What is the general solution for 7cot(z)+tan(z)=7csc(z) ?

The general solution for 7cot(z)+tan(z)=7csc(z) is z=1.40334…+2pin,z=2pi-1.40334…+2pin

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7cot(z)+tan(z)=7csc(z)

Solution

7 cot (z) + tan (z) = 7 csc (z)

Solution

z = 1.40334 \dots + 2 πn, z = 2 π - 1.40334 \dots + 2 πn

Degrees

z = 80.40593 \dots^{\circ} + 36 0^{\circ} n, z = 279.59406 \dots^{\circ} + 36 0^{\circ} n

Solution steps

7 cot (z) + tan (z) = 7 csc (z)

Subtract

7 csc (z)

from both sides

7 cot (z) + tan (z) - 7 csc (z) = 0

Express with sin, cos

tan (z) + 7 cot (z) - 7 csc (z)

Use the basic trigonometric identity:

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

= \frac{sin ( z )}{cos ( z )} + 7 cot (z) - 7 csc (z)

Use the basic trigonometric identity:

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

= \frac{sin ( z )}{cos ( z )} + 7 \cdot \frac{cos ( z )}{sin ( z )} - 7 csc (z)

Use the basic trigonometric identity:

csc (x) = \frac{1}{sin ( x )}

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

Simplify

\frac{sin ( z )}{cos ( z )} + 7 \cdot \frac{cos ( z )}{sin ( z )} - 7 \cdot \frac{1}{sin ( z )} : \frac{sin ^{2} ( z ) + cos ( z ) ( 7 cos ( z ) - 7 )}{cos ( z ) sin ( z )}

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

7 \cdot \frac{cos ( z )}{sin ( z )} = \frac{7 cos ( z )}{sin ( z )}

7 \cdot \frac{cos ( z )}{sin ( z )}

Multiply fractions:

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

= \frac{cos ( z ) \cdot 7}{sin ( z )}

7 \cdot \frac{1}{sin ( z )} = \frac{7}{sin ( z )}

7 \cdot \frac{1}{sin ( z )}

Multiply fractions:

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

= \frac{1 \cdot 7}{sin ( z )}

Multiply the numbers:

1 \cdot 7 = 7

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

= \frac{sin ( z )}{cos ( z )} + \frac{7 cos ( z )}{sin ( z )} - \frac{7}{sin ( z )}

Combine the fractions

\frac{7 cos ( z )}{sin ( z )} - \frac{7}{sin ( z )} : \frac{7 cos ( z ) - 7}{sin ( z )}

Apply rule

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

= \frac{7 cos ( z ) - 7}{sin ( z )}

= \frac{sin ( z )}{cos ( z )} + \frac{7 cos ( z ) - 7}{sin ( z )}

Least Common Multiplier of

cos (z), sin (z) : cos (z) sin (z)

cos (z), sin (z)

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

cos (z)

sin (z)

= cos (z) sin (z)

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

cos (z) sin (z)

For

\frac{sin ( z )}{cos ( z )} :

multiply the denominator and numerator by

sin (z)

\frac{sin ( z )}{cos ( z )} = \frac{sin ( z ) sin ( z )}{cos ( z ) sin ( z )} = \frac{sin ^{2} ( z )}{cos ( z ) sin ( z )}

For

\frac{cos ( z ) \cdot 7 - 7}{sin ( z )} :

multiply the denominator and numerator by

cos (z)

\frac{cos ( z ) \cdot 7 - 7}{sin ( z )} = \frac{( cos ( z ) \cdot 7 - 7 ) cos ( z )}{sin ( z ) cos ( z )}

= \frac{sin ^{2} ( z )}{cos ( z ) sin ( z )} + \frac{( cos ( z ) \cdot 7 - 7 ) cos ( z )}{sin ( z ) cos ( z )}

Since the denominators are equal, combine the fractions:

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

= \frac{sin ^{2} ( z ) + ( cos ( z ) \cdot 7 - 7 ) cos ( z )}{cos ( z ) sin ( z )}

= \frac{sin ^{2} ( z ) + cos ( z ) ( 7 cos ( z ) - 7 )}{cos ( z ) sin ( z )}

\frac{sin ^{2} ( z ) + ( - 7 + 7 cos ( z ) ) cos ( z )}{cos ( z ) sin ( z )} = 0

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

sin^{2} (z) + (- 7 + 7 cos (z)) cos (z) = 0

Rewrite using trig identities

sin^{2} (z) + (- 7 + 7 cos (z)) cos (z)

Use the Pythagorean identity:

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

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

= 1 - cos^{2} (z) + (- 7 + 7 cos (z)) cos (z)

Simplify

1 - cos^{2} (z) + (- 7 + 7 cos (z)) cos (z) : 6 cos^{2} (z) - 7 cos (z) + 1

1 - cos^{2} (z) + (- 7 + 7 cos (z)) cos (z)

= 1 - cos^{2} (z) + cos (z) (- 7 + 7 cos (z))

Expand

cos (z) (- 7 + 7 cos (z)) : - 7 cos (z) + 7 cos^{2} (z)

cos (z) (- 7 + 7 cos (z))

Apply the distributive law:

a (b + c) = ab + a c

a = cos (z), b = - 7, c = 7 cos (z)

= cos (z) (- 7) + cos (z) \cdot 7 cos (z)

Apply minus-plus rules

+ (- a) = - a

= - 7 cos (z) + 7 cos (z) cos (z)

7 cos (z) cos (z) = 7 cos^{2} (z)

7 cos (z) cos (z)

Apply exponent rule:

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

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

= 7 cos^{1 + 1} (z)

Add the numbers:

1 + 1 = 2

= 7 cos^{2} (z)

