What is the general solution for tan(θ)cos(27)=cos(63) ?

The general solution for tan(θ)cos(27)=cos(63) is θ=0.47123…+180n

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tan(θ)cos(27)=cos(63)

Solution

tan (θ) cos (2 7^{\circ}) = cos (6 3^{\circ})

Solution

θ = 0.47123 \dots + 18 0^{\circ} n

Radians

θ = 0.47123 \dots + πn

Solution steps

tan (θ) cos (2 7^{\circ}) = cos (6 3^{\circ})

cos (2 7^{\circ}) = \frac{2 4 + 2 5 - 5}{4}

cos (2 7^{\circ})

Rewrite using trig identities:

\frac{1 + cos ( 5 4 ^{\circ} )}{2}

cos (2 7^{\circ})

Write

cos (2 7^{\circ})

cos (\frac{5 4 ^{\circ}}{2})

= cos (\frac{5 4 ^{\circ}}{2})

Use the Half Angle identity:

cos (\frac{θ}{2}) = \frac{1 + cos ( θ )}{2}

Use the Double Angle identity

cos (2 θ) = 2 cos^{2} (θ) - 1

Substitute

θ

with

\frac{θ}{2}

cos (θ) = 2 cos^{2} (\frac{θ}{2}) - 1

Switch sides

2 cos^{2} (\frac{θ}{2}) = 1 + cos (θ)

Divide both sides by

2

cos^{2} (\frac{θ}{2}) = \frac{( 1 + cos ( θ ) )}{2}

Square root both sides

Choose the root sign according to the quadrant of

\frac{θ}{2}

range [0, 9 0^{\circ}] [9 0^{\circ}, 18 0^{\circ}] [18 0^{\circ}, 27 0^{\circ}] [27 0^{\circ}, 36 0^{\circ}] quadrant I II III I V sin positive positive negative negative cos positive negative negative positive

cos (\frac{θ}{2}) = \frac{( 1 + cos ( θ ) )}{2}

= \frac{1 + cos ( 5 4 ^{\circ} )}{2}

= \frac{1 + cos ( 5 4 ^{\circ} )}{2}

Rewrite using trig identities:

cos (5 4^{\circ}) = \frac{2 5 - 5}{4}

cos (5 4^{\circ})

Rewrite using trig identities:

sin (3 6^{\circ})

cos (5 4^{\circ})

Use the following identity:

cos (x) = sin (9 0^{\circ} - x)

= sin (9 0^{\circ} - 5 4^{\circ})

Simplify:

9 0^{\circ} - 5 4^{\circ} = 3 6^{\circ}

9 0^{\circ} - 5 4^{\circ}

Least Common Multiplier of

2, 10 : 10

2, 10

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Prime factorization of

10 : 2 \cdot 5

10

10

divides by

210 = 5 \cdot 2

= 2 \cdot 5

2, 5

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 5

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

2

10

= 2 \cdot 5

Multiply the numbers:

2 \cdot 5 = 10

= 10

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

10

For

9 0^{\circ} :

multiply the denominator and numerator by

5

9 0^{\circ} = \frac{18 0 ^{\circ} 5}{2 \cdot 5} = 9 0^{\circ}

= 9 0^{\circ} - 5 4^{\circ}

Since the denominators are equal, combine the fractions:

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

= \frac{18 0 ^{\circ} 5 - 54 0 ^{\circ}}{10}

Add similar elements:

90 0^{\circ} - 54 0^{\circ} = 36 0^{\circ}

= 3 6^{\circ}

Cancel the common factor:

2

= 3 6^{\circ}

= sin (3 6^{\circ})

= sin (3 6^{\circ})

Rewrite using trig identities:

\frac{2 5 - 5}{4}

sin (3 6^{\circ})

Show that:

cos (3 6^{\circ}) - sin (1 8^{\circ}) = \frac{1}{2}

Use the following product to sum identity:

2 sin (x) cos (y) = sin (x + y) - sin (x - y)

2 cos (3 6^{\circ}) sin (1 8^{\circ}) = sin (5 4^{\circ}) - sin (1 8^{\circ})

Show that:

2 cos (3 6^{\circ}) sin (1 8^{\circ}) = \frac{1}{2}

Use the Double Angle identity:

sin (2 x) = 2 sin (x) cos (x)

sin (7 2^{\circ}) = 2 sin (3 6^{\circ}) cos (3 6^{\circ})

sin (7 2^{\circ}) sin (3 6^{\circ}) = 4 sin (3 6^{\circ}) sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

Divide both sides by

sin (3 6^{\circ})

sin (7 2^{\circ}) = 4 sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

Use the following identity:

sin (x) = cos (9 0^{\circ} - x)

sin (7 2^{\circ}) = cos (9 0^{\circ} - 7 2^{\circ})

cos (9 0^{\circ} - 7 2^{\circ}) = 4 sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

cos (1 8^{\circ}) = 4 sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

Divide both sides by

cos (1 8^{\circ})

1 = 4 sin (1 8^{\circ}) cos (3 6^{\circ})

Divide both sides by

2

\frac{1}{2} = 2 sin (1 8^{\circ}) cos (3 6^{\circ})

Substitute

\frac{1}{2} = 2 sin (1 8^{\circ}) cos (3 6^{\circ})

\frac{1}{2} = sin (5 4^{\circ}) - sin (1 8^{\circ})

sin (5 4^{\circ}) = cos (9 0^{\circ} - 5 4^{\circ})

\frac{1}{2} = cos (9 0^{\circ} - 5 4^{\circ}) - sin (1 8^{\circ})

\frac{1}{2} = cos (3 6^{\circ}) - sin (1 8^{\circ})

Show that:

cos (3 6^{\circ}) + sin (1 8^{\circ}) = \frac{5}{4}

Use the factorization rule:

a^{2} - b^{2} = (a + b) (a - b)

a = cos (3 6^{\circ}) + sin (1 8^{\circ})

