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Popular Trigonometry >

108cos(0.5a)=sin(90+a)

Solution

108 cos (0.5 a) = sin (9 0^{\circ} + a)

Solution

a = \frac{1.58005 \dots + 36 0 ^{\circ} n}{0.5}, a = \frac{- 1.58005 \dots + 36 0 ^{\circ} n}{0.5}, a = \frac{\emptyset 10}{5}

Radians

a = 0 + \frac{1.58005 \dots + 2 π}{0.5} n, a = 0 + \frac{- 1.58005 \dots + 2 π}{0.5} n, a = \frac{\emptyset 10}{5}

Solution steps

108 cos (0.5 a) = sin (9 0^{\circ} + a)

Rewrite using trig identities

108 cos (0.5 a) = sin (9 0^{\circ} + a)

Rewrite using trig identities

sin (9 0^{\circ} + a)

Use the Angle Sum identity:

sin (s + t) = sin (s) cos (t) + cos (s) sin (t)

= sin (9 0^{\circ}) cos (a) + cos (9 0^{\circ}) sin (a)

Simplify

sin (9 0^{\circ}) cos (a) + cos (9 0^{\circ}) sin (a) : cos (a)

sin (9 0^{\circ}) cos (a) + cos (9 0^{\circ}) sin (a)

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

sin (9 0^{\circ}) cos (a)

Simplify

sin (9 0^{\circ}) : 1

sin (9 0^{\circ})

Use the following trivial identity:

sin (9 0^{\circ}) = 1

sin (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= 1

= 1 \cdot cos (a)

Multiply:

1 \cdot cos (a) = cos (a)

= cos (a)

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

cos (9 0^{\circ}) sin (a)

Simplify

cos (9 0^{\circ}) : 0

cos (9 0^{\circ})

Use the following trivial identity:

cos (9 0^{\circ}) = 0

cos (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= 0

= 0 \cdot sin (a)

Apply rule

0 \cdot a = 0

= 0

= cos (a) + 0

cos (a) + 0 = cos (a)

= cos (a)

= cos (a)

108 cos (0.5 a) = cos (a)

108 cos (0.5 a) = cos (a)

Subtract

cos (a)

from both sides

108 cos (0.5 a) - cos (a) = 0

Let:

u = 0.5 a

108 cos (u) - cos (2 u) = 0

Rewrite using trig identities

- cos (2 u) + 108 cos (u)

Use the Double Angle identity:

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

= - (2 cos^{2} (u) - 1) + 108 cos (u)

- (2 cos^{2} (u) - 1) : - 2 cos^{2} (u) + 1

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

Distribute parentheses

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

Apply minus-plus rules

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

= - 2 cos^{2} (u) + 1

= - 2 cos^{2} (u) + 1 + 108 cos (u)

1 + 108 cos (u) - 2 cos^{2} (u) = 0

Solve by substitution

1 + 108 cos (u) - 2 cos^{2} (u) = 0

Let:

cos (u) = u

1 + 108 u - 2 u^{2} = 0

1 + 108 u - 2 u^{2} = 0 : u = - \frac{- 54 + 2918}{2}, u = \frac{54 + 2918}{2}

1 + 108 u - 2 u^{2} = 0

Write in the standard form

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

- 2 u^{2} + 108 u + 1 = 0

Solve with the quadratic formula

- 2 u^{2} + 108 u + 1 = 0

Quadratic Equation Formula:

For

a = - 2, b = 108, c = 1

u_{1, 2} = \frac{- 108 \pm 10 8 ^{2} - 4 ( - 2 ) \cdot 1}{2 ( - 2 )}

u_{1, 2} = \frac{- 108 \pm 10 8 ^{2} - 4 ( - 2 ) \cdot 1}{2 ( - 2 )}

10 8^{2} - 4 (- 2) \cdot 1 = 22918

10 8^{2} - 4 (- 2) \cdot 1

Apply rule

- (- a) = a

= 10 8^{2} + 4 \cdot 2 \cdot 1

Multiply the numbers:

4 \cdot 2 \cdot 1 = 8

= 10 8^{2} + 8

10 8^{2} = 11664

= 11664 + 8

Add the numbers:

11664 + 8 = 11672

= 11672

Prime factorization of

11672 : 2^{3} \cdot 1459

11672

11672

divides by

211672 = 5836 \cdot 2

= 2 \cdot 5836

5836

divides by

25836 = 2918 \cdot 2

= 2 \cdot 2 \cdot 2918

2918

divides by

22918 = 1459 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 1459

2, 1459

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 2 \cdot 1459

= 2^{3} \cdot 1459

= 2^{3} \cdot 1459

Apply exponent rule:

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

= 2^{2} \cdot 2 \cdot 1459

Apply radical rule:

= 2^{2} 2 \cdot 1459

Apply radical rule:

