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Popular >

(-4cos(x)-6sin(x))^2-20sin^2(x)=16

Solution

(- 4 cos (x) - 6 sin (x))^{2} - 20 sin^{2} (x) = 16

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

(- 4 cos (x) - 6 sin (x))^{2} - 20 sin^{2} (x) = 16

Subtract

16

from both sides

16 cos^{2} (x) + 48 cos (x) sin (x) + 16 sin^{2} (x) - 16 = 0

Factor

16 cos^{2} (x) + 48 cos (x) sin (x) + 16 sin^{2} (x) - 16 : 16 (cos^{2} (x) + 3 sin (x) cos (x) + sin^{2} (x) - 1)

16 cos^{2} (x) + 48 cos (x) sin (x) + 16 sin^{2} (x) - 16

Rewrite

48

3 \cdot 16

= 16 cos^{2} (x) + 3 \cdot 16 sin (x) cos (x) + 16 sin^{2} (x) - 16

Factor out common term

16

= 16 (cos^{2} (x) + 3 sin (x) cos (x) + sin^{2} (x) - 1)

16 (cos^{2} (x) + 3 sin (x) cos (x) + sin^{2} (x) - 1) = 0

Rewrite using trig identities

16 (cos^{2} (x) + 3 sin (x) cos (x) + sin^{2} (x) - 1)

Use the Pythagorean identity:

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

= 16 (- 1 + 3 cos (x) sin (x) + 1)

Simplify

16 (- 1 + 3 cos (x) sin (x) + 1) : 48 cos (x) sin (x)

16 (- 1 + 3 cos (x) sin (x) + 1)

Simplify

- 1 + 3 cos (x) sin (x) + 1 : 3 cos (x) sin (x)

- 1 + 3 cos (x) sin (x) + 1

Group like terms

= 3 cos (x) sin (x) - 1 + 1

- 1 + 1 = 0

= 3 cos (x) sin (x)

= 16 \cdot 3 cos (x) sin (x)

Multiply the numbers:

16 \cdot 3 = 48

= 48 cos (x) sin (x)

= 48 cos (x) sin (x)

48 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

arccos (x) = arcsin (\frac{1}{4}) + arcsin (\frac{7}{8})

3 cot^{2} (2 t - \frac{3 π}{2}) = 1

3 cot (θ) - 2 = 0

sin (u) = \frac{5}{13}

0.8 = cos (x)