(cos (3 6^{\circ}) + sin (1 8^{\circ}))^{2} - (cos (3 6^{\circ}) - sin (1 8^{\circ}))^{2} = ((cos (3 6^{\circ}) + sin (1 8^{\circ})) + (cos (3 6^{\circ}) - sin (1 8^{\circ}))) ((cos (3 6^{\circ}) + sin (1 8^{\circ})) - (cos (3 6^{\circ}) - sin (1 8^{\circ})))

Refine

(cos (3 6^{\circ}) + sin (1 8^{\circ}))^{2} - (cos (3 6^{\circ}) - sin (1 8^{\circ}))^{2} = 2 (2 cos (3 6^{\circ}) sin (1 8^{\circ}))

Show that:

2 cos (3 6^{\circ}) sin (1 8^{\circ}) = \frac{1}{2}

Use the Double Angle identity:

sin (2 x) = 2 sin (x) cos (x)

sin (7 2^{\circ}) = 2 sin (3 6^{\circ}) cos (3 6^{\circ})

sin (7 2^{\circ}) sin (3 6^{\circ}) = 4 sin (3 6^{\circ}) sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

Divide both sides by

sin (3 6^{\circ})

sin (7 2^{\circ}) = 4 sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

Use the following identity:

sin (x) = cos (9 0^{\circ} - x)

sin (7 2^{\circ}) = cos (9 0^{\circ} - 7 2^{\circ})

cos (9 0^{\circ} - 7 2^{\circ}) = 4 sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

cos (1 8^{\circ}) = 4 sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

Divide both sides by

cos (1 8^{\circ})

1 = 4 sin (1 8^{\circ}) cos (3 6^{\circ})

Divide both sides by

2

\frac{1}{2} = 2 sin (1 8^{\circ}) cos (3 6^{\circ})

Substitute

2 cos (3 6^{\circ}) sin (1 8^{\circ}) = \frac{1}{2}

(cos (3 6^{\circ}) + sin (1 8^{\circ}))^{2} - (cos (3 6^{\circ}) - sin (1 8^{\circ}))^{2} = 1

Substitute

cos (3 6^{\circ}) - sin (1 8^{\circ}) = \frac{1}{2}

(cos (3 6^{\circ}) + sin (1 8^{\circ}))^{2} - (\frac{1}{2})^{2} = 1

Refine

(cos (3 6^{\circ}) + sin (1 8^{\circ}))^{2} - \frac{1}{4} = 1

Add

\frac{1}{4}

to both sides

(cos (3 6^{\circ}) + sin (1 8^{\circ}))^{2} - \frac{1}{4} + \frac{1}{4} = 1 + \frac{1}{4}

Refine

(cos (3 6^{\circ}) + sin (1 8^{\circ}))^{2} = \frac{5}{4}

Take the square root of both sides

cos (3 6^{\circ}) + sin (1 8^{\circ}) = \pm \frac{5}{4}

cos (3 6^{\circ})

cannot be negative

sin (1 8^{\circ})

cannot be negative

cos (3 6^{\circ}) + sin (1 8^{\circ}) = \frac{5}{4}

Add the following equations

cos (3 6^{\circ}) + sin (1 8^{\circ}) = \frac{5}{2}

((cos (3 6^{\circ}) + sin (1 8^{\circ})) + (cos (3 6^{\circ}) - sin (1 8^{\circ}))) = (\frac{5}{2} + \frac{1}{2})

Refine

cos (3 6^{\circ}) = \frac{5 + 1}{4}

Square both sides

(cos (3 6^{\circ}))^{2} = (\frac{5 + 1}{4})^{2}

Use the following identity:

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

sin^{2} (3 6^{\circ}) = 1 - cos^{2} (3 6^{\circ})

Substitute

cos (3 6^{\circ}) = \frac{5 + 1}{4}

sin^{2} (3 6^{\circ}) = 1 - (\frac{5 + 1}{4})^{2}

Refine

sin^{2} (3 6^{\circ}) = \frac{5 - 5}{8}

Take the square root of both sides

sin (3 6^{\circ}) = \pm \frac{5 - 5}{8}

sin (3 6^{\circ})

cannot be negative

sin (3 6^{\circ}) = \frac{5 - 5}{8}

Refine

sin (3 6^{\circ}) = \frac{\frac{5 - 5}{2}}{2}

= \frac{\frac{5 - 5}{2}}{2}

\frac{\frac{5 - 5}{2}}{2} = \frac{2 5 - 5}{4}

\frac{\frac{5 - 5}{2}}{2}

\frac{5 - 5}{2} = \frac{5 - 5}{2}

\frac{5 - 5}{2}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{5 - 5}{2}

= \frac{\frac{5 - 5}{2}}{2}

Apply the fraction rule:

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

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

Rationalize

\frac{5 - 5}{2 2} : \frac{2 5 - 5}{4}

\frac{5 - 5}{2 2}

Multiply by the conjugate

\frac{2}{2}

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

2 \cdot 22 = 4

2 \cdot 22

Apply exponent rule:

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

222 = 2 \cdot 2^{\frac{1}{2}} \cdot 2^{\frac{1}{2}} = 2^{1 + \frac{1}{2} + \frac{1}{2}}

= 2^{1 + \frac{1}{2} + \frac{1}{2}}

Add similar elements:

\frac{1}{2} + \frac{1}{2} = 2 \cdot \frac{1}{2}

= 2^{1 + 2 \cdot \frac{1}{2}}

2 \cdot \frac{1}{2} = 1

2 \cdot \frac{1}{2}

Multiply fractions:

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

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 2^{1 + 1}

Add the numbers:

1 + 1 = 2

= 2^{2}

2^{2} = 4

= 4

= \frac{2 5 - 5}{4}

= \frac{2 5 - 5}{4}

= \frac{2 5 - 5}{4}

= \frac{2 5 - 5}{4}

= \frac{1 + \frac{2 5 - 5}{4}}{2}

Simplify

\frac{1 + \frac{2 5 - 5}{4}}{2} : \frac{2 4 + 2 5 - 5}{4}

\frac{1 + \frac{2 5 - 5}{4}}{2}

\frac{1 + \frac{2 5 - 5}{4}}{2} = \frac{4 + 2 5 - 5}{8}

\frac{1 + \frac{2 5 - 5}{4}}{2}

Join

1 + \frac{2 5 - 5}{4} : \frac{4 + 2 5 - 5}{4}

1 + \frac{2 5 - 5}{4}

Convert element to fraction:

1 = \frac{1 \cdot 4}{4}

= \frac{1 \cdot 4}{4} + \frac{2 5 - 5}{4}

Since the denominators are equal, combine the fractions:

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

= \frac{1 \cdot 4 + 2 5 - 5}{4}

Multiply the numbers:

1 \cdot 4 = 4

= \frac{4 + 2 5 - 5}{4}

= \frac{\frac{4 + 2 5 - 5}{4}}{2}

Apply the fraction rule:

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

= \frac{4 + 2 5 - 5}{4 \cdot 2}

Multiply the numbers:

4 \cdot 2 = 8

= \frac{4 + 2 5 - 5}{8}

= \frac{4 + 2 5 - 5}{8}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{4 + 2 5 - 5}{8}

8 = 22

8

Prime factorization of

8 : 2^{3}

8

8

divides by

28 = 4 \cdot 2

= 2 \cdot 4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2 \cdot 2

2

is a prime number, therefore no further factorization is possible

= 2 \cdot 2 \cdot 2

= 2^{3}

= 2^{3}

Apply exponent rule:

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

= 2^{2} \cdot 2

Apply radical rule:

= 2 2^{2}

Apply radical rule:

2^{2} = 2

= 22

= \frac{2 5 - 5 + 4}{2 2}

Rationalize

\frac{4 + 2 5 - 5}{2 2} : \frac{2 2 5 - 5 + 4}{4}

\frac{4 + 2 5 - 5}{2 2}

Multiply by the conjugate

\frac{2}{2}

= \frac{4 + 2 5 - 5 2}{2 2 2}

222 = 4

222

Apply exponent rule:

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

222 = 2 \cdot 2^{\frac{1}{2}} \cdot 2^{\frac{1}{2}} = 2^{1 + \frac{1}{2} + \frac{1}{2}}

= 2^{1 + \frac{1}{2} + \frac{1}{2}}

Add similar elements:

\frac{1}{2} + \frac{1}{2} = 2 \cdot \frac{1}{2}

= 2^{1 + 2 \cdot \frac{1}{2}}

2 \cdot \frac{1}{2} = 1

2 \cdot \frac{1}{2}

Multiply fractions:

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

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 2^{1 + 1}

Add the numbers:

1 + 1 = 2

= 2^{2}

2^{2} = 4

= 4

= \frac{2 4 + 2 5 - 5}{4}

= \frac{2 2 5 - 5 + 4}{4}

= \frac{2 4 + 2 5 - 5}{4}

cos (6 3^{\circ}) = \frac{2 4 - 2 5 - 5}{4}

cos (6 3^{\circ})

Rewrite using trig identities:

\frac{1 + cos ( 12 6 ^{\circ} )}{2}

cos (6 3^{\circ})

Write

cos (6 3^{\circ})

cos (\frac{12 6 ^{\circ}}{2})

= cos (\frac{12 6 ^{\circ}}{2})

Use the Half Angle identity:

cos (\frac{θ}{2}) = \frac{1 + cos ( θ )}{2}

Use the Double Angle identity

cos (2 θ) = 2 cos^{2} (θ) - 1

Substitute

θ

with

\frac{θ}{2}

cos (θ) = 2 cos^{2} (\frac{θ}{2}) - 1

Switch sides

2 cos^{2} (\frac{θ}{2}) = 1 + cos (θ)

Divide both sides by

2

cos^{2} (\frac{θ}{2}) = \frac{( 1 + cos ( θ ) )}{2}

Square root both sides

Choose the root sign according to the quadrant of

\frac{θ}{2}

range [0, 9 0^{\circ}] [9 0^{\circ}, 18 0^{\circ}] [18 0^{\circ}, 27 0^{\circ}] [27 0^{\circ}, 36 0^{\circ}] quadrant I II III I V sin positive positive negative negative cos positive negative negative positive

cos (\frac{θ}{2}) = \frac{( 1 + cos ( θ ) )}{2}

= \frac{1 + cos ( 12 6 ^{\circ} )}{2}

= \frac{1 + cos ( 12 6 ^{\circ} )}{2}

Rewrite using trig identities:

cos (12 6^{\circ}) = - \frac{2 5 - 5}{4}

cos (12 6^{\circ})

Rewrite using trig identities:

- sin (3 6^{\circ})

cos (12 6^{\circ})

Use the following identity:

cos (x) = sin (9 0^{\circ} - x)

= sin (9 0^{\circ} - 12 6^{\circ})

Simplify:

9 0^{\circ} - 12 6^{\circ} = - 3 6^{\circ}

9 0^{\circ} - 12 6^{\circ}

Least Common Multiplier of

2, 10 : 10

2, 10

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Prime factorization of

10 : 2 \cdot 5

10

10

divides by

210 = 5 \cdot 2

= 2 \cdot 5

2, 5

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 5

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

2

10

= 2 \cdot 5

Multiply the numbers:

2 \cdot 5 = 10

= 10

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

10

For

9 0^{\circ} :

multiply the denominator and numerator by

5

9 0^{\circ} = \frac{18 0 ^{\circ} 5}{2 \cdot 5} = 9 0^{\circ}

= 9 0^{\circ} - 12 6^{\circ}

Since the denominators are equal, combine the fractions:

