What is the general solution for sec(x)cot(x)= 7/6 ?

The general solution for sec(x)cot(x)= 7/6 is x=1.02969…+2pin,x=pi-1.02969…+2pin

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sec(x)cot(x)= 7/6

Solution

sec (x) cot (x) = \frac{7}{6}

Solution

x = 1.02969 \dots + 2 πn, x = π - 1.02969 \dots + 2 πn

Degrees

x = 58.99728 \dots^{\circ} + 36 0^{\circ} n, x = 121.00271 \dots^{\circ} + 36 0^{\circ} n

Solution steps

sec (x) cot (x) = \frac{7}{6}

Subtract

\frac{7}{6}

from both sides

sec (x) cot (x) - \frac{7}{6} = 0

Simplify

sec (x) cot (x) - \frac{7}{6} : \frac{6 sec ( x ) cot ( x ) - 7}{6}

sec (x) cot (x) - \frac{7}{6}

Convert element to fraction:

sec (x) cot (x) = \frac{sec ( x ) cot ( x ) 6}{6}

= \frac{sec ( x ) cot ( x ) \cdot 6}{6} - \frac{7}{6}

Since the denominators are equal, combine the fractions:

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

= \frac{sec ( x ) cot ( x ) \cdot 6 - 7}{6}

\frac{6 sec ( x ) cot ( x ) - 7}{6} = 0

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

6 sec (x) cot (x) - 7 = 0

Express with sin, cos

- 7 + 6 cot (x) sec (x)

Use the basic trigonometric identity:

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

= - 7 + 6 \cdot \frac{cos ( x )}{sin ( x )} sec (x)

Use the basic trigonometric identity:

sec (x) = \frac{1}{cos ( x )}

= - 7 + 6 \cdot \frac{cos ( x )}{sin ( x )} \cdot \frac{1}{cos ( x )}

Simplify

- 7 + 6 \cdot \frac{cos ( x )}{sin ( x )} \cdot \frac{1}{cos ( x )} : \frac{- 7 sin ( x ) + 6}{sin ( x )}

- 7 + 6 \cdot \frac{cos ( x )}{sin ( x )} \cdot \frac{1}{cos ( x )}

6 \cdot \frac{cos ( x )}{sin ( x )} \cdot \frac{1}{cos ( x )} = \frac{6}{sin ( x )}

6 \cdot \frac{cos ( x )}{sin ( x )} \cdot \frac{1}{cos ( x )}

Multiply fractions:

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

= \frac{cos ( x ) \cdot 1 \cdot 6}{sin ( x ) cos ( x )}

Cancel the common factor:

cos (x)

= \frac{1 \cdot 6}{sin ( x )}

Multiply the numbers:

1 \cdot 6 = 6

= \frac{6}{sin ( x )}

= - 7 + \frac{6}{sin ( x )}

Convert element to fraction:

7 = \frac{7 sin ( x )}{sin ( x )}

= - \frac{7 sin ( x )}{sin ( x )} + \frac{6}{sin ( x )}

Since the denominators are equal, combine the fractions:

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

= \frac{- 7 sin ( x ) + 6}{sin ( x )}

= \frac{- 7 sin ( x ) + 6}{sin ( x )}

\frac{6 - 7 sin ( x )}{sin ( x )} = 0

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

6 - 7 sin (x) = 0

Move

6

to the right side

6 - 7 sin (x) = 0

Subtract

6

from both sides

6 - 7 sin (x) - 6 = 0 - 6

Simplify

- 7 sin (x) = - 6

- 7 sin (x) = - 6

Divide both sides by

- 7

- 7 sin (x) = - 6

Divide both sides by

- 7

\frac{- 7 sin ( x )}{- 7} = \frac{- 6}{- 7}

Simplify

sin (x) = \frac{6}{7}

sin (x) = \frac{6}{7}

Apply trig inverse properties

sin (x) = \frac{6}{7}

General solutions for

sin (x) = \frac{6}{7}

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π - arcsin (a) + 2 πn

x = arcsin (\frac{6}{7}) + 2 πn, x = π - arcsin (\frac{6}{7}) + 2 πn

x = arcsin (\frac{6}{7}) + 2 πn, x = π - arcsin (\frac{6}{7}) + 2 πn

Show solutions in decimal form

x = 1.02969 \dots + 2 πn, x = π - 1.02969 \dots + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for sec(x)cot(x)= 7/6 ?
The general solution for sec(x)cot(x)= 7/6 is x=1.02969…+2pin,x=pi-1.02969…+2pin