2^{2} = 2

= 2 2 \cdot 1459

Refine

= 22918

u_{1, 2} = \frac{- 108 \pm 2 2918}{2 ( - 2 )}

Separate the solutions

u_{1} = \frac{- 108 + 2 2918}{2 ( - 2 )}, u_{2} = \frac{- 108 - 2 2918}{2 ( - 2 )}

u = \frac{- 108 + 2 2918}{2 ( - 2 )} : - \frac{- 54 + 2918}{2}

\frac{- 108 + 2 2918}{2 ( - 2 )}

Remove parentheses:

(- a) = - a

= \frac{- 108 + 2 2918}{- 2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{- 108 + 2 2918}{- 4}

Apply the fraction rule:

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

= - \frac{- 108 + 2 2918}{4}

Cancel

\frac{- 108 + 2 2918}{4} : \frac{2918 - 54}{2}

\frac{- 108 + 2 2918}{4}

Factor

- 108 + 22918 : 2 (- 54 + 2918)

- 108 + 22918

Rewrite as

= - 2 \cdot 54 + 22918

Factor out common term

2

= 2 (- 54 + 2918)

= \frac{2 ( - 54 + 2918 )}{4}

Cancel the common factor:

2

= \frac{- 54 + 2918}{2}

= - \frac{2918 - 54}{2}

= - \frac{- 54 + 2918}{2}

u = \frac{- 108 - 2 2918}{2 ( - 2 )} : \frac{54 + 2918}{2}

\frac{- 108 - 2 2918}{2 ( - 2 )}

Remove parentheses:

(- a) = - a

= \frac{- 108 - 2 2918}{- 2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{- 108 - 2 2918}{- 4}

Apply the fraction rule:

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

- 108 - 22918 = - (108 + 22918)

= \frac{108 + 2 2918}{4}

Factor

108 + 22918 : 2 (54 + 2918)

108 + 22918

Rewrite as

= 2 \cdot 54 + 22918

Factor out common term

2

= 2 (54 + 2918)

= \frac{2 ( 54 + 2918 )}{4}

Cancel the common factor:

2

= \frac{54 + 2918}{2}

The solutions to the quadratic equation are:

u = - \frac{- 54 + 2918}{2}, u = \frac{54 + 2918}{2}

Substitute back

u = cos (u)

cos (u) = - \frac{- 54 + 2918}{2}, cos (u) = \frac{54 + 2918}{2}

cos (u) = - \frac{- 54 + 2918}{2}, cos (u) = \frac{54 + 2918}{2}

cos (u) = - \frac{- 54 + 2918}{2} : u = arccos (- \frac{- 54 + 2918}{2}) + 36 0^{\circ} n, u = - arccos (- \frac{- 54 + 2918}{2}) + 36 0^{\circ} n

cos (u) = - \frac{- 54 + 2918}{2}

Apply trig inverse properties

cos (u) = - \frac{- 54 + 2918}{2}

General solutions for

cos (u) = - \frac{- 54 + 2918}{2}

cos (x) = - a \Rightarrow x = arccos (- a) + 36 0^{\circ} n, x = - arccos (- a) + 36 0^{\circ} n

u = arccos (- \frac{- 54 + 2918}{2}) + 36 0^{\circ} n, u = - arccos (- \frac{- 54 + 2918}{2}) + 36 0^{\circ} n

u = arccos (- \frac{- 54 + 2918}{2}) + 36 0^{\circ} n, u = - arccos (- \frac{- 54 + 2918}{2}) + 36 0^{\circ} n

cos (u) = \frac{54 + 2918}{2} :

No Solution

cos (u) = \frac{54 + 2918}{2}

- 1 \leq cos (x) \leq 1

No Solution

Combine all the solutions

u = arccos (- \frac{- 54 + 2918}{2}) + 36 0^{\circ} n, u = - arccos (- \frac{- 54 + 2918}{2}) + 36 0^{\circ} n

Substitute back

u = 0.5 a

0.5 a = arccos (- \frac{- 54 + 2918}{2}) + 36 0^{\circ} n : a = \frac{arccos ( - \frac{- 54 + 2918}{2} ) + 36 0 ^{\circ} n}{0.5}

0.5 a = arccos (- \frac{- 54 + 2918}{2}) + 36 0^{\circ} n

Divide both sides by

0.5

0.5 a = arccos (- \frac{- 54 + 2918}{2}) + 36 0^{\circ} n

Divide both sides by

0.5

\frac{0.5 a}{0.5} = \frac{arccos ( - \frac{- 54 + 2918}{2} )}{0.5} + \frac{36 0 ^{\circ} n}{0.5}

Simplify

\frac{0.5 a}{0.5} = \frac{arccos ( - \frac{- 54 + 2918}{2} )}{0.5} + \frac{36 0 ^{\circ} n}{0.5}

