QUESTION 11
11.1 Determine the general solution of :

Ask by Reyes Daniel. in South Africa

Mar 23,2025

Question

QUESTION 11 11.1 Determine the general solution of : $$ \sin \left(x+30^{\circ}\right)=\cos 3 x $$

UpStudy AI · Accepted Answer

1. Start with the equation:
   
   $$
   \sin\left(x+30^\circ\right)=\cos 3x
   $$

2. Express the cosine term in sine form using the identity:
   
   $$
   \cos \theta = \sin\left(90^\circ-\theta\right)
   $$
   
   Thus, we have:
   
   $$
   \cos 3x = \sin\left(90^\circ-3x\right)
   $$
   
   So the equation becomes:
   
   $$
   \sin\left(x+30^\circ\right) = \sin\left(90^\circ-3x\right)
   $$

3. For two sine functions to be equal, the angles must satisfy one of the following:
   
   - Case 1:
     
     $$
     x+30^\circ = 90^\circ-3x+360^\circ k,\quad k\in\mathbb{Z}
     $$
   
   - Case 2:
     
     $$
     x+30^\circ = 180^\circ-\left(90^\circ-3x\right)+360^\circ k,\quad k\in\mathbb{Z}
     $$

4. Solve Case 1:
   
   $$
   x+30^\circ = 90^\circ-3x+360^\circ k
   $$
   
   Add $$3x$$ to both sides:
   
   $$
   4x+30^\circ = 90^\circ+360^\circ k
   $$
   
   Subtract $$30^\circ$$ from both sides:
   
   $$
   4x = 60^\circ+360^\circ k
   $$
   
   Divide by 4:
   
   $$
   x = 15^\circ+90^\circ k
   $$

5. Solve Case 2:
   
   Simplify the right-hand side first:
   
   $$
   180^\circ-\left(90^\circ-3x\right)=180^\circ-90^\circ+3x=90^\circ+3x
   $$
   
   Now, set up the equation:
   
   $$
   x+30^\circ = 90^\circ+3x+360^\circ k
   $$
   
   Subtract $$x$$ from both sides:
   
   $$
   30^\circ = 90^\circ+2x+360^\circ k
   $$
   
   Subtract $$90^\circ+360^\circ k$$:
   
   $$
   2x = 30^\circ-90^\circ-360^\circ k = -60^\circ-360^\circ k
   $$
   
   Divide by 2:
   
   $$
   x = -30^\circ-180^\circ k
   $$
   
   This can also be written by letting $$ k'=-k $$, so an equivalent expression is:
   
   $$
   x = -30^\circ+180^\circ k,\quad k\in\mathbb{Z}.
   $$

6. The general solution is given by the two families:

$$
   \boxed{x = 15^\circ+90^\circ k \quad \text{or} \quad x = -30^\circ+180^\circ k,\quad k\in\mathbb{Z}}
   $$
The general solution is: $$ x = 15^\circ + 90^\circ k \quad \text{or} \quad x = -30^\circ + 180^\circ k,\quad k \in \mathbb{Z} $$

QUESTION 11
11.1 Determine the general solution of :

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