7cos^2(x)-1=5cos(x)

Solution

7 cos^{2} (x) - 1 = 5 cos (x)

x = 0.50090 \dots + 2 πn, x = 2 π - 0.50090 \dots + 2 πn, x = 1.73439 \dots + 2 πn, x = - 1.73439 \dots + 2 πn

Solution steps

7 cos^{2} (x) - 1 = 5 cos (x)

Solve by substitution

cos (x) = \frac{5 + 53}{14}, cos (x) = \frac{5 - 53}{14}

cos (x) = \frac{5 + 53}{14} : x = arccos (\frac{5 + 53}{14}) + 2 πn, x = 2 π - arccos (\frac{5 + 53}{14}) + 2 πn

cos (x) = \frac{5 - 53}{14} : x = arccos (\frac{5 - 53}{14}) + 2 πn, x = - arccos (\frac{5 - 53}{14}) + 2 πn

Combine all the solutions

x = arccos (\frac{5 + 53}{14}) + 2 πn, x = 2 π - arccos (\frac{5 + 53}{14}) + 2 πn, x = arccos (\frac{5 - 53}{14}) + 2 πn, x = - arccos (\frac{5 - 53}{14}) + 2 πn

Show solutions in decimal form

x = 0.50090 \dots + 2 πn, x = 2 π - 0.50090 \dots + 2 πn, x = 1.73439 \dots + 2 πn, x = - 1.73439 \dots + 2 πn

x, sin (2 x) = 0

cot (θ) = - 4

- 4 sin (θ) + 5 = 3 sin (θ) + 5

4 tan^{2} (θ) - 25 = 0

sin^{2} (x) - cos^{2} (x) = - 1