \sqrt{2} \cdot \cos{\left(30 \right)} + \frac{5}{2} \cdot \sqrt{2} \cdot \sin{\left(30 \right)}\end{aligned}

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