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

= \frac{18 0 ^{\circ} 5 - 126 0 ^{\circ}}{10}

Add similar elements:

90 0^{\circ} - 126 0^{\circ} = - 36 0^{\circ}

= \frac{- 36 0 ^{\circ}}{10}

Apply the fraction rule:

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

= - 3 6^{\circ}

Cancel the common factor:

2

= - 3 6^{\circ}

= sin (- 3 6^{\circ})

Use the following property:

sin (- x) = - sin (x)

sin (- 3 6^{\circ}) = - sin (3 6^{\circ})

= - sin (3 6^{\circ})

= - sin (3 6^{\circ})

Rewrite using trig identities:

sin (3 6^{\circ}) = \frac{2 5 - 5}{4}

sin (3 6^{\circ})

Show that:

cos (3 6^{\circ}) - sin (1 8^{\circ}) = \frac{1}{2}

Use the following product to sum identity:

2 sin (x) cos (y) = sin (x + y) - sin (x - y)

2 cos (3 6^{\circ}) sin (1 8^{\circ}) = sin (5 4^{\circ}) - sin (1 8^{\circ})

Show that:

2 cos (3 6^{\circ}) sin (1 8^{\circ}) = \frac{1}{2}

Use the Double Angle identity:

sin (2 x) = 2 sin (x) cos (x)

sin (7 2^{\circ}) = 2 sin (3 6^{\circ}) cos (3 6^{\circ})

sin (7 2^{\circ}) sin (3 6^{\circ}) = 4 sin (3 6^{\circ}) sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

Divide both sides by

sin (3 6^{\circ})

sin (7 2^{\circ}) = 4 sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

Use the following identity:

sin (x) = cos (9 0^{\circ} - x)

sin (7 2^{\circ}) = cos (9 0^{\circ} - 7 2^{\circ})

cos (9 0^{\circ} - 7 2^{\circ}) = 4 sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

cos (1 8^{\circ}) = 4 sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

Divide both sides by

cos (1 8^{\circ})

1 = 4 sin (1 8^{\circ}) cos (3 6^{\circ})

Divide both sides by

2

\frac{1}{2} = 2 sin (1 8^{\circ}) cos (3 6^{\circ})

Substitute

\frac{1}{2} = 2 sin (1 8^{\circ}) cos (3 6^{\circ})

\frac{1}{2} = sin (5 4^{\circ}) - sin (1 8^{\circ})

sin (5 4^{\circ}) = cos (9 0^{\circ} - 5 4^{\circ})

\frac{1}{2} = cos (9 0^{\circ} - 5 4^{\circ}) - sin (1 8^{\circ})

\frac{1}{2} = cos (3 6^{\circ}) - sin (1 8^{\circ})

Show that:

cos (3 6^{\circ}) + sin (1 8^{\circ}) = \frac{5}{4}

Use the factorization rule:

a^{2} - b^{2} = (a + b) (a - b)

a = cos (3 6^{\circ}) + sin (1 8^{\circ})

(cos (3 6^{\circ}) + sin (1 8^{\circ}))^{2} - (cos (3 6^{\circ}) - sin (1 8^{\circ}))^{2} = ((cos (3 6^{\circ}) + sin (1 8^{\circ})) + (cos (3 6^{\circ}) - sin (1 8^{\circ}))) ((cos (3 6^{\circ}) + sin (1 8^{\circ})) - (cos (3 6^{\circ}) - sin (1 8^{\circ})))

Refine

(cos (3 6^{\circ}) + sin (1 8^{\circ}))^{2} - (cos (3 6^{\circ}) - sin (1 8^{\circ}))^{2} = 2 (2 cos (3 6^{\circ}) sin (1 8^{\circ}))

Show that:

2 cos (3 6^{\circ}) sin (1 8^{\circ}) = \frac{1}{2}

Use the Double Angle identity:

sin (2 x) = 2 sin (x) cos (x)

sin (7 2^{\circ}) = 2 sin (3 6^{\circ}) cos (3 6^{\circ})

sin (7 2^{\circ}) sin (3 6^{\circ}) = 4 sin (3 6^{\circ}) sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

Divide both sides by

sin (3 6^{\circ})

sin (7 2^{\circ}) = 4 sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

Use the following identity:

sin (x) = cos (9 0^{\circ} - x)

sin (7 2^{\circ}) = cos (9 0^{\circ} - 7 2^{\circ})

cos (9 0^{\circ} - 7 2^{\circ}) = 4 sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

cos (1 8^{\circ}) = 4 sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

Divide both sides by

cos (1 8^{\circ})

1 = 4 sin (1 8^{\circ}) cos (3 6^{\circ})

Divide both sides by

2

\frac{1}{2} = 2 sin (1 8^{\circ}) cos (3 6^{\circ})

Substitute

2 cos (3 6^{\circ}) sin (1 8^{\circ}) = \frac{1}{2}

(cos (3 6^{\circ}) + sin (1 8^{\circ}))^{2} - (cos (3 6^{\circ}) - sin (1 8^{\circ}))^{2} = 1

Substitute

cos (3 6^{\circ}) - sin (1 8^{\circ}) = \frac{1}{2}

(cos (3 6^{\circ}) + sin (1 8^{\circ}))^{2} - (\frac{1}{2})^{2} = 1

Refine

(cos (3 6^{\circ}) + sin (1 8^{\circ}))^{2} - \frac{1}{4} = 1

Add

\frac{1}{4}

to both sides

(cos (3 6^{\circ}) + sin (1 8^{\circ}))^{2} - \frac{1}{4} + \frac{1}{4} = 1 + \frac{1}{4}

