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(-5cos(x)-6sin(x))^2-11sin^2(x)=25

Solution

(- 5 cos (x) - 6 sin (x))^{2} - 11 sin^{2} (x) = 25

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = 2 πn, x = π + 2 πn

Degrees

x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n, x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n

Solution steps

(- 5 cos (x) - 6 sin (x))^{2} - 11 sin^{2} (x) = 25

Subtract

25

from both sides

25 cos^{2} (x) + 60 cos (x) sin (x) + 25 sin^{2} (x) - 25 = 0

Rewrite using trig identities

- 25 + 25 cos^{2} (x) + 25 sin^{2} (x) + 60 cos (x) sin (x)

Use the Pythagorean identity:

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

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

= - 25 + 25 (1 - sin^{2} (x)) + 25 sin^{2} (x) + 60 cos (x) sin (x)

Simplify

- 25 + 25 (1 - sin^{2} (x)) + 25 sin^{2} (x) + 60 cos (x) sin (x) : 60 cos (x) sin (x)

- 25 + 25 (1 - sin^{2} (x)) + 25 sin^{2} (x) + 60 cos (x) sin (x)

Expand

25 (1 - sin^{2} (x)) : 25 - 25 sin^{2} (x)

25 (1 - sin^{2} (x))

Apply the distributive law:

a (b - c) = ab - a c

a = 25, b = 1, c = sin^{2} (x)

= 25 \cdot 1 - 25 sin^{2} (x)

Multiply the numbers:

25 \cdot 1 = 25

= 25 - 25 sin^{2} (x)

= - 25 + 25 - 25 sin^{2} (x) + 25 sin^{2} (x) + 60 cos (x) sin (x)

Simplify

- 25 + 25 - 25 sin^{2} (x) + 25 sin^{2} (x) + 60 cos (x) sin (x) : 60 cos (x) sin (x)

- 25 + 25 - 25 sin^{2} (x) + 25 sin^{2} (x) + 60 cos (x) sin (x)

Add similar elements:

- 25 sin^{2} (x) + 25 sin^{2} (x) = 0

= - 25 + 25 + 60 cos (x) sin (x)

- 25 + 25 = 0

= 60 cos (x) sin (x)

= 60 cos (x) sin (x)

= 60 cos (x) sin (x)

60 cos (x) sin (x) = 0

Solving each part separately

cos (x) = 0 or sin (x) = 0

cos (x) = 0 : x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

cos (x) = 0

General solutions for

cos (x) = 0

cos (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

sin (x) = 0 : x = 2 πn, x = π + 2 πn

sin (x) = 0

General solutions for

sin (x) = 0

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 2 πn

Solve

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn, x = π + 2 πn

Combine all the solutions

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = 2 πn, x = π + 2 πn