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

The general solution for sin(x)-tan(x)=cos(x)-1 is x=2pin,x= pi/4+pin

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sin(x)-tan(x)=cos(x)-1

Solution

sin (x) - tan (x) = cos (x) - 1

Solution

x = 2 πn, x = \frac{π}{4} + πn

Degrees

x = 0^{\circ} + 36 0^{\circ} n, x = 4 5^{\circ} + 18 0^{\circ} n

Solution steps

sin (x) - tan (x) = cos (x) - 1

Subtract

cos (x) - 1

from both sides

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

Express with sin, cos

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

Use the basic trigonometric identity:

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

= 1 - cos (x) + sin (x) - \frac{sin ( x )}{cos ( x )}

Simplify

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

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

Convert element to fraction:

1 = \frac{1 cos ( x )}{cos ( x )}, cos (x) = \frac{cos ( x ) cos ( x )}{cos ( x )}, sin (x) = \frac{sin ( x ) cos ( x )}{cos ( x )}

= \frac{1 \cdot cos ( x )}{cos ( x )} - \frac{cos ( x ) cos ( x )}{cos ( x )} + \frac{sin ( x ) cos ( x )}{cos ( x )} - \frac{sin ( x )}{cos ( x )}

Since the denominators are equal, combine the fractions:

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

= \frac{1 \cdot cos ( x ) - cos ( x ) cos ( x ) + sin ( x ) cos ( x ) - sin ( x )}{cos ( x )}

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

1 \cdot cos (x) - cos (x) cos (x) + sin (x) cos (x) - sin (x)

1 \cdot cos (x) = cos (x)

1 \cdot cos (x)

Multiply:

1 \cdot cos (x) = cos (x)

= cos (x)

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

cos (x) cos (x)

Apply exponent rule:

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

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

= cos^{1 + 1} (x)

Add the numbers:

1 + 1 = 2

= cos^{2} (x)

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

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

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

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

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

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

Factor

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

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

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

Factor out

sin (x)

from

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

sin (x) cos (x) - sin (x)

Factor out common term

sin (x)

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

Factor out

- cos (x)

from

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

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

Apply exponent rule:

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

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

= - cos (x) cos (x) + cos (x)

Factor out common term

- cos (x)

= - cos (x) (cos (x) - 1)

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

Factor out common term

cos (x) - 1

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

(cos (x) - 1) (sin (x) - cos (x)) = 0

Solving each part separately

cos (x) - 1 = 0 or sin (x) - cos (x) = 0

cos (x) - 1 = 0 : x = 2 πn

cos (x) - 1 = 0

Move

1

to the right side

cos (x) - 1 = 0

Add

1

to both sides

cos (x) - 1 + 1 = 0 + 1

Simplify

cos (x) = 1

cos (x) = 1

General solutions for

cos (x) = 1

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 = 0 + 2 πn

x = 0 + 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

sin (x) - cos (x) = 0 : x = \frac{π}{4} + πn

sin (x) - cos (x) = 0

Rewrite using trig identities

sin (x) - cos (x) = 0

Divide both sides by

cos (x), cos (x) \neq = 0

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

Simplify

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

Use the basic trigonometric identity:

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

tan (x) - 1 = 0

tan (x) - 1 = 0

Move

1

to the right side

tan (x) - 1 = 0

Add

1

to both sides

tan (x) - 1 + 1 = 0 + 1

Simplify

tan (x) = 1

tan (x) = 1

General solutions for

tan (x) = 1

tan (x)

periodicity table with

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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = \frac{π}{4} + πn

x = \frac{π}{4} + πn

Combine all the solutions

x = 2 πn, x = \frac{π}{4} + πn

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Frequently Asked Questions (FAQ)

What is the general solution for sin(x)-tan(x)=cos(x)-1 ?
The general solution for sin(x)-tan(x)=cos(x)-1 is x=2pin,x= pi/4+pin