= - 7 cos (z) + 7 cos^{2} (z)

= 1 - cos^{2} (z) - 7 cos (z) + 7 cos^{2} (z)

Simplify

1 - cos^{2} (z) - 7 cos (z) + 7 cos^{2} (z) : 6 cos^{2} (z) - 7 cos (z) + 1

1 - cos^{2} (z) - 7 cos (z) + 7 cos^{2} (z)

Group like terms

= - cos^{2} (z) - 7 cos (z) + 7 cos^{2} (z) + 1

Add similar elements:

- cos^{2} (z) + 7 cos^{2} (z) = 6 cos^{2} (z)

= 6 cos^{2} (z) - 7 cos (z) + 1

= 6 cos^{2} (z) - 7 cos (z) + 1

= 6 cos^{2} (z) - 7 cos (z) + 1

1 + 6 cos^{2} (z) - 7 cos (z) = 0

Solve by substitution

1 + 6 cos^{2} (z) - 7 cos (z) = 0

Let:

cos (z) = u

1 + 6 u^{2} - 7 u = 0

1 + 6 u^{2} - 7 u = 0 : u = 1, u = \frac{1}{6}

1 + 6 u^{2} - 7 u = 0

Write in the standard form

a x^{2} + b x + c = 0

6 u^{2} - 7 u + 1 = 0

Solve with the quadratic formula

6 u^{2} - 7 u + 1 = 0

Quadratic Equation Formula:

For

a = 6, b = - 7, c = 1

u_{1, 2} = \frac{- ( - 7 ) \pm ( - 7 ) ^{2} - 4 \cdot 6 \cdot 1}{2 \cdot 6}

u_{1, 2} = \frac{- ( - 7 ) \pm ( - 7 ) ^{2} - 4 \cdot 6 \cdot 1}{2 \cdot 6}

(- 7)^{2} - 4 \cdot 6 \cdot 1 = 5

(- 7)^{2} - 4 \cdot 6 \cdot 1

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

(- 7)^{2} = 7^{2}

= 7^{2} - 4 \cdot 6 \cdot 1

Multiply the numbers:

4 \cdot 6 \cdot 1 = 24

= 7^{2} - 24

7^{2} = 49

= 49 - 24

Subtract the numbers:

49 - 24 = 25

= 25

Factor the number:

25 = 5^{2}

= 5^{2}

Apply radical rule:

5^{2} = 5

= 5

u_{1, 2} = \frac{- ( - 7 ) \pm 5}{2 \cdot 6}

Separate the solutions

u_{1} = \frac{- ( - 7 ) + 5}{2 \cdot 6}, u_{2} = \frac{- ( - 7 ) - 5}{2 \cdot 6}

u = \frac{- ( - 7 ) + 5}{2 \cdot 6} : 1

\frac{- ( - 7 ) + 5}{2 \cdot 6}

Apply rule

- (- a) = a

= \frac{7 + 5}{2 \cdot 6}

Add the numbers:

7 + 5 = 12

= \frac{12}{2 \cdot 6}

Multiply the numbers:

2 \cdot 6 = 12

= \frac{12}{12}

Apply rule

\frac{a}{a} = 1

= 1

u = \frac{- ( - 7 ) - 5}{2 \cdot 6} : \frac{1}{6}

\frac{- ( - 7 ) - 5}{2 \cdot 6}

Apply rule

- (- a) = a

= \frac{7 - 5}{2 \cdot 6}

Subtract the numbers:

7 - 5 = 2

= \frac{2}{2 \cdot 6}

Multiply the numbers:

2 \cdot 6 = 12

= \frac{2}{12}

Cancel the common factor:

2

= \frac{1}{6}

The solutions to the quadratic equation are:

u = 1, u = \frac{1}{6}

Substitute back

u = cos (z)

cos (z) = 1, cos (z) = \frac{1}{6}

cos (z) = 1, cos (z) = \frac{1}{6}

cos (z) = 1 : z = 2 πn

cos (z) = 1

General solutions for

cos (z) = 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}

z = 0 + 2 πn

z = 0 + 2 πn

Solve

z = 0 + 2 πn : z = 2 πn

z = 0 + 2 πn

0 + 2 πn = 2 πn

z = 2 πn

z = 2 πn

cos (z) = \frac{1}{6} : z = arccos (\frac{1}{6}) + 2 πn, z = 2 π - arccos (\frac{1}{6}) + 2 πn

cos (z) = \frac{1}{6}

Apply trig inverse properties

cos (z) = \frac{1}{6}

General solutions for

cos (z) = \frac{1}{6}

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

z = arccos (\frac{1}{6}) + 2 πn, z = 2 π - arccos (\frac{1}{6}) + 2 πn

z = arccos (\frac{1}{6}) + 2 πn, z = 2 π - arccos (\frac{1}{6}) + 2 πn

Combine all the solutions

z = 2 πn, z = arccos (\frac{1}{6}) + 2 πn, z = 2 π - arccos (\frac{1}{6}) + 2 πn

Since the equation is undefined for:

2 πn

z = arccos (\frac{1}{6}) + 2 πn, z = 2 π - arccos (\frac{1}{6}) + 2 πn

Show solutions in decimal form

z = 1.40334 \dots + 2 πn, z = 2 π - 1.40334 \dots + 2 πn

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2asin(x)+a=0

2 a sin (x) + a = 0

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tan (θ) = (\frac{2}{3})

tan(θ)=8.1443

tan (θ) = 8.1443

Frequently Asked Questions (FAQ)

What is the general solution for 7cot(z)+tan(z)=7csc(z) ?
The general solution for 7cot(z)+tan(z)=7csc(z) is z=1.40334…+2pin,z=2pi-1.40334…+2pin