Answer :
To solve each trigonometric equation for [tex]\( x \)[/tex] in the range [tex]\( 0^\circ \)[/tex] to [tex]\( 360^\circ \)[/tex], we need to find the values of [tex]\( x \)[/tex] that satisfy each equation.
Let's address each equation one by one.
### Equation 1: [tex]\( 14 \cos x + \cos 5x = 0 \)[/tex]
The equation can be rewritten as:
[tex]\[ 14 \cos x + \cos 5x = 0 \][/tex]
We need to find values of [tex]\( x \)[/tex] where this equation holds true. After analyzing the behavior of this equation over the given domain, we find:
Solutions: [tex]\( x = 90^\circ \)[/tex] and [tex]\( x = 270^\circ \)[/tex]
### Equation 2: [tex]\( 15 \cos 4x - \cos x = 0 \)[/tex]
The equation can be rewritten as:
[tex]\[ 15 \cos 4x - \cos x = 0 \][/tex]
We need to find values of [tex]\( x \)[/tex] where this equation holds true. After analyzing the behavior of this equation over the given domain, we find:
Solutions: There are no values of [tex]\( x \)[/tex] that satisfy this equation in the given interval.
### Equation 3: [tex]\( 16 \sin 3x - \sin x = 0 \)[/tex]
The equation can be rewritten as:
[tex]\[ 16 \sin 3x - \sin x = 0 \][/tex]
We need to find values of [tex]\( x \)[/tex] where this equation holds true. After analyzing the behavior of this equation over the given domain, we find:
Solutions: [tex]\( x = 0^\circ \)[/tex], [tex]\( x = 180^\circ \)[/tex], and [tex]\( x = 360^\circ \)[/tex]
### Equation 4: [tex]\( 18 \sin (x + 10^\circ) + \sin x = 0 \)[/tex]
The equation can be rewritten as:
[tex]\[ 18 \sin (x + 10^\circ) + \sin x = 0 \][/tex]
We need to find values of [tex]\( x \)[/tex] where this equation holds true. After analyzing the behavior of this equation over the given domain, we find:
Solutions: There are no values of [tex]\( x \)[/tex] that satisfy this equation in the given interval.
### Equation 5: [tex]\( 19 \cos (2x + 10^\circ) + \cos (2x - 10^\circ) = 0 \)[/tex]
The equation can be rewritten as:
[tex]\[ 19 \cos (2x + 10^\circ) + \cos (2x - 10^\circ) = 0 \][/tex]
We need to find values of [tex]\( x \)[/tex] where this equation holds true. After analyzing the behavior of this equation over the given domain, we find:
Solutions: There are no values of [tex]\( x \)[/tex] that satisfy this equation in the given interval.
### Equation 6: [tex]\( 20 \cos (x + 20^\circ) - \cos (x - 70^\circ) = 0 \)[/tex]
The equation can be rewritten as:
[tex]\[ 20 \cos (x + 20^\circ) - \cos (x - 70^\circ) = 0 \][/tex]
We need to find values of [tex]\( x \)[/tex] where this equation holds true. After analyzing the behavior of this equation over the given domain, we find:
Solutions: There are no values of [tex]\( x \)[/tex] that satisfy this equation in the given interval.
### Summary of Solutions:
1. [tex]\( 14 \cos x + \cos 5x = 0 \)[/tex]
- [tex]\( x = 90^\circ \)[/tex]
- [tex]\( x = 270^\circ \)[/tex]
2. [tex]\( 15 \cos 4x - \cos x = 0 \)[/tex]
- No solutions.
3. [tex]\( 16 \sin 3x - \sin x = 0 \)[/tex]
- [tex]\( x = 0^\circ \)[/tex]
- [tex]\( x = 180^\circ \)[/tex]
- [tex]\( x = 360^\circ \)[/tex]
4. [tex]\( 18 \sin (x + 10^\circ) + \sin x = 0 \)[/tex]
- No solutions.
5. [tex]\( 19 \cos (2x + 10^\circ) + \cos (2x - 10^\circ) = 0 \)[/tex]
- No solutions.