Refine

(cos (3 6^{\circ}) + sin (1 8^{\circ}))^{2} = \frac{5}{4}

Take the square root of both sides

cos (3 6^{\circ}) + sin (1 8^{\circ}) = \pm \frac{5}{4}

cos (3 6^{\circ})

cannot be negative

sin (1 8^{\circ})

cannot be negative

cos (3 6^{\circ}) + sin (1 8^{\circ}) = \frac{5}{4}

Add the following equations

cos (3 6^{\circ}) + sin (1 8^{\circ}) = \frac{5}{2}

((cos (3 6^{\circ}) + sin (1 8^{\circ})) + (cos (3 6^{\circ}) - sin (1 8^{\circ}))) = (\frac{5}{2} + \frac{1}{2})

Refine

cos (3 6^{\circ}) = \frac{5 + 1}{4}

Square both sides

(cos (3 6^{\circ}))^{2} = (\frac{5 + 1}{4})^{2}

Use the following identity:

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

sin^{2} (3 6^{\circ}) = 1 - cos^{2} (3 6^{\circ})

Substitute

cos (3 6^{\circ}) = \frac{5 + 1}{4}

sin^{2} (3 6^{\circ}) = 1 - (\frac{5 + 1}{4})^{2}

Refine

sin^{2} (3 6^{\circ}) = \frac{5 - 5}{8}

Take the square root of both sides

sin (3 6^{\circ}) = \pm \frac{5 - 5}{8}

sin (3 6^{\circ})

cannot be negative

sin (3 6^{\circ}) = \frac{5 - 5}{8}

Refine

sin (3 6^{\circ}) = \frac{\frac{5 - 5}{2}}{2}

= \frac{\frac{5 - 5}{2}}{2}

\frac{\frac{5 - 5}{2}}{2} = \frac{2 5 - 5}{4}

\frac{\frac{5 - 5}{2}}{2}

\frac{5 - 5}{2} = \frac{5 - 5}{2}

\frac{5 - 5}{2}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{5 - 5}{2}

= \frac{\frac{5 - 5}{2}}{2}

Apply the fraction rule:

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

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

Rationalize

\frac{5 - 5}{2 2} : \frac{2 5 - 5}{4}

\frac{5 - 5}{2 2}

Multiply by the conjugate

\frac{2}{2}

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

2 \cdot 22 = 4

2 \cdot 22

Apply exponent rule:

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

222 = 2 \cdot 2^{\frac{1}{2}} \cdot 2^{\frac{1}{2}} = 2^{1 + \frac{1}{2} + \frac{1}{2}}

= 2^{1 + \frac{1}{2} + \frac{1}{2}}

Add similar elements:

\frac{1}{2} + \frac{1}{2} = 2 \cdot \frac{1}{2}

= 2^{1 + 2 \cdot \frac{1}{2}}

2 \cdot \frac{1}{2} = 1

2 \cdot \frac{1}{2}

Multiply fractions:

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

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 2^{1 + 1}

Add the numbers:

1 + 1 = 2

= 2^{2}

2^{2} = 4

= 4

= \frac{2 5 - 5}{4}

= \frac{2 5 - 5}{4}

= \frac{2 5 - 5}{4}

= - \frac{2 5 - 5}{4}

= \frac{1 - \frac{2 5 - 5}{4}}{2}

Simplify

\frac{1 - \frac{2 5 - 5}{4}}{2} : \frac{2 4 - 2 5 - 5}{4}

\frac{1 - \frac{2 5 - 5}{4}}{2}

\frac{1 - \frac{2 5 - 5}{4}}{2} = \frac{4 - 2 5 - 5}{8}

\frac{1 - \frac{2 5 - 5}{4}}{2}

Join

1 - \frac{2 5 - 5}{4} : \frac{4 - 2 5 - 5}{4}

1 - \frac{2 5 - 5}{4}

Convert element to fraction:

1 = \frac{1 \cdot 4}{4}

= \frac{1 \cdot 4}{4} - \frac{2 5 - 5}{4}

Since the denominators are equal, combine the fractions:

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

= \frac{1 \cdot 4 - 2 5 - 5}{4}

Multiply the numbers:

1 \cdot 4 = 4

= \frac{4 - 2 5 - 5}{4}

= \frac{\frac{4 - 2 5 - 5}{4}}{2}

Apply the fraction rule:

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

= \frac{4 - 2 5 - 5}{4 \cdot 2}

Multiply the numbers:

4 \cdot 2 = 8

= \frac{4 - 2 5 - 5}{8}

= \frac{4 - 2 5 - 5}{8}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{4 - 2 5 - 5}{8}

8 = 22

8

Prime factorization of

8 : 2^{3}

8

8

divides by

28 = 4 \cdot 2

= 2 \cdot 4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2 \cdot 2

2

is a prime number, therefore no further factorization is possible

= 2 \cdot 2 \cdot 2

= 2^{3}

= 2^{3}

Apply exponent rule:

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

= 2^{2} \cdot 2

Apply radical rule:

= 2 2^{2}

Apply radical rule:

2^{2} = 2

= 22

= \frac{- 2 5 - 5 + 4}{2 2}

Rationalize

\frac{4 - 2 5 - 5}{2 2} : \frac{2 - 2 5 - 5 + 4}{4}

\frac{4 - 2 5 - 5}{2 2}

Multiply by the conjugate

\frac{2}{2}

= \frac{4 - 2 5 - 5 2}{2 2 2}

222 = 4

222

Apply exponent rule:

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

222 = 2 \cdot 2^{\frac{1}{2}} \cdot 2^{\frac{1}{2}} = 2^{1 + \frac{1}{2} + \frac{1}{2}}

= 2^{1 + \frac{1}{2} + \frac{1}{2}}

Add similar elements:

\frac{1}{2} + \frac{1}{2} = 2 \cdot \frac{1}{2}

= 2^{1 + 2 \cdot \frac{1}{2}}

2 \cdot \frac{1}{2} = 1

2 \cdot \frac{1}{2}

Multiply fractions:

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

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 2^{1 + 1}

Add the numbers:

1 + 1 = 2

= 2^{2}

2^{2} = 4

= 4

= \frac{2 4 - 2 5 - 5}{4}

= \frac{2 - 2 5 - 5 + 4}{4}

= \frac{2 4 - 2 5 - 5}{4}

tan (θ) \frac{2 4 + 2 5 - 5}{4} = \frac{2 4 - 2 5 - 5}{4}

Multiply both sides by

4

tan (θ) \frac{2 4 + 2 5 - 5}{4} = \frac{2 4 - 2 5 - 5}{4}

Multiply both sides by

4

4 tan (θ) \frac{2 4 + 2 5 - 5}{4} = \frac{4 2 4 - 2 5 - 5}{4}

Simplify

2 4 + 2 5 - 5 tan (θ) = 2 - 2 5 - 5 + 4

2 4 + 2 5 - 5 tan (θ) = 2 - 2 5 - 5 + 4

Divide both sides by

2 4 + 2 5 - 5

2 4 + 2 5 - 5 tan (θ) = 2 - 2 5 - 5 + 4

Divide both sides by

2 4 + 2 5 - 5

\frac{2 4 + 2 5 - 5 tan ( θ )}{2 4 + 2 5 - 5} = \frac{2 - 2 5 - 5 + 4}{2 4 + 2 5 - 5}

Simplify

\frac{2 4 + 2 5 - 5 tan ( θ )}{2 4 + 2 5 - 5} = \frac{2 - 2 5 - 5 + 4}{2 4 + 2 5 - 5}

Simplify

\frac{2 4 + 2 5 - 5 tan ( θ )}{2 4 + 2 5 - 5} : tan (θ)

\frac{2 4 + 2 5 - 5 tan ( θ )}{2 4 + 2 5 - 5}

Cancel the common factor:

2

= \frac{2 5 - 5 + 4 tan ( θ )}{4 + 2 5 - 5}

Cancel the common factor:

4 + 2 5 - 5

= tan (θ)

Simplify

\frac{2 - 2 5 - 5 + 4}{2 4 + 2 5 - 5} : 10 5 - 5 - 32 5 - 5 + 11 - 45

\frac{2 - 2 5 - 5 + 4}{2 4 + 2 5 - 5}

Cancel the common factor:

2

= \frac{- 2 5 - 5 + 4}{4 + 2 5 - 5}

Combine same powers :

\frac{x}{y} = \frac{x}{y}

= \frac{- 2 5 - 5 + 4}{4 + 2 5 - 5}

\frac{- 2 5 - 5 + 4}{4 + 2 5 - 5} = 10 5 - 5 - 32 5 - 5 + 11 - 45

\frac{- 2 5 - 5 + 4}{4 + 2 5 - 5}

Multiply by the conjugate

\frac{4 - 2 5 - 5}{4 - 2 5 - 5}

= \frac{( - 2 5 - 5 + 4 ) ( 4 - 2 5 - 5 )}{( 4 + 2 5 - 5 ) ( 4 - 2 5 - 5 )}

(- 2 5 - 5 + 4) (4 - 2 5 - 5) = - 82 5 - 5 + 26 - 25

(- 2 5 - 5 + 4) (4 - 2 5 - 5)

Apply exponent rule:

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

(- 2 5 - 5 + 4) (4 - 2 5 - 5) = (- 2 5 - 5 + 4)^{1 + 1}

= (- 2 5 - 5 + 4)^{1 + 1}

Add the numbers:

1 + 1 = 2

= (- 2 5 - 5 + 4)^{2}

Apply Perfect Square Formula:

(a + b)^{2} = a^{2} + 2 ab + b^{2}

a = - 2 5 - 5, b = 4

= (- 2 5 - 5)^{2} + 2 (- 2 5 - 5) \cdot 4 + 4^{2}

Simplify

(- 2 5 - 5)^{2} + 2 (- 2 5 - 5) \cdot 4 + 4^{2} : - 82 5 - 5 + 26 - 25

(- 2 5 - 5)^{2} + 2 (- 2 5 - 5) \cdot 4 + 4^{2}

Remove parentheses:

(- a) = - a

= (- 2 5 - 5)^{2} - 22 5 - 5 \cdot 4 + 4^{2}

(- 2 5 - 5)^{2} = 2 (5 - 5)

(- 2 5 - 5)^{2}

Apply exponent rule:

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

n

is even

(- 2 5 - 5)^{2} = (2 5 - 5)^{2}

= (2 5 - 5)^{2}

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

= (2)^{2} (5 - 5)^{2}

(2)^{2} : 2

Apply radical rule:

a = a^{\frac{1}{2}}

= (2^{\frac{1}{2}})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= 2^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiply fractions:

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

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 2

= 2 (5 - 5)^{2}

(5 - 5)^{2} : 5 - 5

Apply radical rule:

a = a^{\frac{1}{2}}

= ((5 - 5)^{\frac{1}{2}})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= (5 - 5)^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiply fractions:

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

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 5 - 5

= 2 (5 - 5)

22 5 - 5 \cdot 4 = 82 5 - 5

22 5 - 5 \cdot 4

Multiply the numbers:

2 \cdot 4 = 8

= 82 5 - 5

4^{2} = 16

4^{2}

4^{2} = 16

= 16

= 2 (5 - 5) - 82 5 - 5 + 16

Expand

2 (5 - 5) : 10 - 25

2 (5 - 5)

Apply the distributive law:

a (b - c) = ab - a c

a = 2, b = 5, c = 5

= 2 \cdot 5 - 25

Multiply the numbers:

2 \cdot 5 = 10

= 10 - 25

= 10 - 25 - 82 5 - 5 + 16

Add the numbers:

10 + 16 = 26

= - 82 5 - 5 + 26 - 25

= - 82 5 - 5 + 26 - 25

(4 + 2 5 - 5) (4 - 2 5 - 5) = 6 + 25

(4 + 2 5 - 5) (4 - 2 5 - 5)

