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Popular Trigonometry >

(tan((3a)/2))tan(a/2)=3

Solution

(tan (\frac{3 a}{2})) tan (\frac{a}{2}) = 3

Solution

a = 2 \cdot 0.46796 \dots + 2 πn, a = - 2 \cdot 0.46796 \dots + 2 πn, a = 2 \cdot 1.28688 \dots + 2 πn, a = - 2 \cdot 1.28688 \dots + 2 πn

Degrees

a = 53.62480 \dots^{\circ} + 36 0^{\circ} n, a = - 53.62480 \dots^{\circ} + 36 0^{\circ} n, a = 147.46577 \dots^{\circ} + 36 0^{\circ} n, a = - 147.46577 \dots^{\circ} + 36 0^{\circ} n

Solution steps

(tan (\frac{3 a}{2})) tan (\frac{a}{2}) = 3

Subtract

3

from both sides

tan (\frac{3 a}{2}) tan (\frac{a}{2}) - 3 = 0

Let:

u = \frac{a}{2}

tan (3 u) tan (u) - 3 = 0

Rewrite using trig identities

- 3 + tan (3 u) tan (u)

tan (3 u) = \frac{3 tan ( u ) - tan ^{3} ( u )}{1 - 3 tan ^{2} ( u )}

tan (3 u)

Rewrite using trig identities

tan (3 u)

Rewrite as

= tan (2 u + u)

Use the Angle Sum identity:

tan (s + t) = \frac{tan ( s ) + tan ( t )}{1 - tan ( s ) tan ( t )}

= \frac{tan ( 2 u ) + tan ( u )}{1 - tan ( 2 u ) tan ( u )}

= \frac{tan ( 2 u ) + tan ( u )}{1 - tan ( 2 u ) tan ( u )}

Use the Double Angle identity:

tan (2 u) = \frac{2 tan ( u )}{1 - tan ^{2} ( u )}

= \frac{\frac{2 t a n ( u )}{1 - t a n ^{2} ( u )} + tan ( u )}{1 - \frac{2 t a n ( u )}{1 - t a n ^{2} ( u )} tan ( u )}

Simplify

\frac{\frac{2 t a n ( u )}{1 - t a n ^{2} ( u )} + tan ( u )}{1 - \frac{2 t a n ( u )}{1 - t a n ^{2} ( u )} tan ( u )} : \frac{3 tan ( u ) - tan ^{3} ( u )}{1 - 3 tan ^{2} ( u )}

\frac{\frac{2 t a n ( u )}{1 - t a n ^{2} ( u )} + tan ( u )}{1 - \frac{2 t a n ( u )}{1 - t a n ^{2} ( u )} tan ( u )}

\frac{2 tan ( u )}{1 - tan ^{2} ( u )} tan (u) = \frac{2 tan ^{2} ( u )}{1 - tan ^{2} ( u )}

\frac{2 tan ( u )}{1 - tan ^{2} ( u )} tan (u)

Multiply fractions:

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

= \frac{2 tan ( u ) tan ( u )}{1 - tan ^{2} ( u )}

2 tan (u) tan (u) = 2 tan^{2} (u)

2 tan (u) tan (u)

Apply exponent rule:

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

tan (u) tan (u) = tan^{1 + 1} (u)

= 2 tan^{1 + 1} (u)

Add the numbers:

1 + 1 = 2

= 2 tan^{2} (u)

= \frac{2 tan ^{2} ( u )}{1 - tan ^{2} ( u )}

= \frac{\frac{2 t a n ( u )}{- t a n ^{2} ( u ) + 1} + tan ( u )}{1 - \frac{2 t a n ^{2} ( u )}{- t a n ^{2} ( u ) + 1}}

Join

\frac{2 tan ( u )}{1 - tan ^{2} ( u )} + tan (u) : \frac{3 tan ( u ) - tan ^{3} ( u )}{1 - tan ^{2} ( u )}

\frac{2 tan ( u )}{1 - tan ^{2} ( u )} + tan (u)

Convert element to fraction:

tan (u) = \frac{tan ( u ) ( 1 - tan ^{2} ( u ) )}{1 - tan ^{2} ( u )}

= \frac{2 tan ( u )}{1 - tan ^{2} ( u )} + \frac{tan ( u ) ( 1 - tan ^{2} ( u ) )}{1 - tan ^{2} ( u )}

Since the denominators are equal, combine the fractions:

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

= \frac{2 tan ( u ) + tan ( u ) ( 1 - tan ^{2} ( u ) )}{1 - tan ^{2} ( u )}

Expand

2 tan (u) + tan (u) (1 - tan^{2} (u)) : 3 tan (u) - tan^{3} (u)

2 tan (u) + tan (u) (1 - tan^{2} (u))

Expand

tan (u) (1 - tan^{2} (u)) : tan (u) - tan^{3} (u)

tan (u) (1 - tan^{2} (u))

Apply the distributive law:

a (b - c) = ab - a c

a = tan (u), b = 1, c = tan^{2} (u)

= tan (u) 1 - tan (u) tan^{2} (u)

= 1 tan (u) - tan^{2} (u) tan (u)

Simplify

1 \cdot tan (u) - tan^{2} (u) tan (u) : tan (u) - tan^{3} (u)

1 tan (u) - tan^{2} (u) tan (u)

1 \cdot tan (u) = tan (u)

1 tan (u)

Multiply:

1 \cdot tan (u) = tan (u)

= tan (u)

tan^{2} (u) tan (u) = tan^{3} (u)

tan^{2} (u) tan (u)

Apply exponent rule:

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

tan^{2} (u) tan (u) = tan^{2 + 1} (u)

= tan^{2 + 1} (u)

Add the numbers:

2 + 1 = 3

= tan^{3} (u)

= tan (u) - tan^{3} (u)

= tan (u) - tan^{3} (u)

= 2 tan (u) + tan (u) - tan^{3} (u)

Add similar elements:

2 tan (u) + tan (u) = 3 tan (u)

= 3 tan (u) - tan^{3} (u)

= \frac{3 tan ( u ) - tan ^{3} ( u )}{1 - tan ^{2} ( u )}

= \frac{\frac{3 t a n ( u ) - t a n ^{3} ( u )}{1 - t a n ^{2} ( u )}}{1 - \frac{2 t a n ^{2} ( u )}{- t a n ^{2} ( u ) + 1}}

Apply the fraction rule:

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

= \frac{3 tan ( u ) - tan ^{3} ( u )}{( 1 - tan ^{2} ( u ) ) ( 1 - \frac{2 t a n ^{2} ( u )}{1 - t a n ^{2} ( u )} )}

Join

1 - \frac{2 tan ^{2} ( u )}{1 - tan ^{2} ( u )} : \frac{1 - 3 tan ^{2} ( u )}{1 - tan ^{2} ( u )}

1 - \frac{2 tan ^{2} ( u )}{1 - tan ^{2} ( u )}

Convert element to fraction:

1 = \frac{1 ( 1 - tan ^{2} ( u ) )}{1 - tan ^{2} ( u )}

= \frac{1 ( 1 - tan ^{2} ( u ) )}{1 - tan ^{2} ( u )} - \frac{2 tan ^{2} ( u )}{1 - tan ^{2} ( u )}

Since the denominators are equal, combine the fractions:

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

= \frac{1 ( 1 - tan ^{2} ( u ) ) - 2 tan ^{2} ( u )}{1 - tan ^{2} ( u )}

1 \cdot (1 - tan^{2} (u)) - 2 tan^{2} (u) = 1 - 3 tan^{2} (u)

1 (1 - tan^{2} (u)) - 2 tan^{2} (u)

1 \cdot (1 - tan^{2} (u)) = 1 - tan^{2} (u)

1 (1 - tan^{2} (u))

Multiply:

1 \cdot (1 - tan^{2} (u)) = (1 - tan^{2} (u))

= 1 - tan^{2} (u)

Remove parentheses:

(a) = a

= 1 - tan^{2} (u)

= 1 - tan^{2} (u) - 2 tan^{2} (u)

Add similar elements:

- tan^{2} (u) - 2 tan^{2} (u) = - 3 tan^{2} (u)

= 1 - 3 tan^{2} (u)

= \frac{1 - 3 tan ^{2} ( u )}{1 - tan ^{2} ( u )}

= \frac{3 tan ( u ) - tan ^{3} ( u )}{\frac{- 3 t a n ^{2} ( u ) + 1}{- t a n ^{2} ( u ) + 1} ( - tan ^{2} ( u ) + 1 )}

Multiply

(1 - tan^{2} (u)) \frac{1 - 3 tan ^{2} ( u )}{1 - tan ^{2} ( u )} : 1 - 3 tan^{2} (u)

(1 - tan^{2} (u)) \frac{1 - 3 tan ^{2} ( u )}{1 - tan ^{2} ( u )}

Multiply fractions:

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

= \frac{( 1 - 3 tan ^{2} ( u ) ) ( 1 - tan ^{2} ( u ) )}{1 - tan ^{2} ( u )}

Cancel the common factor:

1 - tan^{2} (u)

= 1 - 3 tan^{2} (u)

= \frac{3 tan ( u ) - tan ^{3} ( u )}{1 - 3 tan ^{2} ( u )}

= \frac{3 tan ( u ) - tan ^{3} ( u )}{1 - 3 tan ^{2} ( u )}

= - 3 + \frac{3 tan ( u ) - tan ^{3} ( u )}{1 - 3 tan ^{2} ( u )} tan (u)

Multiply fractions:

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

= - 3 + \frac{tan ( u ) ( 3 tan ( u ) - tan ^{3} ( u ) )}{1 - 3 tan ^{2} ( u )}

- 3 + \frac{( - tan ^{3} ( u ) + 3 tan ( u ) ) tan ( u )}{1 - 3 tan ^{2} ( u )} = 0

Solve by substitution

- 3 + \frac{( - tan ^{3} ( u ) + 3 tan ( u ) ) tan ( u )}{1 - 3 tan ^{2} ( u )} = 0

Let:

tan (u) = u

- 3 + \frac{( - u ^{3} + 3 u ) u}{1 - 3 u ^{2}} = 0

- 3 + \frac{( - u ^{3} + 3 u ) u}{1 - 3 u ^{2}} = 0 : u = 6 - 33, u = - 6 - 33, u = 6 + 33, u = - 6 + 33

- 3 + \frac{( - u ^{3} + 3 u ) u}{1 - 3 u ^{2}} = 0

Multiply both sides by

1 - 3 u^{2}

- 3 + \frac{( - u ^{3} + 3 u ) u}{1 - 3 u ^{2}} = 0

Multiply both sides by

1 - 3 u^{2}

- 3 (1 - 3 u^{2}) + \frac{( - u ^{3} + 3 u ) u}{1 - 3 u ^{2}} (1 - 3 u^{2}) = 0 \cdot (1 - 3 u^{2})

Simplify

- 3 (1 - 3 u^{2}) + \frac{( - u ^{3} + 3 u ) u}{1 - 3 u ^{2}} (1 - 3 u^{2}) = 0 \cdot (1 - 3 u^{2})

Simplify

\frac{( - u ^{3} + 3 u ) u}{1 - 3 u ^{2}} (1 - 3 u^{2}) : u (- u^{3} + 3 u)

\frac{( - u ^{3} + 3 u ) u}{1 - 3 u ^{2}} (1 - 3 u^{2})

Multiply fractions:

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

= \frac{( - u ^{3} + 3 u ) u ( 1 - 3 u ^{2} )}{1 - 3 u ^{2}}

Cancel the common factor:

1 - 3 u^{2}

= (- u^{3} + 3 u) u

Simplify

0 \cdot (1 - 3 u^{2}) : 0

0 \cdot (1 - 3 u^{2})

Apply rule

0 \cdot a = 0

= 0

- 3 (1 - 3 u^{2}) + u (- u^{3} + 3 u) = 0

- 3 (1 - 3 u^{2}) + u (- u^{3} + 3 u) = 0

- 3 (1 - 3 u^{2}) + u (- u^{3} + 3 u) = 0

Solve

- 3 (1 - 3 u^{2}) + u (- u^{3} + 3 u) = 0 : u = 6 - 33, u = - 6 - 33, u = 6 + 33, u = - 6 + 33

- 3 (1 - 3 u^{2}) + u (- u^{3} + 3 u) = 0

Expand

- 3 (1 - 3 u^{2}) + u (- u^{3} + 3 u) : - u^{4} + 12 u^{2} - 3

- 3 (1 - 3 u^{2}) + u (- u^{3} + 3 u)

Expand

- 3 (1 - 3 u^{2}) : - 3 + 9 u^{2}

- 3 (1 - 3 u^{2})

Apply the distributive law:

a (b - c) = ab - a c

a = - 3, b = 1, c = 3 u^{2}

= - 3 \cdot 1 - (- 3) \cdot 3 u^{2}

Apply minus-plus rules

- (- a) = a

= - 3 \cdot 1 + 3 \cdot 3 u^{2}

Simplify

- 3 \cdot 1 + 3 \cdot 3 u^{2} : - 3 + 9 u^{2}

- 3 \cdot 1 + 3 \cdot 3 u^{2}

Multiply the numbers:

3 \cdot 1 = 3

= - 3 + 3 \cdot 3 u^{2}

Multiply the numbers:

3 \cdot 3 = 9

= - 3 + 9 u^{2}

= - 3 + 9 u^{2}

= - 3 + 9 u^{2} + u (- u^{3} + 3 u)

Expand

u (- u^{3} + 3 u) : - u^{4} + 3 u^{2}

u (- u^{3} + 3 u)

Apply the distributive law:

a (b + c) = ab + a c

a = u, b = - u^{3}, c = 3 u

= u (- u^{3}) + u \cdot 3 u

Apply minus-plus rules

+ (- a) = - a

= - u^{3} u + 3 uu

Simplify

- u^{3} u + 3 uu : - u^{4} + 3 u^{2}

- u^{3} u + 3 uu

u^{3} u = u^{4}

u^{3} u

Apply exponent rule:

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

u^{3} u = u^{3 + 1}

= u^{3 + 1}

Add the numbers:

3 + 1 = 4

= u^{4}

3 uu = 3 u^{2}

3 uu

Apply exponent rule:

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

uu = u^{1 + 1}

= 3 u^{1 + 1}

Add the numbers:

1 + 1 = 2

= 3 u^{2}

= - u^{4} + 3 u^{2}

= - u^{4} + 3 u^{2}

= - 3 + 9 u^{2} - u^{4} + 3 u^{2}

Simplify

- 3 + 9 u^{2} - u^{4} + 3 u^{2} : - u^{4} + 12 u^{2} - 3

- 3 + 9 u^{2} - u^{4} + 3 u^{2}

Group like terms

= - u^{4} + 9 u^{2} + 3 u^{2} - 3

Add similar elements:

9 u^{2} + 3 u^{2} = 12 u^{2}

= - u^{4} + 12 u^{2} - 3

= - u^{4} + 12 u^{2} - 3

- u^{4} + 12 u^{2} - 3 = 0

Rewrite the equation with

v = u^{2}

and

v^{2} = u^{4}

- v^{2} + 12 v - 3 = 0

Solve

- v^{2} + 12 v - 3 = 0 : v = 6 - 33, v = 6 + 33

- v^{2} + 12 v - 3 = 0

Solve with the quadratic formula

- v^{2} + 12 v - 3 = 0

Quadratic Equation Formula:

For

a = - 1, b = 12, c = - 3

v_{1, 2} = \frac{- 12 \pm 1 2 ^{2} - 4 ( - 1 ) ( - 3 )}{2 ( - 1 )}

v_{1, 2} = \frac{- 12 \pm 1 2 ^{2} - 4 ( - 1 ) ( - 3 )}{2 ( - 1 )}

1 2^{2} - 4 (- 1) (- 3) = 233

1 2^{2} - 4 (- 1) (- 3)

Apply rule

- (- a) = a

= 1 2^{2} - 4 \cdot 1 \cdot 3

Multiply the numbers:

4 \cdot 1 \cdot 3 = 12

= 1 2^{2} - 12

1 2^{2} = 144

= 144 - 12

Subtract the numbers:

144 - 12 = 132

= 132

Prime factorization of

132 : 2^{2} \cdot 3 \cdot 11

132

132

divides by

2132 = 66 \cdot 2

= 2 \cdot 66

66

divides by

266 = 33 \cdot 2

= 2 \cdot 2 \cdot 33

33

divides by

333 = 11 \cdot 3

= 2 \cdot 2 \cdot 3 \cdot 11

2, 3, 11

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 3 \cdot 11

= 2^{2} \cdot 3 \cdot 11

= 2^{2} \cdot 3 \cdot 11

Apply radical rule:

= 2^{2} 3 \cdot 11

Apply radical rule:

2^{2} = 2

= 2 3 \cdot 11

Refine

= 233

v_{1, 2} = \frac{- 12 \pm 2 33}{2 ( - 1 )}

Separate the solutions

v_{1} = \frac{- 12 + 2 33}{2 ( - 1 )}, v_{2} = \frac{- 12 - 2 33}{2 ( - 1 )}

v = \frac{- 12 + 2 33}{2 ( - 1 )} : 6 - 33

\frac{- 12 + 2 33}{2 ( - 1 )}

Remove parentheses:

(- a) = - a

= \frac{- 12 + 2 33}{- 2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{- 12 + 2 33}{- 2}

Apply the fraction rule:

\frac{- a}{- b} = \frac{a}{b}

- 12 + 233 = - (12 - 233)