6. [tex]\( 20 \cos (x + 20^\circ) - \cos (x - 70^\circ) = 0 \)[/tex]
- No solutions.
Let's address each equation one by one.
### Equation 1: [tex]\( 14 \cos x + \cos 5x = 0 \)[/tex]
The equation can be rewritten as:
[tex]\[ 14 \cos x + \cos 5x = 0 \][/tex]
We need to find values of [tex]\( x \)[/tex] where this equation holds true. After analyzing the behavior of this equation over the given domain, we find:
Solutions: [tex]\( x = 90^\circ \)[/tex] and [tex]\( x = 270^\circ \)[/tex]
### Equation 2: [tex]\( 15 \cos 4x - \cos x = 0 \)[/tex]
The equation can be rewritten as:
[tex]\[ 15 \cos 4x - \cos x = 0 \][/tex]
We need to find values of [tex]\( x \)[/tex] where this equation holds true. After analyzing the behavior of this equation over the given domain, we find:
Solutions: There are no values of [tex]\( x \)[/tex] that satisfy this equation in the given interval.
### Equation 3: [tex]\( 16 \sin 3x - \sin x = 0 \)[/tex]
The equation can be rewritten as:
[tex]\[ 16 \sin 3x - \sin x = 0 \][/tex]
We need to find values of [tex]\( x \)[/tex] where this equation holds true. After analyzing the behavior of this equation over the given domain, we find:
Solutions: [tex]\( x = 0^\circ \)[/tex], [tex]\( x = 180^\circ \)[/tex], and [tex]\( x = 360^\circ \)[/tex]
### Equation 4: [tex]\( 18 \sin (x + 10^\circ) + \sin x = 0 \)[/tex]
The equation can be rewritten as:
[tex]\[ 18 \sin (x + 10^\circ) + \sin x = 0 \][/tex]
We need to find values of [tex]\( x \)[/tex] where this equation holds true. After analyzing the behavior of this equation over the given domain, we find:
Solutions: There are no values of [tex]\( x \)[/tex] that satisfy this equation in the given interval.
### Equation 5: [tex]\( 19 \cos (2x + 10^\circ) + \cos (2x - 10^\circ) = 0 \)[/tex]
The equation can be rewritten as:
[tex]\[ 19 \cos (2x + 10^\circ) + \cos (2x - 10^\circ) = 0 \][/tex]
We need to find values of [tex]\( x \)[/tex] where this equation holds true. After analyzing the behavior of this equation over the given domain, we find:
Solutions: There are no values of [tex]\( x \)[/tex] that satisfy this equation in the given interval.
### Equation 6: [tex]\( 20 \cos (x + 20^\circ) - \cos (x - 70^\circ) = 0 \)[/tex]
The equation can be rewritten as:
[tex]\[ 20 \cos (x + 20^\circ) - \cos (x - 70^\circ) = 0 \][/tex]
We need to find values of [tex]\( x \)[/tex] where this equation holds true. After analyzing the behavior of this equation over the given domain, we find:
Solutions: There are no values of [tex]\( x \)[/tex] that satisfy this equation in the given interval.
### Summary of Solutions:
1. [tex]\( 14 \cos x + \cos 5x = 0 \)[/tex]
- [tex]\( x = 90^\circ \)[/tex]
- [tex]\( x = 270^\circ \)[/tex]
2. [tex]\( 15 \cos 4x - \cos x = 0 \)[/tex]
- No solutions.
3. [tex]\( 16 \sin 3x - \sin x = 0 \)[/tex]
- [tex]\( x = 0^\circ \)[/tex]
- [tex]\( x = 180^\circ \)[/tex]
- [tex]\( x = 360^\circ \)[/tex]
4. [tex]\( 18 \sin (x + 10^\circ) + \sin x = 0 \)[/tex]
- No solutions.
5. [tex]\( 19 \cos (2x + 10^\circ) + \cos (2x - 10^\circ) = 0 \)[/tex]
- No solutions.
6. [tex]\( 20 \cos (x + 20^\circ) - \cos (x - 70^\circ) = 0 \)[/tex]
- No solutions.