2 5 - 5 = 10 - 25

2 5 - 5

Apply radical rule:

a b = a \cdot b

2 5 - 5 = 2 (5 - 5)

= 2 (5 - 5)

Expand

2 (5 - 5) : 10 - 25

2 (5 - 5)

Apply the distributive law:

a (b - c) = ab - a c

a = 2, b = 5, c = 5

= 2 \cdot 5 - 25

Multiply the numbers:

2 \cdot 5 = 10

= 10 - 25

= 10 - 25

= (10 - 25 + 4) (- 2 5 - 5 + 4)

2 5 - 5 = 10 - 25

2 5 - 5

Apply radical rule:

a b = a \cdot b

2 5 - 5 = 2 (5 - 5)

= 2 (5 - 5)

Expand

2 (5 - 5) : 10 - 25

2 (5 - 5)

Apply the distributive law:

a (b - c) = ab - a c

a = 2, b = 5, c = 5

= 2 \cdot 5 - 25

Multiply the numbers:

2 \cdot 5 = 10

= 10 - 25

= 10 - 25

= (10 - 25 + 4) (- 10 - 25 + 4)

Apply Difference of Two Squares Formula:

(a + b) (a - b) = a^{2} - b^{2}

a = 4, b = 10 - 25

= 4^{2} - (10 - 25)^{2}

Simplify

4^{2} - (10 - 25)^{2} : 6 + 25

4^{2} - (10 - 25)^{2}

4^{2} = 16

4^{2}

4^{2} = 16

= 16

(10 - 25)^{2} = 10 - 25

(10 - 25)^{2}

Apply radical rule:

a = a^{\frac{1}{2}}

= ((10 - 25)^{\frac{1}{2}})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= (10 - 25)^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiply fractions:

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

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 10 - 25

= 16 - (10 - 25)

- (10 - 25) : - 10 + 25

- (10 - 25)

Distribute parentheses

= - (10) - (- 25)

Apply minus-plus rules

- (- a) = a, - (a) = - a

= - 10 + 25

= 16 - 10 + 25

Subtract the numbers:

16 - 10 = 6

= 6 + 25

= 6 + 25

= \frac{- 8 2 5 - 5 + 26 - 2 5}{6 + 2 5}

Factor

- 82 5 - 5 + 26 - 25 : 2 (- 42 - 5 + 5 + 13 - 5)

- 82 5 - 5 + 26 - 25

Rewrite as

= - 2 \cdot 42 5 - 5 + 2 \cdot 13 - 25

Factor out common term

2

= 2 (- 42 5 - 5 + 13 - 5)

Expand

- 42 5 - 5 + 13 - 5 : - 42 - 5 + 5 + 13 - 5

- 42 5 - 5 + 13 - 5

42 5 - 5 = 42 - 5 + 5

42 5 - 5

Apply radical rule:

a b = a \cdot b

2 5 - 5 = 2 (5 - 5)

= 4 2 (5 - 5)

Factor

5 - 5 : - (5 - 5)

5 - 5

Factor out common term

- 1

= - (5 - 5)

= 4 - 2 (5 - 5)

Apply radical rule:

assuming

a \geq 0, b \geq 0

- 2 (5 - 5) = 2 - (5 - 5)

= 42 - (5 - 5)

Expand

- (5 - 5) : - 5 + 5

- (5 - 5)

Distribute parentheses

= - (5) - (- 5)

Apply minus-plus rules

- (- a) = a, - (a) = - a

= - 5 + 5

= 42 5 - 5

= - 42 5 - 5 + 13 - 5

= 2 (- 42 5 - 5 + 13 - 5)

= \frac{2 ( - 4 2 - 5 + 5 + 13 - 5 )}{6 + 2 5}

Factor

6 + 25 : 2 (3 + 5)

6 + 25

Rewrite as

= 2 \cdot 3 + 25

Factor out common term

2

= 2 (3 + 5)

= \frac{2 ( - 4 2 - 5 + 5 + 13 - 5 )}{2 ( 3 + 5 )}

Divide the numbers:

\frac{2}{2} = 1

= \frac{- 4 2 5 - 5 + 13 - 5}{( 3 + 5 )}

Remove parentheses:

(a) = a

= \frac{- 4 2 5 - 5 + 13 - 5}{3 + 5}

Multiply by the conjugate

\frac{3 - 5}{3 - 5}

= \frac{( - 4 2 5 - 5 + 13 - 5 ) ( 3 - 5 )}{( 3 + 5 ) ( 3 - 5 )}

(- 42 5 - 5 + 13 - 5) (3 - 5) = 410 5 - 5 - 122 5 - 5 + 44 - 165

(- 42 5 - 5 + 13 - 5) (3 - 5)

Distribute parentheses

= (- 42 5 - 5) \cdot 3 + (- 42 5 - 5) (- 5) + 13 \cdot 3 + 13 (- 5) + (- 5) \cdot 3 + (- 5) (- 5)

Apply minus-plus rules

+ (- a) = - a, (- a) (- b) = ab

= - 4 \cdot 32 5 - 5 + 425 5 - 5 + 13 \cdot 3 - 135 - 35 + 55

Simplify

- 4 \cdot 32 5 - 5 + 425 5 - 5 + 13 \cdot 3 - 135 - 35 + 55 : 410 5 - 5 - 122 5 - 5 + 44 - 165

- 4 \cdot 32 5 - 5 + 425 5 - 5 + 13 \cdot 3 - 135 - 35 + 55

Add similar elements:

- 135 - 35 = - 165

= - 4 \cdot 32 5 - 5 + 425 5 - 5 + 13 \cdot 3 - 165 + 55

4 \cdot 32 5 - 5 = 122 5 - 5

4 \cdot 32 5 - 5

Multiply the numbers:

