(4) দেওয়া আছে, 3 sec 4 θ + 8 = 10 sec 2 θ 3\sec^4\theta + 8 = 10\sec^2\theta 3 sec 4 θ + 8 = 10 sec 2 θ
ধরি, sec 2 θ = s \sec^2\theta = s sec 2 θ = s 3 s 2 − 10 s + 8 = 0 3s^2 - 10s + 8 = 0 3 s 2 − 10 s + 8 = 0
বা, 3 s 2 − 6 s − 4 s + 8 = 0 3s^2 - 6s - 4s + 8 = 0 3 s 2 − 6 s − 4 s + 8 = 0 ( s − 2 ) ( 3 s − 4 ) = 0 (s-2)(3s-4) = 0 ( s − 2 ) ( 3 s − 4 ) = 0
অতএব, s = 2 s = 2 s = 2 s = 4 3 s = \frac{4}{3} s = 3 4
(i) sec 2 θ = 2 \sec^2\theta = 2 sec 2 θ = 2 cos 2 θ = 1 2 \cos^2\theta = \frac{1}{2} cos 2 θ = 2 1 cos θ = ± 1 2 \cos\theta = \pm\frac{1}{\sqrt{2}} cos θ = ± 2 1
0 < θ < 2 π 0 < \theta < 2\pi 0 < θ < 2 π θ = π 4 , 3 π 4 , 5 π 4 , 7 π 4 \theta = \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4} θ = 4 π , 4 3 π , 4 5 π , 4 7 π
(ii) sec 2 θ = 4 3 \sec^2\theta = \frac{4}{3} sec 2 θ = 3 4 cos 2 θ = 3 4 \cos^2\theta = \frac{3}{4} cos 2 θ = 4 3 cos θ = ± 3 2 \cos\theta = \pm\frac{\sqrt{3}}{2} cos θ = ± 2 3
0 < θ < 2 π 0 < \theta < 2\pi 0 < θ < 2 π θ = π 6 , 5 π 6 , 7 π 6 , 11 π 6 \theta = \frac{\pi}{6}, \frac{5\pi}{6}, \frac{7\pi}{6}, \frac{11\pi}{6} θ = 6 π , 6 5 π , 6 7 π , 6 11 π
উত্তর: θ = π 6 , π 4 , 3 π 4 , 5 π 6 , 7 π 6 , 5 π 4 , 7 π 4 , 11 π 6 \theta = \frac{\pi}{6}, \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{6}, \frac{7\pi}{6}, \frac{5\pi}{4}, \frac{7\pi}{4}, \frac{11\pi}{6} θ = 6 π , 4 π , 4 3 π , 6 5 π , 6 7 π , 4 5 π , 4 7 π , 6 11 π