= \frac{12 - 2 33}{2}

Factor

12 - 233 : 2 (6 - 33)

12 - 233

Rewrite as

= 2 \cdot 6 - 233

Factor out common term

2

= 2 (6 - 33)

= \frac{2 ( 6 - 33 )}{2}

Divide the numbers:

\frac{2}{2} = 1

= 6 - 33

v = \frac{- 12 - 2 33}{2 ( - 1 )} : 6 + 33

\frac{- 12 - 2 33}{2 ( - 1 )}

Remove parentheses:

(- a) = - a

= \frac{- 12 - 2 33}{- 2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{- 12 - 2 33}{- 2}

Apply the fraction rule:

\frac{- a}{- b} = \frac{a}{b}

- 12 - 233 = - (12 + 233)

= \frac{12 + 2 33}{2}

Factor

12 + 233 : 2 (6 + 33)

12 + 233

Rewrite as

= 2 \cdot 6 + 233

Factor out common term

2

= 2 (6 + 33)

= \frac{2 ( 6 + 33 )}{2}

Divide the numbers:

\frac{2}{2} = 1

= 6 + 33

The solutions to the quadratic equation are:

v = 6 - 33, v = 6 + 33

v = 6 - 33, v = 6 + 33

Substitute back

v = u^{2},

solve for

u

Solve

u^{2} = 6 - 33 : u = 6 - 33, u = - 6 - 33

u^{2} = 6 - 33

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

u = 6 - 33, u = - 6 - 33

Solve

u^{2} = 6 + 33 : u = 6 + 33, u = - 6 + 33

u^{2} = 6 + 33

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

u = 6 + 33, u = - 6 + 33

The solutions are

u = 6 - 33, u = - 6 - 33, u = 6 + 33, u = - 6 + 33

u = 6 - 33, u = - 6 - 33, u = 6 + 33, u = - 6 + 33

Verify Solutions

Find undefined (singularity) points:

u = \frac{1}{3}, u = - \frac{1}{3}

Take the denominator(s) of

- 3 + \frac{( - u ^{3} + 3 u ) u}{1 - 3 u ^{2}}

and compare to zero

Solve

1 - 3 u^{2} = 0 : u = \frac{1}{3}, u = - \frac{1}{3}

1 - 3 u^{2} = 0

Move

1

to the right side

1 - 3 u^{2} = 0

Subtract

1

from both sides

1 - 3 u^{2} - 1 = 0 - 1

Simplify

- 3 u^{2} = - 1

- 3 u^{2} = - 1

Divide both sides by

- 3

- 3 u^{2} = - 1

Divide both sides by

- 3

\frac{- 3 u ^{2}}{- 3} = \frac{- 1}{- 3}

Simplify

u^{2} = \frac{1}{3}

u^{2} = \frac{1}{3}

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

u = \frac{1}{3}, u = - \frac{1}{3}

\frac{1}{3} = \frac{1}{3}

\frac{1}{3}

Apply radical rule:

\frac{a}{b} = \frac{a}{b}, a \geq 0, b \geq 0

= \frac{1}{3}

Apply radical rule:

1 = 1

1 = 1

= \frac{1}{3}

- \frac{1}{3} = - \frac{1}{3}

- \frac{1}{3}

Apply radical rule:

\frac{a}{b} = \frac{a}{b}, a \geq 0, b \geq 0

= - \frac{1}{3}

Apply radical rule:

1 = 1

1 = 1

= - \frac{1}{3}

u = \frac{1}{3}, u = - \frac{1}{3}

The following points are undefined

u = \frac{1}{3}, u = - \frac{1}{3}

Combine undefined points with solutions:

u = 6 - 33, u = - 6 - 33, u = 6 + 33, u = - 6 + 33

Substitute back

u = tan (u)

