What is the general solution for (0.9)/(cos(x))+(0.7*sin(x))=1.1 ?

The general solution for (0.9)/(cos(x))+(0.7*sin(x))=1.1 is x=0.24774…+2pin,x=-1.01026…+2pin

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(0.9)/(cos(x))+(0.7*sin(x))=1.1

Solution

\frac{0.9}{cos ( x )} + (0.7 \cdot sin (x)) = 1.1

Solution

x = 0.24774 \dots + 2 πn, x = - 1.01026 \dots + 2 πn

Degrees

x = 14.19491 \dots^{\circ} + 36 0^{\circ} n, x = - 57.88377 \dots^{\circ} + 36 0^{\circ} n

Solution steps

\frac{0.9}{cos ( x )} + (0.7 sin (x)) = 1.1

Subtract

0.7 sin (x)

from both sides

\frac{0.9}{cos ( x )} = 1.1 - 0.7 sin (x)

Square both sides

(\frac{0.9}{cos ( x )})^{2} = (1.1 - 0.7 sin (x))^{2}

Subtract

(1.1 - 0.7 sin (x))^{2}

from both sides

\frac{0.81}{cos ^{2} ( x )} - 1.21 + 1.54 sin (x) - 0.49 sin^{2} (x) = 0

Rewrite using trig identities

- 1.21 + \frac{0.81}{cos ^{2} ( x )} - 0.49 sin^{2} (x) + 1.54 sin (x)

Use the Pythagorean identity:

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

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

= - 1.21 + \frac{0.81}{1 - sin ^{2} ( x )} - 0.49 sin^{2} (x) + 1.54 sin (x)

- 1.21 + \frac{0.81}{1 - sin ^{2} ( x )} - 0.49 sin^{2} (x) + 1.54 sin (x) = 0

Solve by substitution

- 1.21 + \frac{0.81}{1 - sin ^{2} ( x )} - 0.49 sin^{2} (x) + 1.54 sin (x) = 0

Let:

sin (x) = u

- 1.21 + \frac{0.81}{1 - u ^{2}} - 0.49 u^{2} + 1.54 u = 0

- 1.21 + \frac{0.81}{1 - u ^{2}} - 0.49 u^{2} + 1.54 u = 0 : u \approx 0.24522 \dots, u \approx - 0.84697 \dots

- 1.21 + \frac{0.81}{1 - u ^{2}} - 0.49 u^{2} + 1.54 u = 0

Multiply both sides by

100

- 1.21 + \frac{0.81}{1 - u ^{2}} - 0.49 u^{2} + 1.54 u = 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

- 1.21 \cdot 100 + \frac{0.81}{1 - u ^{2}} \cdot 100 - 0.49 u^{2} \cdot 100 + 1.54 u \cdot 100 = 0 \cdot 100

Refine

- 121 + \frac{81}{1 - u ^{2}} - 49 u^{2} + 154 u = 0

- 121 + \frac{81}{1 - u ^{2}} - 49 u^{2} + 154 u = 0

Multiply both sides by

1 - u^{2}

- 121 + \frac{81}{1 - u ^{2}} - 49 u^{2} + 154 u = 0

Multiply both sides by

1 - u^{2}

- 121 (1 - u^{2}) + \frac{81}{1 - u ^{2}} (1 - u^{2}) - 49 u^{2} (1 - u^{2}) + 154 u (1 - u^{2}) = 0 \cdot (1 - u^{2})

Simplify

- 121 (1 - u^{2}) + \frac{81}{1 - u ^{2}} (1 - u^{2}) - 49 u^{2} (1 - u^{2}) + 154 u (1 - u^{2}) = 0 \cdot (1 - u^{2})

Simplify

\frac{81}{1 - u ^{2}} (1 - u^{2}) : 81

\frac{81}{1 - u ^{2}} (1 - u^{2})

Multiply fractions:

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

= \frac{81 ( 1 - u ^{2} )}{1 - u ^{2}}

Cancel the common factor:

1 - u^{2}

= 81

Simplify

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

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

Apply rule

0 \cdot a = 0

= 0

- 121 (1 - u^{2}) + 81 - 49 u^{2} (1 - u^{2}) + 154 u (1 - u^{2}) = 0

- 121 (1 - u^{2}) + 81 - 49 u^{2} (1 - u^{2}) + 154 u (1 - u^{2}) = 0

- 121 (1 - u^{2}) + 81 - 49 u^{2} (1 - u^{2}) + 154 u (1 - u^{2}) = 0

Solve

- 121 (1 - u^{2}) + 81 - 49 u^{2} (1 - u^{2}) + 154 u (1 - u^{2}) = 0 : u \approx 0.24522 \dots, u \approx - 0.84697 \dots

