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O lvl A Maths: Trigonometry

Solve each of the following equations for the domain 0° ≤ x ≤ 360°.

(a) 3 cos 2x = cos x + 2
(b) tan 2y = 4 tan y

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Answer:

(a) 3 cos 2x = cos x + 2
Sub cos 2x = 2 cos² x - 1

3 (2 cos² x - 1) = cos x + 2
6 cos² x - 3 - cos x - 2 = 0
(6 cos x + 5) (cos x - 1) = 0
cos x = -5/6 or cos x = 1

For cos x = -5/6, x = 146.4°, 213.6°
For cos x = 1, x = 0°, 360°

Thus, x = 0°, 146.4°, 213.6°, 360°


(b) tan 2y = 4 tan y
Sub tan 2y = (2 tan y)/(1 - tan² y)

(2 tan y)/(1 - tan² y) = 4 tan y
2 tan y = 4 tan y (1 - tan² y)
2 tan y = 4 tan y - 4 tan³ y
4 tan³ y - 4 tan y + 2 tan y = 0
4 tan³ y - 2 tan y = 0
2 tan³ y - tan y = 0
tan y (2 tan² y - 1) = 0
tan y = 0 or tan² y = 0.5

For tan y = 0, y = 0°, 180°, 360°

For tan² y = 0.5,
\tan y=\pm \frac{1}{\sqrt{2}}
\tan y=\frac{1}{\sqrt{2}} or \tan y=-\frac{1}{\sqrt{2}}

y = 35.3°, 215.3° or y = 144.7°, 324.7°

Combining,
y = 0°, 35.3°, 144.7°, 180°, 215.3°, 324.7°, 360°



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