What is the general solution for sec(x-pi/7)cos(x-pi/7)=sin(x-pi/7) ?

The general solution for sec(x-pi/7)cos(x-pi/7)=sin(x-pi/7) is No Solution for x\in\mathbb{R}

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Popular Trigonometry >

sec(x-pi/7)cos(x-pi/7)=sin(x-pi/7)

Solution

sec (x - \frac{π}{7}) cos (x - \frac{π}{7}) = sin (x - \frac{π}{7})

Solution

No Solution for x \in R

Solution steps

sec (x - \frac{π}{7}) cos (x - \frac{π}{7}) = sin (x - \frac{π}{7})

Rewrite using trig identities

sec (x - \frac{π}{7}) cos (x - \frac{π}{7})

sec (- \frac{π}{7} + x) cos (- \frac{π}{7} + x) = 1

sec (- \frac{π}{7} + x) cos (- \frac{π}{7} + x)

Express with sin, cos

sec (- \frac{π}{7} + x) cos (- \frac{π}{7} + x)

Use the basic trigonometric identity:

sec (- \frac{π}{7} + x) = \frac{1}{cos ( - \frac{π}{7} + x )}

= \frac{1}{cos ( - \frac{π}{7} + x )} cos (- \frac{π}{7} + x)

\frac{1}{cos ( - \frac{π}{7} + x )} cos (- \frac{π}{7} + x) = 1

\frac{1}{cos ( - \frac{π}{7} + x )} cos (- \frac{π}{7} + x)

Multiply fractions:

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

= \frac{1 cos ( - \frac{π}{7} + x )}{cos ( - \frac{π}{7} + x )}

Cancel the common factor:

cos (- \frac{π}{7} + x)

= 1

= 1

= 1

1 = sin (x - \frac{π}{7})

Switch sides

sin (x - \frac{π}{7}) = 1

General solutions for

sin (x - \frac{π}{7}) = 1

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 - \frac{π}{7} = \frac{π}{2} + 2 πn

x - \frac{π}{7} = \frac{π}{2} + 2 πn

Solve

x - \frac{π}{7} = \frac{π}{2} + 2 πn : x = 2 πn + \frac{9 π}{14}

x - \frac{π}{7} = \frac{π}{2} + 2 πn

Move

\frac{π}{7}

to the right side

x - \frac{π}{7} = \frac{π}{2} + 2 πn

Add

\frac{π}{7}

to both sides

x - \frac{π}{7} + \frac{π}{7} = \frac{π}{2} + 2 πn + \frac{π}{7}

Simplify

x - \frac{π}{7} + \frac{π}{7} = \frac{π}{2} + 2 πn + \frac{π}{7}

Simplify

x - \frac{π}{7} + \frac{π}{7} : x

x - \frac{π}{7} + \frac{π}{7}

Add similar elements:

- \frac{π}{7} + \frac{π}{7} = 0

= x

Simplify

\frac{π}{2} + 2 πn + \frac{π}{7} : 2 πn + \frac{9 π}{14}

\frac{π}{2} + 2 πn + \frac{π}{7}

Group like terms

= 2 πn + \frac{π}{2} + \frac{π}{7}

Least Common Multiplier of

2, 7 : 14

2, 7

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Prime factorization of

7 : 7

7

7

is a prime number, therefore no factorization is possible

= 7

Multiply each factor the greatest number of times it occurs in either

2

7

= 2 \cdot 7

Multiply the numbers:

2 \cdot 7 = 14

= 14

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

14

For

\frac{π}{2} :

multiply the denominator and numerator by

7

\frac{π}{2} = \frac{π 7}{2 \cdot 7} = \frac{π 7}{14}

For

\frac{π}{7} :

multiply the denominator and numerator by

2

\frac{π}{7} = \frac{π 2}{7 \cdot 2} = \frac{π 2}{14}

= \frac{π 7}{14} + \frac{π 2}{14}

Since the denominators are equal, combine the fractions:

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

= \frac{π 7 + π 2}{14}

Add similar elements:

7 π + 2 π = 9 π

= 2 πn + \frac{9 π}{14}

x = 2 πn + \frac{9 π}{14}

x = 2 πn + \frac{9 π}{14}

x = 2 πn + \frac{9 π}{14}

x = 2 πn + \frac{9 π}{14}

Since the equation is undefined for:

2 πn + \frac{9 π}{14}

No Solution for x \in R

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for sec(x-pi/7)cos(x-pi/7)=sin(x-pi/7) ?
The general solution for sec(x-pi/7)cos(x-pi/7)=sin(x-pi/7) is No Solution for x\in\mathbb{R}