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cos^2(x)+cos^2(30)+cos^2(75)=1

Solution

cos^{2} (x) + cos^{2} (30) + cos^{2} (75) = 1

x = 1.20704 \dots + 2 πn, x = 2 π - 1.20704 \dots + 2 πn, x = 1.93454 \dots + 2 πn, x = - 1.93454 \dots + 2 πn

Degrees

x = 69.15859 \dots^{\circ} + 36 0^{\circ} n, x = 290.84140 \dots^{\circ} + 36 0^{\circ} n, x = 110.84140 \dots^{\circ} + 36 0^{\circ} n, x = - 110.84140 \dots^{\circ} + 36 0^{\circ} n

Solution steps

cos^{2} (x) + cos^{2} (30) + cos^{2} (75) = 1

Solve by substitution

cos^{2} (x) + cos^{2} (30) + cos^{2} (75) = 1

Let:

cos (x) = u

u^{2} + cos^{2} (30) + cos^{2} (75) = 1

u^{2} + cos^{2} (30) + cos^{2} (75) = 1 : u = 1 - cos^{2} (30) - cos^{2} (75), u = - 1 - cos^{2} (30) - cos^{2} (75)

u^{2} + cos^{2} (30) + cos^{2} (75) = 1

Move

cos^{2} (30)

to the right side

u^{2} + cos^{2} (30) + cos^{2} (75) = 1

Subtract

cos^{2} (30)

from both sides

u^{2} + cos^{2} (30) + cos^{2} (75) - cos^{2} (30) = 1 - cos^{2} (30)

Simplify

u^{2} + cos^{2} (75) = 1 - cos^{2} (30)

u^{2} + cos^{2} (75) = 1 - cos^{2} (30)

Move

cos^{2} (75)

to the right side

u^{2} + cos^{2} (75) = 1 - cos^{2} (30)

Subtract

cos^{2} (75)

from both sides

u^{2} + cos^{2} (75) - cos^{2} (75) = 1 - cos^{2} (30) - cos^{2} (75)

Simplify

u^{2} = 1 - cos^{2} (30) - cos^{2} (75)

u^{2} = 1 - cos^{2} (30) - cos^{2} (75)

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

u = 1 - cos^{2} (30) - cos^{2} (75), u = - 1 - cos^{2} (30) - cos^{2} (75)

Substitute back

u = cos (x)

cos (x) = 1 - cos^{2} (30) - cos^{2} (75), cos (x) = - 1 - cos^{2} (30) - cos^{2} (75)

cos (x) = 1 - cos^{2} (30) - cos^{2} (75), cos (x) = - 1 - cos^{2} (30) - cos^{2} (75)

cos (x) = 1 - cos^{2} (30) - cos^{2} (75) : x = arccos (1 - cos^{2} (30) - cos^{2} (75)) + 2 πn, x = 2 π - arccos (1 - cos^{2} (30) - cos^{2} (75)) + 2 πn

cos (x) = 1 - cos^{2} (30) - cos^{2} (75)

Apply trig inverse properties

cos (x) = 1 - cos^{2} (30) - cos^{2} (75)

General solutions for

cos (x) = 1 - cos^{2} (30) - cos^{2} (75)

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

x = arccos (1 - cos^{2} (30) - cos^{2} (75)) + 2 πn, x = 2 π - arccos (1 - cos^{2} (30) - cos^{2} (75)) + 2 πn

x = arccos (1 - cos^{2} (30) - cos^{2} (75)) + 2 πn, x = 2 π - arccos (1 - cos^{2} (30) - cos^{2} (75)) + 2 πn

cos (x) = - 1 - cos^{2} (30) - cos^{2} (75) : x = arccos (- 1 - cos^{2} (30) - cos^{2} (75)) + 2 πn, x = - arccos (- 1 - cos^{2} (30) - cos^{2} (75)) + 2 πn

cos (x) = - 1 - cos^{2} (30) - cos^{2} (75)

Apply trig inverse properties

cos (x) = - 1 - cos^{2} (30) - cos^{2} (75)

General solutions for

cos (x) = - 1 - cos^{2} (30) - cos^{2} (75)

cos (x) = - a \Rightarrow x = arccos (- a) + 2 πn, x = - arccos (- a) + 2 πn

x = arccos (- 1 - cos^{2} (30) - cos^{2} (75)) + 2 πn, x = - arccos (- 1 - cos^{2} (30) - cos^{2} (75)) + 2 πn

x = arccos (- 1 - cos^{2} (30) - cos^{2} (75)) + 2 πn, x = - arccos (- 1 - cos^{2} (30) - cos^{2} (75)) + 2 πn

Combine all the solutions

x = arccos (1 - cos^{2} (30) - cos^{2} (75)) + 2 πn, x = 2 π - arccos (1 - cos^{2} (30) - cos^{2} (75)) + 2 πn, x = arccos (- 1 - cos^{2} (30) - cos^{2} (75)) + 2 πn, x = - arccos (- 1 - cos^{2} (30) - cos^{2} (75)) + 2 πn

Show solutions in decimal form

x = 1.20704 \dots + 2 πn, x = 2 π - 1.20704 \dots + 2 πn, x = 1.93454 \dots + 2 πn, x = - 1.93454 \dots + 2 πn

\frac{sin ( 1.3785 )}{70} = \frac{sin ( a )}{7}

cos^{2} (x) - 6 cos (x) - 7 = 0

cos (θ) = 0, 0 \leq θ \leq 2 π

- 2 sin (x) - 2 + 4 sin^{2} (x) = 0

cos (x) = - \frac{3}{5}, π < x < \frac{3 π}{2}, sin (\frac{x}{2})