- 121 (1 - u^{2}) + 81 - 49 u^{2} (1 - u^{2}) + 154 u (1 - u^{2}) = 0

Expand

- 121 (1 - u^{2}) + 81 - 49 u^{2} (1 - u^{2}) + 154 u (1 - u^{2}) : 49 u^{4} - 154 u^{3} + 72 u^{2} + 154 u - 40

- 121 (1 - u^{2}) + 81 - 49 u^{2} (1 - u^{2}) + 154 u (1 - u^{2})

Expand

- 121 (1 - u^{2}) : - 121 + 121 u^{2}

- 121 (1 - u^{2})

Apply the distributive law:

a (b - c) = ab - a c

a = - 121, b = 1, c = u^{2}

= - 121 \cdot 1 - (- 121) u^{2}

Apply minus-plus rules

- (- a) = a

= - 121 \cdot 1 + 121 u^{2}

Multiply the numbers:

121 \cdot 1 = 121

= - 121 + 121 u^{2}

= - 121 + 121 u^{2} + 81 - 49 u^{2} (1 - u^{2}) + 154 u (1 - u^{2})

Expand

- 49 u^{2} (1 - u^{2}) : - 49 u^{2} + 49 u^{4}

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

Apply the distributive law:

a (b - c) = ab - a c

a = - 49 u^{2}, b = 1, c = u^{2}

= - 49 u^{2} \cdot 1 - (- 49 u^{2}) u^{2}

Apply minus-plus rules

- (- a) = a

= - 49 \cdot 1 \cdot u^{2} + 49 u^{2} u^{2}

Simplify

- 49 \cdot 1 \cdot u^{2} + 49 u^{2} u^{2} : - 49 u^{2} + 49 u^{4}

- 49 \cdot 1 \cdot u^{2} + 49 u^{2} u^{2}

49 \cdot 1 \cdot u^{2} = 49 u^{2}

49 \cdot 1 \cdot u^{2}

Multiply the numbers:

49 \cdot 1 = 49

= 49 u^{2}

49 u^{2} u^{2} = 49 u^{4}

49 u^{2} u^{2}

Apply exponent rule:

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

u^{2} u^{2} = u^{2 + 2}

= 49 u^{2 + 2}

Add the numbers:

2 + 2 = 4

= 49 u^{4}

= - 49 u^{2} + 49 u^{4}

= - 49 u^{2} + 49 u^{4}

= - 121 + 121 u^{2} + 81 - 49 u^{2} + 49 u^{4} + 154 u (1 - u^{2})

Expand

154 u (1 - u^{2}) : 154 u - 154 u^{3}

154 u (1 - u^{2})

Apply the distributive law:

a (b - c) = ab - a c

a = 154 u, b = 1, c = u^{2}

= 154 u \cdot 1 - 154 u u^{2}

= 154 \cdot 1 \cdot u - 154 u^{2} u

Simplify

154 \cdot 1 \cdot u - 154 u^{2} u : 154 u - 154 u^{3}

154 \cdot 1 \cdot u - 154 u^{2} u

154 \cdot 1 \cdot u = 154 u

154 \cdot 1 \cdot u

Multiply the numbers:

154 \cdot 1 = 154

= 154 u

154 u^{2} u = 154 u^{3}

154 u^{2} u

Apply exponent rule:

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

u^{2} u = u^{2 + 1}

= 154 u^{2 + 1}

Add the numbers:

2 + 1 = 3

= 154 u^{3}

= 154 u - 154 u^{3}

= 154 u - 154 u^{3}

= - 121 + 121 u^{2} + 81 - 49 u^{2} + 49 u^{4} + 154 u - 154 u^{3}

Simplify

- 121 + 121 u^{2} + 81 - 49 u^{2} + 49 u^{4} + 154 u - 154 u^{3} : 49 u^{4} - 154 u^{3} + 72 u^{2} + 154 u - 40

- 121 + 121 u^{2} + 81 - 49 u^{2} + 49 u^{4} + 154 u - 154 u^{3}

Group like terms

= 49 u^{4} - 154 u^{3} + 121 u^{2} - 49 u^{2} + 154 u - 121 + 81

Add similar elements:

121 u^{2} - 49 u^{2} = 72 u^{2}

= 49 u^{4} - 154 u^{3} + 72 u^{2} + 154 u - 121 + 81

Add/Subtract the numbers:

- 121 + 81 = - 40

= 49 u^{4} - 154 u^{3} + 72 u^{2} + 154 u - 40

= 49 u^{4} - 154 u^{3} + 72 u^{2} + 154 u - 40

49 u^{4} - 154 u^{3} + 72 u^{2} + 154 u - 40 = 0

Find one solution for

49 u^{4} - 154 u^{3} + 72 u^{2} + 154 u - 40 = 0

using Newton-Raphson:

u \approx 0.24522 \dots

49 u^{4} - 154 u^{3} + 72 u^{2} + 154 u - 40 = 0

Newton-Raphson Approximation Definition

f (u) = 49 u^{4} - 154 u^{3} + 72 u^{2} + 154 u - 40

Find

f^{^{'}} (u) : 196 u^{3} - 462 u^{2} + 144 u + 154

\frac{d}{d u} (49 u^{4} - 154 u^{3} + 72 u^{2} + 154 u - 40)

Apply the Sum/Difference Rule:

(f \pm g)^{'} = f^{'} \pm g^{'}

= \frac{d}{d u} (49 u^{4}) - \frac{d}{d u} (154 u^{3}) + \frac{d}{d u} (72 u^{2}) + \frac{d}{d u} (154 u) - \frac{d}{d u} (40)

\frac{d}{d u} (49 u^{4}) = 196 u^{3}

\frac{d}{d u} (49 u^{4})

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 49 \frac{d}{d u} (u^{4})

Apply the Power Rule:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 49 \cdot 4 u^{4 - 1}

Simplify

= 196 u^{3}

\frac{d}{d u} (154 u^{3}) = 462 u^{2}

\frac{d}{d u} (154 u^{3})

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 154 \frac{d}{d u} (u^{3})

Apply the Power Rule:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 154 \cdot 3 u^{3 - 1}

Simplify

= 462 u^{2}

\frac{d}{d u} (72 u^{2}) = 144 u

\frac{d}{d u} (72 u^{2})

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 72 \frac{d}{d u} (u^{2})

Apply the Power Rule:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 72 \cdot 2 u^{2 - 1}

Simplify

= 144 u

\frac{d}{d u} (154 u) = 154

\frac{d}{d u} (154 u)

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 154 \frac{d u}{d u}

Apply the common derivative:

\frac{d u}{d u} = 1

= 154 \cdot 1

Simplify

= 154

\frac{d}{d u} (40) = 0

\frac{d}{d u} (40)

Derivative of a constant:

\frac{d}{d x} (a) = 0

= 0

= 196 u^{3} - 462 u^{2} + 144 u + 154 - 0

Simplify

= 196 u^{3} - 462 u^{2} + 144 u + 154

Let

u_{0} = 0

Compute

u_{n + 1}

until

Δ u_{n + 1} < 0.000001

u_{1} = 0.25974 \dots : Δ u_{1} = 0.25974 \dots

f (u_{0}) = 49 \cdot 0^{4} - 154 \cdot 0^{3} + 72 \cdot 0^{2} + 154 \cdot 0 - 40 = - 40

f^{^{'}} (u_{0}) = 196 \cdot 0^{3} - 462 \cdot 0^{2} + 144 \cdot 0 + 154 = 154

u_{1} = 0.25974 \dots

Δ u_{1} = ∣ 0.25974 \dots - 0 ∣ = 0.25974 \dots

Δ u_{1} = 0.25974 \dots

u_{2} = 0.24518 \dots : Δ u_{2} = 0.01455 \dots

f (u_{1}) = 49 \cdot 0.25974 \dots^{4} - 154 \cdot 0.25974 \dots^{3} + 72 \cdot 0.25974 \dots^{2} + 154 \cdot 0.25974 \dots - 40 = 2.38190 \dots

f^{^{'}} (u_{1}) = 196 \cdot 0.25974 \dots^{3} - 462 \cdot 0.25974 \dots^{2} + 144 \cdot 0.25974 \dots + 154 = 163.66834 \dots

u_{2} = 0.24518 \dots

Δ u_{2} = ∣ 0.24518 \dots - 0.25974 \dots ∣ = 0.01455 \dots

Δ u_{2} = 0.01455 \dots

u_{3} = 0.24522 \dots : Δ u_{3} = 0.00003 \dots

f (u_{2}) = 49 \cdot 0.24518 \dots^{4} - 154 \cdot 0.24518 \dots^{3} + 72 \cdot 0.24518 \dots^{2} + 154 \cdot 0.24518 \dots - 40 = - 0.00564 \dots

f^{^{'}} (u_{2}) = 196 \cdot 0.24518 \dots^{3} - 462 \cdot 0.24518 \dots^{2} + 144 \cdot 0.24518 \dots + 154 = 164.42203 \dots

