12cos^2(x)+sin(x)-6=0

Solution

12 cos^{2} (x) + sin (x) - 6 = 0

x = - 0.72972 \dots + 2 πn, x = π + 0.72972 \dots + 2 πn, x = 0.84806 \dots + 2 πn, x = π - 0.84806 \dots + 2 πn

Solution steps

12 cos^{2} (x) + sin (x) - 6 = 0

Rewrite using trig identities

6 + sin (x) - 12 sin^{2} (x) = 0

Solve by substitution

sin (x) = - \frac{2}{3}, sin (x) = \frac{3}{4}

sin (x) = - \frac{2}{3} : x = arcsin (- \frac{2}{3}) + 2 πn, x = π + arcsin (\frac{2}{3}) + 2 πn

sin (x) = \frac{3}{4} : x = arcsin (\frac{3}{4}) + 2 πn, x = π - arcsin (\frac{3}{4}) + 2 πn

Combine all the solutions

x = arcsin (- \frac{2}{3}) + 2 πn, x = π + arcsin (\frac{2}{3}) + 2 πn, x = arcsin (\frac{3}{4}) + 2 πn, x = π - arcsin (\frac{3}{4}) + 2 πn

Show solutions in decimal form

x = - 0.72972 \dots + 2 πn, x = π + 0.72972 \dots + 2 πn, x = 0.84806 \dots + 2 πn, x = π - 0.84806 \dots + 2 πn

sin (A) = \frac{3}{5}

tan (θ) = 0.2

sin (x) tan (x) = sin (x), 0 \leq x \leq 2 π

2 tan (x) + sec^{2} (x) = 4

cos (a) + sin (a) = 1