What is the general solution for cos^2(90-θ)=1872 ?

The general solution for cos^2(90-θ)=1872 is No Solution for θ\in\mathbb{R}

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Popular >

cos^2(90-θ)=1872

Solution

cos^{2} (9 0^{\circ} - θ) = 1872

Solution

No Solution for θ \in R

Solution steps

cos^{2} (9 0^{\circ} - θ) = 1872

Solve by substitution

cos^{2} (9 0^{\circ} - θ) = 1872

Let:

cos (9 0^{\circ} - θ) = u

u^{2} = 1872

u^{2} = 1872 : u = 1213, u = - 1213

u^{2} = 1872

For

x^{2} = f (a)

the solutions are

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

u = 1872, u = - 1872

1872 = 1213

1872

Prime factorization of

1872 : 2^{4} \cdot 3^{2} \cdot 13

1872

1872

divides by

21872 = 936 \cdot 2

= 2 \cdot 936

936

divides by

2936 = 468 \cdot 2

= 2 \cdot 2 \cdot 468

468

divides by

2468 = 234 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 234

234

divides by

2234 = 117 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 117

117

divides by

3117 = 39 \cdot 3

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 39

39

divides by

339 = 13 \cdot 3

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 \cdot 13

2, 3, 13

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 \cdot 13

= 2^{4} \cdot 3^{2} \cdot 13

= 2^{4} \cdot 3^{2} \cdot 13

Apply radical rule:

= 13 2^{4} 3^{2}

Apply radical rule:

2^{4} = 2^{\frac{4}{2}} = 2^{2}

= 2^{2} 13 3^{2}

Apply radical rule:

3^{2} = 3

= 2^{2} \cdot 313

Refine

= 1213

- 1872 = - 1213

- 1872

1872 = 1213

1872

Prime factorization of

1872 : 2^{4} \cdot 3^{2} \cdot 13

1872

1872

divides by

21872 = 936 \cdot 2

= 2 \cdot 936

936

divides by

2936 = 468 \cdot 2

= 2 \cdot 2 \cdot 468

468

divides by

2468 = 234 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 234

234

divides by

2234 = 117 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 117

117

divides by

3117 = 39 \cdot 3

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 39

39

divides by

339 = 13 \cdot 3

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 \cdot 13

2, 3, 13

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 \cdot 13

= 2^{4} \cdot 3^{2} \cdot 13

= 2^{4} \cdot 3^{2} \cdot 13

Apply radical rule:

= 13 2^{4} 3^{2}

Apply radical rule:

2^{4} = 2^{\frac{4}{2}} = 2^{2}

= 2^{2} 13 3^{2}

Apply radical rule:

3^{2} = 3

= 2^{2} \cdot 313

Refine

= 1213

= - 1213

u = 1213, u = - 1213

Substitute back

u = cos (9 0^{\circ} - θ)

cos (9 0^{\circ} - θ) = 1213, cos (9 0^{\circ} - θ) = - 1213

cos (9 0^{\circ} - θ) = 1213, cos (9 0^{\circ} - θ) = - 1213

cos (9 0^{\circ} - θ) = 1213 :

No Solution

cos (9 0^{\circ} - θ) = 1213

- 1 \leq cos (x) \leq 1

No Solution

cos (9 0^{\circ} - θ) = - 1213 :

No Solution

cos (9 0^{\circ} - θ) = - 1213

- 1 \leq cos (x) \leq 1

No Solution

Combine all the solutions

No Solution for θ \in R

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Frequently Asked Questions (FAQ)

What is the general solution for cos^2(90-θ)=1872 ?
The general solution for cos^2(90-θ)=1872 is No Solution for θ\in\mathbb{R}