English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

cos^2(111)+cos^2(69.3)+cos^2(x)=1

Solution

cos^{2} (11 1^{\circ}) + cos^{2} (69. 3^{\circ}) + cos^{2} (x) = 1

x = 0.52748 \dots + 36 0^{\circ} n, x = 36 0^{\circ} - 0.52748 \dots + 36 0^{\circ} n, x = 2.61410 \dots + 36 0^{\circ} n, x = - 2.61410 \dots + 36 0^{\circ} n

Radians

x = 0.52748 \dots + 2 πn, x = 2 π - 0.52748 \dots + 2 πn, x = 2.61410 \dots + 2 πn, x = - 2.61410 \dots + 2 πn

Solution steps

cos^{2} (11 1^{\circ}) + cos^{2} (69. 3^{\circ}) + cos^{2} (x) = 1

Solve by substitution

cos^{2} (11 1^{\circ}) + cos^{2} (69. 3^{\circ}) + cos^{2} (x) = 1

Let:

cos (x) = u

cos^{2} (11 1^{\circ}) + cos^{2} (69. 3^{\circ}) + u^{2} = 1

cos^{2} (11 1^{\circ}) + cos^{2} (69. 3^{\circ}) + u^{2} = 1 : u = 1 - cos^{2} (11 1^{\circ}) - cos^{2} (69. 3^{\circ}), u = - 1 - cos^{2} (11 1^{\circ}) - cos^{2} (69. 3^{\circ})

cos^{2} (11 1^{\circ}) + cos^{2} (69. 3^{\circ}) + u^{2} = 1

Move

cos^{2} (11 1^{\circ})

to the right side

cos^{2} (11 1^{\circ}) + cos^{2} (69. 3^{\circ}) + u^{2} = 1

Subtract

cos^{2} (11 1^{\circ})

from both sides

cos^{2} (11 1^{\circ}) + cos^{2} (69. 3^{\circ}) + u^{2} - cos^{2} (11 1^{\circ}) = 1 - cos^{2} (11 1^{\circ})

Simplify

cos^{2} (69. 3^{\circ}) + u^{2} = 1 - cos^{2} (11 1^{\circ})

cos^{2} (69. 3^{\circ}) + u^{2} = 1 - cos^{2} (11 1^{\circ})

Move

cos^{2} (69. 3^{\circ})

to the right side

cos^{2} (69. 3^{\circ}) + u^{2} = 1 - cos^{2} (11 1^{\circ})

Subtract

cos^{2} (69. 3^{\circ})

from both sides

cos^{2} (69. 3^{\circ}) + u^{2} - cos^{2} (69. 3^{\circ}) = 1 - cos^{2} (11 1^{\circ}) - cos^{2} (69. 3^{\circ})

Simplify

u^{2} = 1 - cos^{2} (11 1^{\circ}) - cos^{2} (69. 3^{\circ})

u^{2} = 1 - cos^{2} (11 1^{\circ}) - cos^{2} (69. 3^{\circ})

For

x^{2} = f (a)

the solutions are

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

u = 1 - cos^{2} (11 1^{\circ}) - cos^{2} (69. 3^{\circ}), u = - 1 - cos^{2} (11 1^{\circ}) - cos^{2} (69. 3^{\circ})

Substitute back

u = cos (x)

cos (x) = 1 - cos^{2} (11 1^{\circ}) - cos^{2} (69. 3^{\circ}), cos (x) = - 1 - cos^{2} (11 1^{\circ}) - cos^{2} (69. 3^{\circ})

cos (x) = 1 - cos^{2} (11 1^{\circ}) - cos^{2} (69. 3^{\circ}), cos (x) = - 1 - cos^{2} (11 1^{\circ}) - cos^{2} (69. 3^{\circ})

cos (x) = 1 - cos^{2} (11 1^{\circ}) - cos^{2} (69. 3^{\circ}) : x = arccos (1 - cos^{2} (11 1^{\circ}) - cos^{2} (69. 3^{\circ})) + 36 0^{\circ} n, x = 36 0^{\circ} - arccos (1 - cos^{2} (11 1^{\circ}) - cos^{2} (69. 3^{\circ})) + 36 0^{\circ} n

cos (x) = 1 - cos^{2} (11 1^{\circ}) - cos^{2} (69. 3^{\circ})

