Using equivalent angles, the solutions are given as follows:
a) [tex]x = \frac{15\pi}{23}[/tex].
b) [tex]x = \frac{9\pi}{62}, x = \frac{53\pi}{62}[/tex].
c) [tex]x = \frac{3\pi}{8}, \frac{11\pi}{8}[/tex]
What are equivalent angles?
Each angle on the second, third and fourth quadrants will have an equivalent on the first quadrant.
For item a, we have to find the equivalent angle on the 2nd quadrant, where the sine is also positive.
Hence:
[tex]\pi - \frac{8\pi}{23} = \frac{23\pi}{23} - \frac{8\pi}{23} = \frac{15\pi}{23}[/tex]
Hence [tex]x = \frac{15\pi}{23}[/tex].
For item b, if two angles are complementary, the sine of one is the cosine of the other.
Complementary angles add to 90º = 0.5pi, hence:
[tex]x + \frac{11\pi}{31} = \frac{\pi}{2}[/tex]
[tex]x = \frac{31\pi}{62} - \frac{22\pi}{62}[/tex]
[tex]x = \frac{9\pi}{62}[/tex]
The equivalent angle on the second quadrant is:
[tex]\pi - \frac{9\pi}{62} = \frac{62\pi}{62} - \frac{9\pi}{62} = \frac{53\pi}{62}[/tex]
Hence the solutions are:
[tex]x = \frac{9\pi}{62}, x = \frac{53\pi}{62}[/tex]
For item c, the angles are also complementary, hence:
[tex]x + \frac{\pi}{8} = \frac{\pi}{2}[/tex]
[tex]x = \frac{4\pi}{8} - \frac{\pi}{8}[/tex]
[tex]x = \frac{3\pi}{8}[/tex]
The tangent is also positive on the third quadrant, hence the equivalent angle is:
[tex]x = \pi + \frac{3\pi}{8} = \frac{8\pi}{8} + \frac{3\pi}{8} = \frac{11\pi}{8}[/tex]
Hence the solutions are:
[tex]x = \frac{3\pi}{8}, \frac{11\pi}{8}[/tex]
More can be learned about equivalent angles at https://brainly.com/question/28163477
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