What is the general solution for (sin(x))(cos(x))(tan(x))+cos(2x)=1 ?

The general solution for (sin(x))(cos(x))(tan(x))+cos(2x)=1 is x=2pin,x=pi+2pin

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Popular >

(sin(x))(cos(x))(tan(x))+cos(2x)=1

Solution

(sin (x)) (cos (x)) (tan (x)) + cos (2 x) = 1

Solution

x = 2 πn, x = π + 2 πn

Degrees

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

Solution steps

(sin (x)) (cos (x)) (tan (x)) + cos (2 x) = 1

Subtract

1

from both sides

sin (x) cos (x) tan (x) + cos (2 x) - 1 = 0

Rewrite using trig identities

- 1 + cos (2 x) + cos (x) sin (x) tan (x)

Use the Double Angle identity:

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

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

Simplify

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

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

Factor

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

- 2 sin^{2} (x) + cos (x) sin (x) tan (x)

Apply exponent rule:

a^{b + c} = a^{b} a^{c}

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

= - 2 sin (x) sin (x) + cos (x) sin (x) tan (x)

Factor out common term

sin (x)

= sin (x) (- 2 sin (x) + cos (x) tan (x))

sin (x) (- 2 sin (x) + cos (x) tan (x)) = 0

Solving each part separately

sin (x) = 0 or - 2 sin (x) + cos (x) tan (x) = 0

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

- 2 sin (x) + cos (x) tan (x) = 0 : x = 2 πn, x = π + 2 πn

- 2 sin (x) + cos (x) tan (x) = 0

Rewrite using trig identities

- 2 sin (x) + cos (x) tan (x)

Use the basic trigonometric identity:

tan (x) = \frac{sin ( x )}{cos ( x )}

= - 2 sin (x) + cos (x) \frac{sin ( x )}{cos ( x )}

Simplify

- 2 sin (x) + cos (x) \frac{sin ( x )}{cos ( x )} : - sin (x)

- 2 sin (x) + cos (x) \frac{sin ( x )}{cos ( x )}

cos (x) \frac{sin ( x )}{cos ( x )} = sin (x)

cos (x) \frac{sin ( x )}{cos ( x )}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{sin ( x ) cos ( x )}{cos ( x )}

Cancel the common factor:

cos (x)

= sin (x)

= - 2 sin (x) + sin (x)

Add similar elements:

- 2 sin (x) + sin (x) = - sin (x)

= - sin (x)

= - sin (x)

- sin (x) = 0

Divide both sides by

- 1

- sin (x) = 0

Divide both sides by

- 1

\frac{- sin ( x )}{- 1} = \frac{0}{- 1}

Simplify

sin (x) = 0

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 = 2 πn, x = π + 2 πn

Popular Examples

tan(x)= 20/30

tan (x) = \frac{20}{30}

sin(x)=sin(27)

sin (x) = sin (2 7^{\circ})

cos(a)= 12/37

cos (a) = \frac{12}{37}

120sin(x-120)=120sin(x-240)

120 sin (x - 12 0^{\circ}) = 120 sin (x - 24 0^{\circ})

sin^2(x)=1-sin^2(x)

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

Frequently Asked Questions (FAQ)

What is the general solution for (sin(x))(cos(x))(tan(x))+cos(2x)=1 ?
The general solution for (sin(x))(cos(x))(tan(x))+cos(2x)=1 is x=2pin,x=pi+2pin