Simplify

\frac{0.5 a}{0.5} : a

\frac{0.5 a}{0.5}

Cancel the common factor:

0.5

= a

Simplify

\frac{arccos ( - \frac{- 54 + 2918}{2} )}{0.5} + \frac{36 0 ^{\circ} n}{0.5} : \frac{arccos ( - \frac{- 54 + 2918}{2} ) + 36 0 ^{\circ} n}{0.5}

\frac{arccos ( - \frac{- 54 + 2918}{2} )}{0.5} + \frac{36 0 ^{\circ} n}{0.5}

Apply rule

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

= \frac{arccos ( - \frac{2918 - 54}{2} ) + 36 0 ^{\circ} n}{0.5}

a = \frac{arccos ( - \frac{- 54 + 2918}{2} ) + 36 0 ^{\circ} n}{0.5}

a = \frac{arccos ( - \frac{- 54 + 2918}{2} ) + 36 0 ^{\circ} n}{0.5}

a = \frac{arccos ( - \frac{- 54 + 2918}{2} ) + 36 0 ^{\circ} n}{0.5}

0.5 a = - arccos (- \frac{- 54 + 2918}{2}) + 36 0^{\circ} n : a = \frac{- arccos ( - \frac{- 54 + 2918}{2} ) + 36 0 ^{\circ} n}{0.5}

0.5 a = - arccos (- \frac{- 54 + 2918}{2}) + 36 0^{\circ} n

Divide both sides by

0.5

0.5 a = - arccos (- \frac{- 54 + 2918}{2}) + 36 0^{\circ} n

Divide both sides by

0.5

\frac{0.5 a}{0.5} = - \frac{arccos ( - \frac{- 54 + 2918}{2} )}{0.5} + \frac{36 0 ^{\circ} n}{0.5}

Simplify

\frac{0.5 a}{0.5} = - \frac{arccos ( - \frac{- 54 + 2918}{2} )}{0.5} + \frac{36 0 ^{\circ} n}{0.5}

Simplify

\frac{0.5 a}{0.5} : a

\frac{0.5 a}{0.5}

Cancel the common factor:

0.5

= a

Simplify

- \frac{arccos ( - \frac{- 54 + 2918}{2} )}{0.5} + \frac{36 0 ^{\circ} n}{0.5} : \frac{- arccos ( - \frac{- 54 + 2918}{2} ) + 36 0 ^{\circ} n}{0.5}

- \frac{arccos ( - \frac{- 54 + 2918}{2} )}{0.5} + \frac{36 0 ^{\circ} n}{0.5}

Apply rule

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

= \frac{- arccos ( - \frac{2918 - 54}{2} ) + 36 0 ^{\circ} n}{0.5}

a = \frac{- arccos ( - \frac{- 54 + 2918}{2} ) + 36 0 ^{\circ} n}{0.5}

a = \frac{- arccos ( - \frac{- 54 + 2918}{2} ) + 36 0 ^{\circ} n}{0.5}

a = \frac{- arccos ( - \frac{- 54 + 2918}{2} ) + 36 0 ^{\circ} n}{0.5}

0.5 a = \emptyset : a = \frac{\emptyset 10}{5}

0.5 a = \emptyset

Multiply both sides by

10

0.5 a = \emptyset

To eliminate decimal points, multiply by 10 for every digit after the decimal pointThere is one digit to the right of the decimal point, therefore multiply by

10

0.5 a \cdot 10 = \emptyset 10

Refine

5 a = \emptyset 10

5 a = \emptyset 10

Divide both sides by

5

5 a = \emptyset 10

Divide both sides by

5

\frac{5 a}{5} = \frac{\emptyset 10}{5}

Simplify

a = \frac{\emptyset 10}{5}

a = \frac{\emptyset 10}{5}

a = \frac{arccos ( - \frac{- 54 + 2918}{2} ) + 36 0 ^{\circ} n}{0.5}, a = \frac{- arccos ( - \frac{- 54 + 2918}{2} ) + 36 0 ^{\circ} n}{0.5}, a = \frac{\emptyset 10}{5}

Since the equation is undefined for:

\frac{\emptyset 10}{5}

a = \frac{arccos ( - \frac{- 54 + 2918}{2} ) + 36 0 ^{\circ} n}{0.5}, a = \frac{- arccos ( - \frac{- 54 + 2918}{2} ) + 36 0 ^{\circ} n}{0.5}, a = \frac{\emptyset 10}{5}

Show solutions in decimal form

a = \frac{1.58005 \dots + 36 0 ^{\circ} n}{0.5}, a = \frac{- 1.58005 \dots + 36 0 ^{\circ} n}{0.5}, a = \frac{\emptyset 10}{5}

Graph

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