Question

Solve the equation 3sin (2x)=5cosx

Answer 1

`\bf \textit{Double Angle Identities} \\ \quad \\ \boxed{sin(2\theta)=2sin(\theta)cos(\theta)} \\ \quad \\ cos(2\theta)= \begin{cases} cos^2(\theta)-sin^2(\theta)\\ 1-2sin^2(\theta)\\ 2cos^2(\theta)-1 \end{cases} \\ \quad \\ tan(2\theta)=\cfrac{2tan(\theta)}{1-tan^2(\theta)}\\\\ -----------------------------\\\\`


`\bf 3sin(2x)=5cos(x)\implies 3sin(x)cos(x)=5cos(x) \\\\\\ 3sin(x)cos(x)-5cos(x)=0\implies cos(x)[\ 3sin(x)-5\ ]=0 \\\\\\ thus\quad \begin{cases} cos(x)=0\implies \measuredangle x=cos^(-1)(0)\\ ---------------\\ 3sin(x)-5=0\implies 3sin(x)=5\\\\ sin(x)=(5)/(3)\implies \measuredangle x=sin^(-1)\left( (5)/(3) \right) \end{cases}`

the first angle(s), are easy to see, where cosine is 0
the second one, you can plug that in your calculator

Answer 2

x = 56.442690238079284707119200844376

x = 56.44° (Rounding)

Explanation:

Solve the equation 3sin(2x) = 5cos(x)

Step 1: Identity sin(2x) = 2sin(x)cos(x)

Step 2: Apply the identity to the equation, substituting sin(2x):

3[2sin(x)cos(x)] = 5cos(x)

Step 3: Multiply left side:

6sin(x)cos(x) = 5cos(x)

Step 4: Move all terms to the left side to get an equation equal to zero:

6sin(x)cos(x) - 5cos(x) = 0

Step 5: Factorize (common factor):

Cos(x)[6sin(x) - 5] = 0

Step 6: Divide the equation by cos(x) (both sides):

Cos(x)[6sin(x) - 5] / cos(x) = 0 / cos(x)

6sin(x) – 5 = 0

Step 7: Isolate sin(x):

6sin(x) = 5

sin(x) = 5/6

sin(x) = 0.8333333333333

Step 8: Calculate x value applying arcsin (Inverse of sin):

Since sin(x) = 0.8333333333333, then

x = arcsin(0.8333333333333)

x = 56.442690238079284707119200844376

x = 56.44° (Rounding)

Step 9: Testing the equation:

3sin(2x) = 5cos(x)

3sin(2*56.44) = 5cos(56.44)

3sin(112.88) = 5(0.55281)

3(0.92132) = 2.764

2.764 = 2.764