Activity

Find the angles between [tex]$0^{\circ}[tex]$[/tex] and [tex]$[/tex]360^{\circ}$[/tex] (inclusive) which satisfy the equation:

a) [tex]$2 \sin \left(x+50^{\circ}\right)=1$[/tex]

b) [tex]$\cos 3x = \sin \frac{\pi}{6}$[/tex]

c) [tex]$\cos \left(2x + 20^{\circ}\right) = -\frac{1}{2}$[/tex]



Answer :

To find the angles between \(0^\circ\) and \(360^\circ\) (inclusive) that satisfy the given equations, we will go through each equation step-by-step.

### Part a) \(2 \sin (x + 50^\circ) = 1\)

1. Isolate the sine term:
[tex]\[ 2 \sin (x + 50^\circ) = 1 \][/tex]
[tex]\[ \sin (x + 50^\circ) = \frac{1}{2} \][/tex]

2. Find the general solutions for the sine function:
Since \(\sin \theta = \frac{1}{2}\) at \(\theta = 30^\circ + 360^\circ n\) and \(150^\circ + 360^\circ n\) for integer values of \(n\).

3. Adjust for the phase shift:
[tex]\[ x + 50^\circ = 30^\circ + 360^\circ n \][/tex]
[tex]\[ x + 50^\circ = 150^\circ + 360^\circ n \][/tex]

4. Solve for \(x\):
[tex]\[ x = 30^\circ - 50^\circ + 360^\circ n = -20^\circ + 360^\circ n \][/tex]
[tex]\[ x = 150^\circ - 50^\circ + 360^\circ n = 100^\circ + 360^\circ n \][/tex]

[tex]\[ x = 30^\circ + 360^\circ n - 50^\circ = -20^\circ + 360^\circ n \][/tex]
[tex]\[ x = 150^\circ + 360^\circ n - 50^\circ = 100^\circ + 360^\circ n \][/tex]

5. Plug in values of \(n\) to find solutions within \(0^\circ\) to \(360^\circ\):

For \(n = 0\):
[tex]\[ x = -20^\circ \][/tex] (Not in the range 0° to 360°)
[tex]\[ x = 100^\circ \][/tex]

For \(n = 1\):
[tex]\[ x = 340^\circ \][/tex]

Therefore, the solutions are \(100^\circ\) and \(340^\circ\).

### Part b) \(\cos 3x = \sin \frac{\pi}{6}\)

1. Recognize that \(\sin \frac{\pi}{6} = \frac{1}{2}\):
[tex]\[ \cos 3x = \frac{1}{2} \][/tex]

2. Find the general solutions for the cosine function:
Since \(\cos \theta = \frac{1}{2}\) at \(\theta = 60^\circ + 360^\circ n\) and \(300^\circ + 360^\circ n\) for integer values of \(n\).

3. Adjust for the coefficient \(3x\):
[tex]\[ 3x = 60^\circ + 360^\circ n \][/tex]
[tex]\[ 3x = 300^\circ + 360^\circ n \][/tex]

4. Solve for \(x\):
[tex]\[ x = \frac{60^\circ + 360^\circ n}{3} = 20^\circ + 120^\circ n \][/tex]
[tex]\[ x = \frac{300^\circ + 360^\circ n}{3} = 100^\circ + 120^\circ n \][/tex]

5. Plug in values of \(n\) to find solutions within \(0^\circ\) to \(360^\circ\):

For \(n = 0\),
[tex]\[ x = 20^\circ \][/tex]
[tex]\[ x = 100^\circ \][/tex]

For \(n = 1\),
[tex]\[ x = 140^\circ \][/tex]
[tex]\[ x = 220^\circ \][/tex]

For \(n = 2\),
[tex]\[ x = 260^\circ \][/tex]
[tex]\[ x = 340^\circ \][/tex]

Therefore, the solutions are \(20^\circ, 100^\circ, 140^\circ, 220^\circ, 260^\circ, \) and \(340^\circ\).

### Part c) \(\cos (2x + 20^\circ) = -\frac{1}{2}\)

1. Isolate the cosine term:
[tex]\[ \cos (2x + 20^\circ) = -\frac{1}{2} \][/tex]

2. Find the general solutions for the cosine function:
Since \(\cos \theta = -\frac{1}{2}\) at \(\theta = 120^\circ + 360^\circ n\) and \(240^\circ + 360^\circ n\) for integer values of \(n\).

3. Adjust for the phase shift:
[tex]\[ 2x + 20^\circ = 120^\circ + 360^\circ n \][/tex]
[tex]\[ 2x + 20^\circ = 240^\circ + 360^\circ n \][/tex]

4. Solve for \(x\):
[tex]\[ 2x = 120^\circ - 20^\circ + 360^\circ n = 100^\circ + 360^\circ n \][/tex]
[tex]\[ 2x = 240^\circ - 20^\circ + 360^\circ n = 220^\circ + 360^\circ n \][/tex]

[tex]\[ x = \frac{100^\circ + 360^\circ n}{2} = 50^\circ + 180^\circ n \][/tex]
[tex]\[ x = \frac{220^\circ + 360^\circ n}{2} = 110^\circ + 180^\circ n \][/tex]

5. Plug in values of \(n\) to find solutions within \(0^\circ\) to \(360^\circ\):

For \(n = 0\),
[tex]\[ x = 50^\circ \][/tex]
[tex]\[ x = 110^\circ \][/tex]

For \(n = 1\),
[tex]\[ x = 230^\circ \][/tex]
[tex]\[ x = 290^\circ \][/tex]

Therefore, the solutions are \(50^\circ, 110^\circ, 230^\circ,\) and \(290^\circ\).

### Summary

- For part a), the angles that satisfy the equation are: \(100^\circ\) and \(340^\circ\).
- For part b), the angles that satisfy the equation are: \(20^\circ, 100^\circ, 140^\circ, 220^\circ, 260^\circ,\) and \(340^\circ\).
- For part c), the angles that satisfy the equation are: [tex]\(50^\circ, 110^\circ, 230^\circ,\)[/tex] and [tex]\(290^\circ\)[/tex].