u_{3} = 0.24522 \dots

Δ u_{3} = ∣ 0.24522 \dots - 0.24518 \dots ∣ = 0.00003 \dots

Δ u_{3} = 0.00003 \dots

u_{4} = 0.24522 \dots : Δ u_{4} = 1.69249 E - 10

f (u_{3}) = 49 \cdot 0.24522 \dots^{4} - 154 \cdot 0.24522 \dots^{3} + 72 \cdot 0.24522 \dots^{2} + 154 \cdot 0.24522 \dots - 40 = - 2.78279 E - 8

f^{^{'}} (u_{3}) = 196 \cdot 0.24522 \dots^{3} - 462 \cdot 0.24522 \dots^{2} + 144 \cdot 0.24522 \dots + 154 = 164.42041 \dots

u_{4} = 0.24522 \dots

Δ u_{4} = ∣ 0.24522 \dots - 0.24522 \dots ∣ = 1.69249 E - 10

Δ u_{4} = 1.69249 E - 10

u \approx 0.24522 \dots

Apply long division:

\frac{49 u ^{4} - 154 u ^{3} + 72 u ^{2} + 154 u - 40}{u - 0.24522 \dots} = 49 u^{3} - 141.98415 \dots u^{2} + 37.18245 \dots u + 163.11793 \dots

49 u^{3} - 141.98415 \dots u^{2} + 37.18245 \dots u + 163.11793 \dots \approx 0

Find one solution for

49 u^{3} - 141.98415 \dots u^{2} + 37.18245 \dots u + 163.11793 \dots = 0

using Newton-Raphson:

u \approx - 0.84697 \dots

49 u^{3} - 141.98415 \dots u^{2} + 37.18245 \dots u + 163.11793 \dots = 0

Newton-Raphson Approximation Definition

f (u) = 49 u^{3} - 141.98415 \dots u^{2} + 37.18245 \dots u + 163.11793 \dots

Find

f^{^{'}} (u) : 147 u^{2} - 283.96830 \dots u + 37.18245 \dots

\frac{d}{d u} (49 u^{3} - 141.98415 \dots u^{2} + 37.18245 \dots u + 163.11793 \dots)

Apply the Sum/Difference Rule:

(f \pm g)^{'} = f^{'} \pm g^{'}

= \frac{d}{d u} (49 u^{3}) - \frac{d}{d u} (141.98415 \dots u^{2}) + \frac{d}{d u} (37.18245 \dots u) + \frac{d}{d u} (163.11793 \dots)

\frac{d}{d u} (49 u^{3}) = 147 u^{2}

\frac{d}{d u} (49 u^{3})

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 49 \frac{d}{d u} (u^{3})

Apply the Power Rule:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 49 \cdot 3 u^{3 - 1}

Simplify

= 147 u^{2}

\frac{d}{d u} (141.98415 \dots u^{2}) = 283.96830 \dots u

\frac{d}{d u} (141.98415 \dots u^{2})

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 141.98415 \dots \frac{d}{d u} (u^{2})

Apply the Power Rule:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 141.98415 \dots \cdot 2 u^{2 - 1}

Simplify

= 283.96830 \dots u

\frac{d}{d u} (37.18245 \dots u) = 37.18245 \dots

\frac{d}{d u} (37.18245 \dots u)

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 37.18245 \dots \frac{d u}{d u}

Apply the common derivative:

\frac{d u}{d u} = 1

= 37.18245 \dots \cdot 1

Simplify

= 37.18245 \dots

\frac{d}{d u} (163.11793 \dots) = 0

\frac{d}{d u} (163.11793 \dots)

Derivative of a constant:

\frac{d}{d x} (a) = 0

= 0

= 147 u^{2} - 283.96830 \dots u + 37.18245 \dots + 0

Simplify

= 147 u^{2} - 283.96830 \dots u + 37.18245 \dots

Let

u_{0} = - 4

Compute

u_{n + 1}

until

Δ u_{n + 1} < 0.000001

u_{1} = - 2.46999 \dots : Δ u_{1} = 1.53000 \dots

f (u_{0}) = 49 (- 4)^{3} - 141.98415 \dots (- 4)^{2} + 37.18245 \dots (- 4) + 163.11793 \dots = - 5393.35834 \dots

f^{^{'}} (u_{0}) = 147 (- 4)^{2} - 283.96830 \dots (- 4) + 37.18245 \dots = 3525.05568 \dots

u_{1} = - 2.46999 \dots

Δ u_{1} = ∣ - 2.46999 \dots - (- 4) ∣ = 1.53000 \dots

Δ u_{1} = 1.53000 \dots

u_{2} = - 1.53240 \dots : Δ u_{2} = 0.93758 \dots

f (u_{1}) = 49 (- 2.46999 \dots)^{3} - 141.98415 \dots (- 2.46999 \dots)^{2} + 37.18245 \dots (- 2.46999 \dots) + 163.11793 \dots = - 1533.33504 \dots

