\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: \(\frac{5}{\sqrt{2}} \cdot \cos\left( 30 \right) + \frac{6}{\sqrt{2}} \cdot \cos\left( 60 \right) + \frac{5}{\sqrt{2}} \cdot \sin\left( 30 \right) + \frac{6}{\sqrt{2}} \cdot \sin\left( 60 \right)\)

MathBot Answer:

\[\frac{5}{\sqrt{2}} \cdot \cos\left( 30 \right) + \frac{6}{\sqrt{2}} \cdot \cos\left( 60 \right) + \frac{5}{\sqrt{2}} \cdot \sin\left( 30 \right) + \frac{6}{\sqrt{2}} \cdot \sin\left( 60 \right) = 3 \sqrt{2} \cos{\left(60 \right)} + \frac{5 \sqrt{2} \sin{\left(30 \right)}}{2} + 3 \sqrt{2} \sin{\left(60 \right)} + \frac{5 \sqrt{2} \cos{\left(30 \right)}}{2} \approx -8.28180608060151373548635646962202\]

\[\begin{aligned}\frac{5}{\sqrt{2}} \cdot \cos\left( 30 \right) + \frac{6}{\sqrt{2}} \cdot \cos\left( 60 \right) + \frac{5}{\sqrt{2}} \cdot \sin\left( 30 \right) + \frac{6}{\sqrt{2}} \cdot \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}\]

+ Submit Question