tan (u) = 6 - 33, tan (u) = - 6 - 33, tan (u) = 6 + 33, tan (u) = - 6 + 33

tan (u) = 6 - 33, tan (u) = - 6 - 33, tan (u) = 6 + 33, tan (u) = - 6 + 33

tan (u) = 6 - 33 : u = arctan (6 - 33) + πn

tan (u) = 6 - 33

Apply trig inverse properties

tan (u) = 6 - 33

General solutions for

tan (u) = 6 - 33

tan (x) = a \Rightarrow x = arctan (a) + πn

u = arctan (6 - 33) + πn

u = arctan (6 - 33) + πn

tan (u) = - 6 - 33 : u = arctan (- 6 - 33) + πn

tan (u) = - 6 - 33

Apply trig inverse properties

tan (u) = - 6 - 33

General solutions for

tan (u) = - 6 - 33

tan (x) = - a \Rightarrow x = arctan (- a) + πn

u = arctan (- 6 - 33) + πn

u = arctan (- 6 - 33) + πn

tan (u) = 6 + 33 : u = arctan (6 + 33) + πn

tan (u) = 6 + 33

Apply trig inverse properties

tan (u) = 6 + 33

General solutions for

tan (u) = 6 + 33

tan (x) = a \Rightarrow x = arctan (a) + πn

u = arctan (6 + 33) + πn

u = arctan (6 + 33) + πn

tan (u) = - 6 + 33 : u = arctan (- 6 + 33) + πn

tan (u) = - 6 + 33

Apply trig inverse properties

tan (u) = - 6 + 33

General solutions for

tan (u) = - 6 + 33

tan (x) = - a \Rightarrow x = arctan (- a) + πn

u = arctan (- 6 + 33) + πn

u = arctan (- 6 + 33) + πn

Combine all the solutions

u = arctan (6 - 33) + πn, u = arctan (- 6 - 33) + πn, u = arctan (6 + 33) + πn, u = arctan (- 6 + 33) + πn

Substitute back

u = \frac{a}{2}

\frac{a}{2} = arctan (6 - 33) + πn : a = 2 arctan (6 - 33) + 2 πn

\frac{a}{2} = arctan (6 - 33) + πn

Multiply both sides by

2

\frac{a}{2} = arctan (6 - 33) + πn

Multiply both sides by

2

\frac{2 a}{2} = 2 arctan (6 - 33) + 2 πn

Simplify

a = 2 arctan (6 - 33) + 2 πn

a = 2 arctan (6 - 33) + 2 πn

\frac{a}{2} = arctan (- 6 - 33) + πn : a = - 2 arctan (6 - 33) + 2 πn

\frac{a}{2} = arctan (- 6 - 33) + πn

Simplify

arctan (- 6 - 33) + πn : - arctan (6 - 33) + πn

arctan (- 6 - 33) + πn

Use the following property:

arctan (- x) = - arctan (x)

arctan (- 6 - 33) = - arctan (6 - 33)

= - arctan (6 - 33) + πn

\frac{a}{2} = - arctan (6 - 33) + πn

Multiply both sides by

2

\frac{a}{2} = - arctan (6 - 33) + πn

Multiply both sides by

2

\frac{2 a}{2} = - 2 arctan (6 - 33) + 2 πn

Simplify

a = - 2 arctan (6 - 33) + 2 πn

a = - 2 arctan (6 - 33) + 2 πn

\frac{a}{2} = arctan (6 + 33) + πn : a = 2 arctan (6 + 33) + 2 πn

\frac{a}{2} = arctan (6 + 33) + πn

Multiply both sides by

2

\frac{a}{2} = arctan (6 + 33) + πn

Multiply both sides by

2

\frac{2 a}{2} = 2 arctan (6 + 33) + 2 πn

Simplify

a = 2 arctan (6 + 33) + 2 πn

a = 2 arctan (6 + 33) + 2 πn

\frac{a}{2} = arctan (- 6 + 33) + πn : a = - 2 arctan (6 + 33) + 2 πn

\frac{a}{2} = arctan (- 6 + 33) + πn

Simplify

arctan (- 6 + 33) + πn : - arctan (6 + 33) + πn

arctan (- 6 + 33) + πn

Use the following property:

arctan (- x) = - arctan (x)

arctan (- 6 + 33) = - arctan (6 + 33)

= - arctan (6 + 33) + πn

\frac{a}{2} = - arctan (6 + 33) + πn

Multiply both sides by

2

\frac{a}{2} = - arctan (6 + 33) + πn

Multiply both sides by

2

\frac{2 a}{2} = - 2 arctan (6 + 33) + 2 πn

Simplify

a = - 2 arctan (6 + 33) + 2 πn

a = - 2 arctan (6 + 33) + 2 πn

a = 2 arctan (6 - 33) + 2 πn, a = - 2 arctan (6 - 33) + 2 πn, a = 2 arctan (6 + 33) + 2 πn, a = - 2 arctan (6 + 33) + 2 πn

Show solutions in decimal form

a = 2 \cdot 0.46796 \dots + 2 πn, a = - 2 \cdot 0.46796 \dots + 2 πn, a = 2 \cdot 1.28688 \dots + 2 πn, a = - 2 \cdot 1.28688 \dots + 2 πn

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