f^{^{'}} (u_{1}) = 147 (- 2.46999 \dots)^{2} - 283.96830 \dots (- 2.46999 \dots) + 37.18245 \dots = 1635.40983 \dots

u_{2} = - 1.53240 \dots

Δ u_{2} = ∣ - 1.53240 \dots - (- 2.46999 \dots) ∣ = 0.93758 \dots

Δ u_{2} = 0.93758 \dots

u_{3} = - 1.03872 \dots : Δ u_{3} = 0.49368 \dots

f (u_{2}) = 49 (- 1.53240 \dots)^{3} - 141.98415 \dots (- 1.53240 \dots)^{2} + 37.18245 \dots (- 1.53240 \dots) + 163.11793 \dots = - 403.60639 \dots

f^{^{'}} (u_{2}) = 147 (- 1.53240 \dots)^{2} - 283.96830 \dots (- 1.53240 \dots) + 37.18245 \dots = 817.53470 \dots

u_{3} = - 1.03872 \dots

Δ u_{3} = ∣ - 1.03872 \dots - (- 1.53240 \dots) ∣ = 0.49368 \dots

Δ u_{3} = 0.49368 \dots

u_{4} = - 0.86834 \dots : Δ u_{4} = 0.17037 \dots

f (u_{3}) = 49 (- 1.03872 \dots)^{3} - 141.98415 \dots (- 1.03872 \dots)^{2} + 37.18245 \dots (- 1.03872 \dots) + 163.11793 \dots = - 83.61240 \dots

f^{^{'}} (u_{3}) = 147 (- 1.03872 \dots)^{2} - 283.96830 \dots (- 1.03872 \dots) + 37.18245 \dots = 490.75107 \dots

u_{4} = - 0.86834 \dots

Δ u_{4} = ∣ - 0.86834 \dots - (- 1.03872 \dots) ∣ = 0.17037 \dots

Δ u_{4} = 0.17037 \dots

u_{5} = - 0.84728 \dots : Δ u_{5} = 0.02106 \dots

f (u_{4}) = 49 (- 0.86834 \dots)^{3} - 141.98415 \dots (- 0.86834 \dots)^{2} + 37.18245 \dots (- 0.86834 \dots) + 163.11793 \dots = - 8.31155 \dots

f^{^{'}} (u_{4}) = 147 (- 0.86834 \dots)^{2} - 283.96830 \dots (- 0.86834 \dots) + 37.18245 \dots = 394.60645 \dots

u_{5} = - 0.84728 \dots

Δ u_{5} = ∣ - 0.84728 \dots - (- 0.86834 \dots) ∣ = 0.02106 \dots

Δ u_{5} = 0.02106 \dots

u_{6} = - 0.84697 \dots : Δ u_{6} = 0.00031 \dots

f (u_{5}) = 49 (- 0.84728 \dots)^{3} - 141.98415 \dots (- 0.84728 \dots)^{2} + 37.18245 \dots (- 0.84728 \dots) + 163.11793 \dots = - 0.11916 \dots

f^{^{'}} (u_{5}) = 147 (- 0.84728 \dots)^{2} - 283.96830 \dots (- 0.84728 \dots) + 37.18245 \dots = 383.31325 \dots

u_{6} = - 0.84697 \dots

Δ u_{6} = ∣ - 0.84697 \dots - (- 0.84728 \dots) ∣ = 0.00031 \dots

Δ u_{6} = 0.00031 \dots

u_{7} = - 0.84697 \dots : Δ u_{7} = 6.72258 E - 8

f (u_{6}) = 49 (- 0.84697 \dots)^{3} - 141.98415 \dots (- 0.84697 \dots)^{2} + 37.18245 \dots (- 0.84697 \dots) + 163.11793 \dots = - 0.00002 \dots

f^{^{'}} (u_{6}) = 147 (- 0.84697 \dots)^{2} - 283.96830 \dots (- 0.84697 \dots) + 37.18245 \dots = 383.14754 \dots

u_{7} = - 0.84697 \dots

Δ u_{7} = ∣ - 0.84697 \dots - (- 0.84697 \dots) ∣ = 6.72258 E - 8

Δ u_{7} = 6.72258 E - 8

u \approx - 0.84697 \dots

Apply long division:

\frac{49 u ^{3} - 141.98415 \dots u ^{2} + 37.18245 \dots u + 163.11793 \dots}{u + 0.84697 \dots} = 49 u^{2} - 183.48575 \dots u + 192.58964 \dots

49 u^{2} - 183.48575 \dots u + 192.58964 \dots \approx 0

Find one solution for

49 u^{2} - 183.48575 \dots u + 192.58964 \dots = 0

using Newton-Raphson:

No Solution for

u \in R

49 u^{2} - 183.48575 \dots u + 192.58964 \dots = 0

Newton-Raphson Approximation Definition

f (u) = 49 u^{2} - 183.48575 \dots u + 192.58964 \dots

Find

f^{^{'}} (u) : 98 u - 183.48575 \dots

\frac{d}{d u} (49 u^{2} - 183.48575 \dots u + 192.58964 \dots)

Apply the Sum/Difference Rule:

(f \pm g)^{'} = f^{'} \pm g^{'}

= \frac{d}{d u} (49 u^{2}) - \frac{d}{d u} (183.48575 \dots u) + \frac{d}{d u} (192.58964 \dots)

\frac{d}{d u} (49 u^{2}) = 98 u

\frac{d}{d u} (49 u^{2})

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 49 \frac{d}{d u} (u^{2})

Apply the Power Rule:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 49 \cdot 2 u^{2 - 1}

Simplify

= 98 u

\frac{d}{d u} (183.48575 \dots u) = 183.48575 \dots

\frac{d}{d u} (183.48575 \dots u)

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 183.48575 \dots \frac{d u}{d u}

Apply the common derivative:

\frac{d u}{d u} = 1

= 183.48575 \dots \cdot 1

Simplify

= 183.48575 \dots

\frac{d}{d u} (192.58964 \dots) = 0

\frac{d}{d u} (192.58964 \dots)

Derivative of a constant:

\frac{d}{d x} (a) = 0

= 0

= 98 u - 183.48575 \dots + 0

Simplify

= 98 u - 183.48575 \dots

Let

u_{0} = 1

Compute

u_{n + 1}

until

Δ u_{n + 1} < 0.000001

u_{1} = 1.67969 \dots : Δ u_{1} = 0.67969 \dots

f (u_{0}) = 49 \cdot 1^{2} - 183.48575 \dots \cdot 1 + 192.58964 \dots = 58.10389 \dots

f^{^{'}} (u_{0}) = 98 \cdot 1 - 183.48575 \dots = - 85.48575 \dots

u_{1} = 1.67969 \dots

Δ u_{1} = ∣ 1.67969 \dots - 1 ∣ = 0.67969 \dots

Δ u_{1} = 0.67969 \dots

u_{2} = 2.87893 \dots : Δ u_{2} = 1.19924 \dots

f (u_{1}) = 49 \cdot 1.67969 \dots^{2} - 183.48575 \dots \cdot 1.67969 \dots + 192.58964 \dots = 22.63700 \dots

f^{^{'}} (u_{1}) = 98 \cdot 1.67969 \dots - 183.48575 \dots = - 18.87604 \dots

u_{2} = 2.87893 \dots

Δ u_{2} = ∣ 2.87893 \dots - 1.67969 \dots ∣ = 1.19924 \dots

Δ u_{2} = 1.19924 \dots

u_{3} = 2.16457 \dots : Δ u_{3} = 0.71435 \dots

f (u_{2}) = 49 \cdot 2.87893 \dots^{2} - 183.48575 \dots \cdot 2.87893 \dots + 192.58964 \dots = 70.47130 \dots

f^{^{'}} (u_{2}) = 98 \cdot 2.87893 \dots - 183.48575 \dots = 98.65002 \dots

u_{3} = 2.16457 \dots

Δ u_{3} = ∣ 2.16457 \dots - 2.87893 \dots ∣ = 0.71435 \dots

Δ u_{3} = 0.71435 \dots

u_{4} = 1.29159 \dots : Δ u_{4} = 0.87298 \dots

f (u_{3}) = 49 \cdot 2.16457 \dots^{2} - 183.48575 \dots \cdot 2.16457 \dots + 192.58964 \dots = 25.00497 \dots

f^{^{'}} (u_{3}) = 98 \cdot 2.16457 \dots - 183.48575 \dots = 28.64306 \dots

u_{4} = 1.29159 \dots

Δ u_{4} = ∣ 1.29159 \dots - 2.16457 \dots ∣ = 0.87298 \dots

Δ u_{4} = 0.87298 \dots

u_{5} = 1.94777 \dots : Δ u_{5} = 0.65618 \dots

f (u_{4}) = 49 \cdot 1.29159 \dots^{2} - 183.48575 \dots \cdot 1.29159 \dots + 192.58964 \dots = 37.34304 \dots

f^{^{'}} (u_{4}) = 98 \cdot 1.29159 \dots - 183.48575 \dots = - 56.90947 \dots

u_{5} = 1.94777 \dots

Δ u_{5} = ∣ 1.94777 \dots - 1.29159 \dots ∣ = 0.65618 \dots

Δ u_{5} = 0.65618 \dots

u_{6} = - 0.90468 \dots : Δ u_{6} = 2.85246 \dots

f (u_{5}) = 49 \cdot 1.94777 \dots^{2} - 183.48575 \dots \cdot 1.94777 \dots + 192.58964 \dots = 21.09824 \dots

