What is the general solution for (sin^2(x))/(cos(x))=16.33 ?

The general solution for (sin^2(x))/(cos(x))=16.33 is x=1.50974…+2pin,x=2pi-1.50974…+2pin

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(sin^2(x))/(cos(x))=16.33

Solution

\frac{sin ^{2} ( x )}{cos ( x )} = 16.33

Solution

x = 1.50974 \dots + 2 πn, x = 2 π - 1.50974 \dots + 2 πn

Degrees

x = 86.50226 \dots^{\circ} + 36 0^{\circ} n, x = 273.49773 \dots^{\circ} + 36 0^{\circ} n

Solution steps

\frac{sin ^{2} ( x )}{cos ( x )} = 16.33

Subtract

16.33

from both sides

\frac{sin ^{2} ( x )}{cos ( x )} - 16.33 = 0

Rewrite using trig identities

- 16.33 + \frac{sin ^{2} ( x )}{cos ( x )}

Use the Pythagorean identity:

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

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

= - 16.33 + \frac{1 - cos ^{2} ( x )}{cos ( x )}

- 16.33 + \frac{1 - cos ^{2} ( x )}{cos ( x )} = 0

Solve by substitution

- 16.33 + \frac{1 - cos ^{2} ( x )}{cos ( x )} = 0

Let:

cos (x) = u

- 16.33 + \frac{1 - u ^{2}}{u} = 0

- 16.33 + \frac{1 - u ^{2}}{u} = 0 : u = - \frac{1633 + 2706689}{200}, u = \frac{2706689 - 1633}{200}

- 16.33 + \frac{1 - u ^{2}}{u} = 0

Multiply both sides by

u

- 16.33 + \frac{1 - u ^{2}}{u} = 0

Multiply both sides by

u

- 16.33 u + \frac{1 - u ^{2}}{u} u = 0 \cdot u

Simplify

- 16.33 u + \frac{1 - u ^{2}}{u} u = 0 \cdot u

Simplify

\frac{1 - u ^{2}}{u} u : 1 - u^{2}

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

Multiply fractions:

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

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

Cancel the common factor:

u

= 1 - u^{2}

Simplify

0 \cdot u : 0

0 \cdot u

Apply rule

0 \cdot a = 0

= 0

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

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

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

Solve

- 16.33 u + 1 - u^{2} = 0 : u = - \frac{1633 + 2706689}{200}, u = \frac{2706689 - 1633}{200}

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

Multiply both sides by

100

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

To eliminate decimal points, multiply by 10 for every digit after the decimal pointThere are

2

digits to the right of the decimal point, therefore multiply by

100

- 16.33 u \cdot 100 + 1 \cdot 100 - u^{2} \cdot 100 = 0 \cdot 100

Refine

- 1633 u + 100 - 100 u^{2} = 0

- 1633 u + 100 - 100 u^{2} = 0

Write in the standard form

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

- 100 u^{2} - 1633 u + 100 = 0

Solve with the quadratic formula

- 100 u^{2} - 1633 u + 100 = 0

Quadratic Equation Formula:

For

a = - 100, b = - 1633, c = 100

u_{1, 2} = \frac{- ( - 1633 ) \pm ( - 1633 ) ^{2} - 4 ( - 100 ) \cdot 100}{2 ( - 100 )}

u_{1, 2} = \frac{- ( - 1633 ) \pm ( - 1633 ) ^{2} - 4 ( - 100 ) \cdot 100}{2 ( - 100 )}

(- 1633)^{2} - 4 (- 100) \cdot 100 = 2706689

(- 1633)^{2} - 4 (- 100) \cdot 100

Apply rule

- (- a) = a

= (- 1633)^{2} + 4 \cdot 100 \cdot 100

Apply exponent rule:

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

n

is even

(- 1633)^{2} = 163 3^{2}

= 163 3^{2} + 4 \cdot 100 \cdot 100

Multiply the numbers:

4 \cdot 100 \cdot 100 = 40000

= 163 3^{2} + 40000

163 3^{2} = 2666689

= 2666689 + 40000

Add the numbers:

2666689 + 40000 = 2706689

= 2706689

u_{1, 2} = \frac{- ( - 1633 ) \pm 2706689}{2 ( - 100 )}

Separate the solutions

u_{1} = \frac{- ( - 1633 ) + 2706689}{2 ( - 100 )}, u_{2} = \frac{- ( - 1633 ) - 2706689}{2 ( - 100 )}

u = \frac{- ( - 1633 ) + 2706689}{2 ( - 100 )} : - \frac{1633 + 2706689}{200}

\frac{- ( - 1633 ) + 2706689}{2 ( - 100 )}

Remove parentheses:

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

= \frac{1633 + 2706689}{- 2 \cdot 100}

Multiply the numbers:

2 \cdot 100 = 200

= \frac{1633 + 2706689}{- 200}

Apply the fraction rule:

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

= - \frac{1633 + 2706689}{200}

u = \frac{- ( - 1633 ) - 2706689}{2 ( - 100 )} : \frac{2706689 - 1633}{200}

\frac{- ( - 1633 ) - 2706689}{2 ( - 100 )}

Remove parentheses:

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

= \frac{1633 - 2706689}{- 2 \cdot 100}

Multiply the numbers:

2 \cdot 100 = 200

= \frac{1633 - 2706689}{- 200}

Apply the fraction rule:

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

1633 - 2706689 = - (2706689 - 1633)

= \frac{2706689 - 1633}{200}

The solutions to the quadratic equation are:

u = - \frac{1633 + 2706689}{200}, u = \frac{2706689 - 1633}{200}

u = - \frac{1633 + 2706689}{200}, u = \frac{2706689 - 1633}{200}

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

- 16.33 + \frac{1 - u ^{2}}{u}

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u = - \frac{1633 + 2706689}{200}, u = \frac{2706689 - 1633}{200}

Substitute back

u = cos (x)

cos (x) = - \frac{1633 + 2706689}{200}, cos (x) = \frac{2706689 - 1633}{200}

cos (x) = - \frac{1633 + 2706689}{200}, cos (x) = \frac{2706689 - 1633}{200}

cos (x) = - \frac{1633 + 2706689}{200} :

No Solution

cos (x) = - \frac{1633 + 2706689}{200}

- 1 \leq cos (x) \leq 1

No Solution

cos (x) = \frac{2706689 - 1633}{200} : x = arccos (\frac{2706689 - 1633}{200}) + 2 πn, x = 2 π - arccos (\frac{2706689 - 1633}{200}) + 2 πn

cos (x) = \frac{2706689 - 1633}{200}

Apply trig inverse properties

cos (x) = \frac{2706689 - 1633}{200}

General solutions for

cos (x) = \frac{2706689 - 1633}{200}

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

x = arccos (\frac{2706689 - 1633}{200}) + 2 πn, x = 2 π - arccos (\frac{2706689 - 1633}{200}) + 2 πn

x = arccos (\frac{2706689 - 1633}{200}) + 2 πn, x = 2 π - arccos (\frac{2706689 - 1633}{200}) + 2 πn

Combine all the solutions

x = arccos (\frac{2706689 - 1633}{200}) + 2 πn, x = 2 π - arccos (\frac{2706689 - 1633}{200}) + 2 πn

Show solutions in decimal form

x = 1.50974 \dots + 2 πn, x = 2 π - 1.50974 \dots + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for (sin^2(x))/(cos(x))=16.33 ?
The general solution for (sin^2(x))/(cos(x))=16.33 is x=1.50974…+2pin,x=2pi-1.50974…+2pin