Apply trig inverse properties

cos (x) = 1 - cos^{2} (11 1^{\circ}) - cos^{2} (69. 3^{\circ})

General solutions for

cos (x) = 1 - cos^{2} (11 1^{\circ}) - cos^{2} (69. 3^{\circ})

cos (x) = a \Rightarrow x = arccos (a) + 36 0^{\circ} n, x = 36 0^{\circ} - arccos (a) + 36 0^{\circ} n

x = arccos (1 - cos^{2} (11 1^{\circ}) - cos^{2} (69. 3^{\circ})) + 36 0^{\circ} n, x = 36 0^{\circ} - arccos (1 - cos^{2} (11 1^{\circ}) - cos^{2} (69. 3^{\circ})) + 36 0^{\circ} n

x = arccos (1 - cos^{2} (11 1^{\circ}) - cos^{2} (69. 3^{\circ})) + 36 0^{\circ} n, x = 36 0^{\circ} - arccos (1 - cos^{2} (11 1^{\circ}) - cos^{2} (69. 3^{\circ})) + 36 0^{\circ} n

cos (x) = - 1 - cos^{2} (11 1^{\circ}) - cos^{2} (69. 3^{\circ}) : x = arccos (- 1 - cos^{2} (11 1^{\circ}) - cos^{2} (69. 3^{\circ})) + 36 0^{\circ} n, x = - arccos (- 1 - cos^{2} (11 1^{\circ}) - cos^{2} (69. 3^{\circ})) + 36 0^{\circ} n

cos (x) = - 1 - cos^{2} (11 1^{\circ}) - cos^{2} (69. 3^{\circ})

Apply trig inverse properties

cos (x) = - 1 - cos^{2} (11 1^{\circ}) - cos^{2} (69. 3^{\circ})

General solutions for

cos (x) = - 1 - cos^{2} (11 1^{\circ}) - cos^{2} (69. 3^{\circ})

cos (x) = - a \Rightarrow x = arccos (- a) + 36 0^{\circ} n, x = - arccos (- a) + 36 0^{\circ} n

x = arccos (- 1 - cos^{2} (11 1^{\circ}) - cos^{2} (69. 3^{\circ})) + 36 0^{\circ} n, x = - arccos (- 1 - cos^{2} (11 1^{\circ}) - cos^{2} (69. 3^{\circ})) + 36 0^{\circ} n

x = arccos (- 1 - cos^{2} (11 1^{\circ}) - cos^{2} (69. 3^{\circ})) + 36 0^{\circ} n, x = - arccos (- 1 - cos^{2} (11 1^{\circ}) - cos^{2} (69. 3^{\circ})) + 36 0^{\circ} n

Combine all the solutions

x = arccos (1 - cos^{2} (11 1^{\circ}) - cos^{2} (69. 3^{\circ})) + 36 0^{\circ} n, x = 36 0^{\circ} - arccos (1 - cos^{2} (11 1^{\circ}) - cos^{2} (69. 3^{\circ})) + 36 0^{\circ} n, x = arccos (- 1 - cos^{2} (11 1^{\circ}) - cos^{2} (69. 3^{\circ})) + 36 0^{\circ} n, x = - arccos (- 1 - cos^{2} (11 1^{\circ}) - cos^{2} (69. 3^{\circ})) + 36 0^{\circ} n

Show solutions in decimal form

x = 0.52748 \dots + 36 0^{\circ} n, x = 36 0^{\circ} - 0.52748 \dots + 36 0^{\circ} n, x = 2.61410 \dots + 36 0^{\circ} n, x = - 2.61410 \dots + 36 0^{\circ} n

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

4 + 2 cos (2 x) = 6 cos (x)

cos (x) - tan (x) cos (x) = 0

sin (θ) + cos (2 θ) = 0

3 sec (θ) - 4 = 0