f^{^{'}} (u_{5}) = 98 \cdot 1.94777 \dots - 183.48575 \dots = 7.39649 \dots

u_{6} = - 0.90468 \dots

Δ u_{6} = ∣ - 0.90468 \dots - 1.94777 \dots ∣ = 2.85246 \dots

Δ u_{6} = 2.85246 \dots

u_{7} = 0.56030 \dots : Δ u_{7} = 1.46499 \dots

f (u_{6}) = 49 (- 0.90468 \dots)^{2} - 183.48575 \dots (- 0.90468 \dots) + 192.58964 \dots = 398.69183 \dots

f^{^{'}} (u_{6}) = 98 (- 0.90468 \dots) - 183.48575 \dots = - 272.14527 \dots

u_{7} = 0.56030 \dots

Δ u_{7} = ∣ 0.56030 \dots - (- 0.90468 \dots) ∣ = 1.46499 \dots

Δ u_{7} = 1.46499 \dots

u_{8} = 1.37822 \dots : Δ u_{8} = 0.81791 \dots

f (u_{7}) = 49 \cdot 0.56030 \dots^{2} - 183.48575 \dots \cdot 0.56030 \dots + 192.58964 \dots = 105.16450 \dots

f^{^{'}} (u_{7}) = 98 \cdot 0.56030 \dots - 183.48575 \dots = - 128.57563 \dots

u_{8} = 1.37822 \dots

Δ u_{8} = ∣ 1.37822 \dots - 0.56030 \dots ∣ = 0.81791 \dots

Δ u_{8} = 0.81791 \dots

u_{9} = 2.05523 \dots : Δ u_{9} = 0.67701 \dots

f (u_{8}) = 49 \cdot 1.37822 \dots^{2} - 183.48575 \dots \cdot 1.37822 \dots + 192.58964 \dots = 32.78061 \dots

f^{^{'}} (u_{8}) = 98 \cdot 1.37822 \dots - 183.48575 \dots = - 48.41953 \dots

u_{9} = 2.05523 \dots

Δ u_{9} = ∣ 2.05523 \dots - 1.37822 \dots ∣ = 0.67701 \dots

Δ u_{9} = 0.67701 \dots

u_{10} = 0.80248 \dots : Δ u_{10} = 1.25275 \dots

f (u_{9}) = 49 \cdot 2.05523 \dots^{2} - 183.48575 \dots \cdot 2.05523 \dots + 192.58964 \dots = 22.45892 \dots

f^{^{'}} (u_{9}) = 98 \cdot 2.05523 \dots - 183.48575 \dots = 17.92765 \dots

u_{10} = 0.80248 \dots

Δ u_{10} = ∣ 0.80248 \dots - 2.05523 \dots ∣ = 1.25275 \dots

Δ u_{10} = 1.25275 \dots

Cannot find solution

The solutions are

u \approx 0.24522 \dots, u \approx - 0.84697 \dots

u \approx 0.24522 \dots, u \approx - 0.84697 \dots

Verify Solutions

Find undefined (singularity) points:

u = 1, u = - 1

Take the denominator(s) of

- 1.21 + \frac{0.81}{1 - u ^{2}} - 0.49 u^{2} + 1.54 u

and compare to zero

Solve

1 - u^{2} = 0 : u = 1, u = - 1

1 - u^{2} = 0

Move

1

to the right side

1 - u^{2} = 0

Subtract

1

from both sides

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

Simplify

- u^{2} = - 1

- u^{2} = - 1

Divide both sides by

- 1

- u^{2} = - 1

Divide both sides by

- 1

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

Simplify

u^{2} = 1

u^{2} = 1

For

x^{2} = f (a)

the solutions are

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

u = 1, u = - 1

1 = 1

1

Apply radical rule:

1 = 1

= 1

- 1 = - 1

- 1

Apply radical rule:

1 = 1

1 = 1

= - 1

u = 1, u = - 1

The following points are undefined

u = 1, u = - 1

Combine undefined points with solutions:

u \approx 0.24522 \dots, u \approx - 0.84697 \dots

Substitute back

u = sin (x)

sin (x) \approx 0.24522 \dots, sin (x) \approx - 0.84697 \dots

sin (x) \approx 0.24522 \dots, sin (x) \approx - 0.84697 \dots

sin (x) = 0.24522 \dots : x = arcsin (0.24522 \dots) + 2 πn, x = π - arcsin (0.24522 \dots) + 2 πn

sin (x) = 0.24522 \dots

Apply trig inverse properties

sin (x) = 0.24522 \dots

General solutions for

sin (x) = 0.24522 \dots

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π - arcsin (a) + 2 πn

x = arcsin (0.24522 \dots) + 2 πn, x = π - arcsin (0.24522 \dots) + 2 πn

x = arcsin (0.24522 \dots) + 2 πn, x = π - arcsin (0.24522 \dots) + 2 πn

sin (x) = - 0.84697 \dots : x = arcsin (- 0.84697 \dots) + 2 πn, x = π + arcsin (0.84697 \dots) + 2 πn

sin (x) = - 0.84697 \dots

Apply trig inverse properties

sin (x) = - 0.84697 \dots

General solutions for

sin (x) = - 0.84697 \dots

sin (x) = - a \Rightarrow x = arcsin (- a) + 2 πn, x = π + arcsin (a) + 2 πn

x = arcsin (- 0.84697 \dots) + 2 πn, x = π + arcsin (0.84697 \dots) + 2 πn

x = arcsin (- 0.84697 \dots) + 2 πn, x = π + arcsin (0.84697 \dots) + 2 πn

Combine all the solutions

x = arcsin (0.24522 \dots) + 2 πn, x = π - arcsin (0.24522 \dots) + 2 πn, x = arcsin (- 0.84697 \dots) + 2 πn, x = π + arcsin (0.84697 \dots) + 2 πn

Verify solutions by plugging them into the original equation

Check the solutions by plugging them into

\frac{0.9}{cos ( x )} + 0.7 sin (x) = 1.1

Remove the ones that don't agree with the equation.

Check the solution

arcsin (0.24522 \dots) + 2 πn :

True

arcsin (0.24522 \dots) + 2 πn

Plug in

n = 1

arcsin (0.24522 \dots) + 2 π 1

For

\frac{0.9}{cos ( x )} + 0.7 sin (x) = 1.1

plug in

x = arcsin (0.24522 \dots) + 2 π 1

\frac{0.9}{cos ( arcsin ( 0.24522 \dots ) + 2 π 1 )} + 0.7 sin (arcsin (0.24522 \dots) + 2 π 1) = 1.1

Refine

1.1 = 1.1

\Rightarrow True

Check the solution

π - arcsin (0.24522 \dots) + 2 πn :

False

π - arcsin (0.24522 \dots) + 2 πn

Plug in

n = 1

π - arcsin (0.24522 \dots) + 2 π 1

For

\frac{0.9}{cos ( x )} + 0.7 sin (x) = 1.1

plug in

x = π - arcsin (0.24522 \dots) + 2 π 1

\frac{0.9}{cos ( π - arcsin ( 0.24522 \dots ) + 2 π 1 )} + 0.7 sin (π - arcsin (0.24522 \dots) + 2 π 1) = 1.1

Refine

- 0.75669 \dots = 1.1

\Rightarrow False

Check the solution

arcsin (- 0.84697 \dots) + 2 πn :

True

arcsin (- 0.84697 \dots) + 2 πn

Plug in

n = 1

arcsin (- 0.84697 \dots) + 2 π 1

For

\frac{0.9}{cos ( x )} + 0.7 sin (x) = 1.1

plug in

x = arcsin (- 0.84697 \dots) + 2 π 1

\frac{0.9}{cos ( arcsin ( - 0.84697 \dots ) + 2 π 1 )} + 0.7 sin (arcsin (- 0.84697 \dots) + 2 π 1) = 1.1

Refine

1.1 = 1.1

\Rightarrow True

Check the solution

π + arcsin (0.84697 \dots) + 2 πn :

False

π + arcsin (0.84697 \dots) + 2 πn

Plug in

n = 1

π + arcsin (0.84697 \dots) + 2 π 1

For

\frac{0.9}{cos ( x )} + 0.7 sin (x) = 1.1

plug in

x = π + arcsin (0.84697 \dots) + 2 π 1

\frac{0.9}{cos ( π + arcsin ( 0.84697 \dots ) + 2 π 1 )} + 0.7 sin (π + arcsin (0.84697 \dots) + 2 π 1) = 1.1

Refine

- 2.28576 \dots = 1.1

\Rightarrow False

x = arcsin (0.24522 \dots) + 2 πn, x = arcsin (- 0.84697 \dots) + 2 πn

Show solutions in decimal form

x = 0.24774 \dots + 2 πn, x = - 1.01026 \dots + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for (0.9)/(cos(x))+(0.7*sin(x))=1.1 ?
The general solution for (0.9)/(cos(x))+(0.7*sin(x))=1.1 is x=0.24774…+2pin,x=-1.01026…+2pin