4 \cdot 3 = 12

= 122 5 - 5

425 5 - 5 = 410 5 - 5

425 5 - 5

Apply radical rule:

a b = a \cdot b

25 5 - 5 = 2 \cdot 5 (5 - 5)

= 4 2 \cdot 5 (5 - 5)

Multiply the numbers:

2 \cdot 5 = 10

= 4 10 (5 - 5)

Apply radical rule:

assuming

a \geq 0, b \geq 0

10 (5 - 5) = 10 5 - 5

= 410 5 - 5

13 \cdot 3 = 39

13 \cdot 3

Multiply the numbers:

13 \cdot 3 = 39

= 39

55 = 5

55

Apply radical rule:

a a = a

55 = 5

= 5

= - 122 5 - 5 + 410 5 - 5 + 39 - 165 + 5

Add the numbers:

39 + 5 = 44

= 410 5 - 5 - 122 5 - 5 + 44 - 165

= 410 5 - 5 - 122 5 - 5 + 44 - 165

(3 + 5) (3 - 5) = 4

(3 + 5) (3 - 5)

Apply Difference of Two Squares Formula:

(a + b) (a - b) = a^{2} - b^{2}

a = 3, b = 5

= 3^{2} - (5)^{2}

Simplify

3^{2} - (5)^{2} : 4

3^{2} - (5)^{2}

3^{2} = 9

3^{2}

3^{2} = 9

= 9

(5)^{2} = 5

(5)^{2}

Apply radical rule:

a = a^{\frac{1}{2}}

= (5^{\frac{1}{2}})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= 5^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiply fractions:

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

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 5

= 9 - 5

Subtract the numbers:

9 - 5 = 4

= 4

= 4

= \frac{4 10 5 - 5 - 12 2 5 - 5 + 44 - 16 5}{4}

Factor

410 5 - 5 - 122 5 - 5 + 44 - 165 : 4 (10 - 5 + 5 - 32 - 5 + 5 + 11 - 45)

410 5 - 5 - 122 5 - 5 + 44 - 165

Rewrite as

= 410 5 - 5 - 4 \cdot 32 5 - 5 + 4 \cdot 11 - 4 \cdot 45

Factor out common term

4

= 4 (10 5 - 5 - 32 5 - 5 + 11 - 45)

Expand

10 5 - 5 - 32 5 - 5 + 11 - 45 : 10 - 5 + 5 - 32 - 5 + 5 + 11 - 45

10 5 - 5 - 32 5 - 5 + 11 - 45

10 5 - 5 = 10 - 5 + 5

10 5 - 5

Apply radical rule:

a b = a \cdot b

10 5 - 5 = 10 (5 - 5)

= 10 (5 - 5)

Factor

5 - 5 : - (5 - 5)

5 - 5

Factor out common term

- 1

= - (5 - 5)

= - 10 (5 - 5)

Apply radical rule:

assuming

a \geq 0, b \geq 0

= 10 - (5 - 5)

Expand

- (5 - 5) : - 5 + 5

- (5 - 5)

Distribute parentheses

= - (5) - (- 5)

Apply minus-plus rules

- (- a) = a, - (a) = - a

= - 5 + 5

= 10 5 - 5

32 5 - 5 = 32 - 5 + 5

32 5 - 5

Apply radical rule:

a b = a \cdot b

2 5 - 5 = 2 (5 - 5)

= 3 2 (5 - 5)

Factor

5 - 5 : - (5 - 5)

5 - 5

Factor out common term

- 1

= - (5 - 5)

= 3 - 2 (5 - 5)

Apply radical rule:

assuming

a \geq 0, b \geq 0

- 2 (5 - 5) = 2 - (5 - 5)

= 32 - (5 - 5)

Expand

- (5 - 5) : - 5 + 5

- (5 - 5)

Distribute parentheses

= - (5) - (- 5)

Apply minus-plus rules

- (- a) = a, - (a) = - a

= - 5 + 5

= 32 5 - 5

= 10 5 - 5 - 32 5 - 5 + 11 - 45

= 4 (10 5 - 5 - 32 5 - 5 + 11 - 45)

= \frac{4 ( 10 - 5 + 5 - 3 2 - 5 + 5 + 11 - 4 5 )}{4}

Divide the numbers:

\frac{4}{4} = 1

= 10 5 - 5 - 32 5 - 5 + 11 - 45

= 10 5 - 5 - 32 5 - 5 + 11 - 45

tan (θ) = 10 5 - 5 - 32 5 - 5 + 11 - 45

tan (θ) = 10 5 - 5 - 32 5 - 5 + 11 - 45

tan (θ) = 10 5 - 5 - 32 5 - 5 + 11 - 45

Apply trig inverse properties

tan (θ) = 10 5 - 5 - 32 5 - 5 + 11 - 45

General solutions for

tan (θ) = 10 5 - 5 - 32 5 - 5 + 11 - 45

tan (x) = a \Rightarrow x = arctan (a) + 18 0^{\circ} n

θ = arctan (10 5 - 5 - 32 5 - 5 + 11 - 45) + 18 0^{\circ} n

θ = arctan (10 5 - 5 - 32 5 - 5 + 11 - 45) + 18 0^{\circ} n

Show solutions in decimal form

θ = 0.47123 \dots + 18 0^{\circ} n

Popular Examples

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3 - sin (x) = cos (2 x)

cot^2(x)-cot(x)-2=0

cot^{2} (x) - cot (x) - 2 = 0

cos^2(θ)=0.1831

cos^{2} (θ) = 0.1831

tan(a+5)=sqrt(2sin(30)+sec(245))

tan (a + 5^{\circ}) = 2 sin (3 0^{\circ}) + sec (24 5^{\circ})

cot(x)=tan(25)

cot (x) = tan (2 5^{\circ})

Frequently Asked Questions (FAQ)

What is the general solution for tan(θ)cos(27)=cos(63) ?
The general solution for tan(θ)cos(27)=cos(63) is θ=0.47123…+180n