What is the general solution for sec^2(x)+cos(x)=-1 ?

The general solution for sec^2(x)+cos(x)=-1 is No Solution for x\in\mathbb{R}

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sec^2(x)+cos(x)=-1

Solution

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

Solution

No Solution for x \in R

Solution steps

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

Subtract

- 1

from both sides

sec^{2} (x) + cos (x) + 1 = 0

Rewrite using trig identities

1 + cos (x) + sec^{2} (x)

Use the basic trigonometric identity:

cos (x) = \frac{1}{sec ( x )}

= 1 + \frac{1}{sec ( x )} + sec^{2} (x)

1 + \frac{1}{sec ( x )} + sec^{2} (x) = 0

Solve by substitution

1 + \frac{1}{sec ( x )} + sec^{2} (x) = 0

Let:

sec (x) = u

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

1 + \frac{1}{u} + u^{2} = 0 : u \approx - 0.68232 \dots

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

Multiply both sides by

u

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

Multiply both sides by

u

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

Simplify

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

Simplify

1 \cdot u : u

1 \cdot u

Multiply:

1 \cdot u = u

= u

Simplify

\frac{1}{u} u : 1

\frac{1}{u} u

Multiply fractions:

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

= \frac{1 \cdot u}{u}

Cancel the common factor:

u

= 1

Simplify

u^{2} u : u^{3}

u^{2} u

Apply exponent rule:

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

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

= u^{2 + 1}

Add the numbers:

2 + 1 = 3

= u^{3}

Simplify

0 \cdot u : 0

0 \cdot u

Apply rule

0 \cdot a = 0

= 0

u + 1 + u^{3} = 0

u + 1 + u^{3} = 0

u + 1 + u^{3} = 0

Solve

u + 1 + u^{3} = 0 : u \approx - 0.68232 \dots

u + 1 + u^{3} = 0

Write in the standard form

a_{n} x^{n} + \dots + a_{1} x + a_{0} = 0

u^{3} + u + 1 = 0

Find one solution for

u^{3} + u + 1 = 0

using Newton-Raphson:

u \approx - 0.68232 \dots

u^{3} + u + 1 = 0

Newton-Raphson Approximation Definition

f (u) = u^{3} + u + 1

Find

f^{^{'}} (u) : 3 u^{2} + 1

\frac{d}{d u} (u^{3} + u + 1)

Apply the Sum/Difference Rule:

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

= \frac{d}{d u} (u^{3}) + \frac{d u}{d u} + \frac{d}{d u} (1)

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

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

Apply the Power Rule:

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

= 3 u^{3 - 1}

Simplify

= 3 u^{2}

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

\frac{d u}{d u}

Apply the common derivative:

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

= 1

\frac{d}{d u} (1) = 0

\frac{d}{d u} (1)

Derivative of a constant:

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

= 0

= 3 u^{2} + 1 + 0

Simplify

= 3 u^{2} + 1

Let

u_{0} = - 1

Compute

u_{n + 1}

until

Δ u_{n + 1} < 0.000001

u_{1} = - 0.75 : Δ u_{1} = 0.25

f (u_{0}) = (- 1)^{3} + (- 1) + 1 = - 1

f^{^{'}} (u_{0}) = 3 (- 1)^{2} + 1 = 4

u_{1} = - 0.75

Δ u_{1} = ∣ - 0.75 - (- 1) ∣ = 0.25

Δ u_{1} = 0.25

u_{2} = - 0.68604 \dots : Δ u_{2} = 0.06395 \dots

f (u_{1}) = (- 0.75)^{3} + (- 0.75) + 1 = - 0.171875

f^{^{'}} (u_{1}) = 3 (- 0.75)^{2} + 1 = 2.6875

u_{2} = - 0.68604 \dots

Δ u_{2} = ∣ - 0.68604 \dots - (- 0.75) ∣ = 0.06395 \dots

Δ u_{2} = 0.06395 \dots

u_{3} = - 0.68233 \dots : Δ u_{3} = 0.00370 \dots

f (u_{2}) = (- 0.68604 \dots)^{3} + (- 0.68604 \dots) + 1 = - 0.00894 \dots

f^{^{'}} (u_{2}) = 3 (- 0.68604 \dots)^{2} + 1 = 2.41197 \dots

u_{3} = - 0.68233 \dots

Δ u_{3} = ∣ - 0.68233 \dots - (- 0.68604 \dots) ∣ = 0.00370 \dots

Δ u_{3} = 0.00370 \dots

u_{4} = - 0.68232 \dots : Δ u_{4} = 0.00001 \dots

f (u_{3}) = (- 0.68233 \dots)^{3} + (- 0.68233 \dots) + 1 = - 0.00002 \dots

f^{^{'}} (u_{3}) = 3 (- 0.68233 \dots)^{2} + 1 = 2.39676 \dots

u_{4} = - 0.68232 \dots

Δ u_{4} = ∣ - 0.68232 \dots - (- 0.68233 \dots) ∣ = 0.00001 \dots

Δ u_{4} = 0.00001 \dots

u_{5} = - 0.68232 \dots : Δ u_{5} = 1.18493 E - 10

f (u_{4}) = (- 0.68232 \dots)^{3} + (- 0.68232 \dots) + 1 = - 2.83995 E - 10

f^{^{'}} (u_{4}) = 3 (- 0.68232 \dots)^{2} + 1 = 2.39671 \dots

u_{5} = - 0.68232 \dots

Δ u_{5} = ∣ - 0.68232 \dots - (- 0.68232 \dots) ∣ = 1.18493 E - 10

Δ u_{5} = 1.18493 E - 10

u \approx - 0.68232 \dots

Apply long division:

\frac{u ^{3} + u + 1}{u + 0.68232 \dots} = u^{2} - 0.68232 \dots u + 1.46557 \dots

u^{2} - 0.68232 \dots u + 1.46557 \dots \approx 0

Find one solution for

u^{2} - 0.68232 \dots u + 1.46557 \dots = 0

using Newton-Raphson:

No Solution for

u \in R

u^{2} - 0.68232 \dots u + 1.46557 \dots = 0

Newton-Raphson Approximation Definition

f (u) = u^{2} - 0.68232 \dots u + 1.46557 \dots

Find

f^{^{'}} (u) : 2 u - 0.68232 \dots

\frac{d}{d u} (u^{2} - 0.68232 \dots u + 1.46557 \dots)

Apply the Sum/Difference Rule:

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

= \frac{d}{d u} (u^{2}) - \frac{d}{d u} (0.68232 \dots u) + \frac{d}{d u} (1.46557 \dots)

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

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

Apply the Power Rule:

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

= 2 u^{2 - 1}

Simplify

= 2 u

\frac{d}{d u} (0.68232 \dots u) = 0.68232 \dots

\frac{d}{d u} (0.68232 \dots u)

Take the constant out:

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

= 0.68232 \dots \frac{d u}{d u}

Apply the common derivative:

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

= 0.68232 \dots \cdot 1

Simplify

= 0.68232 \dots

\frac{d}{d u} (1.46557 \dots) = 0

\frac{d}{d u} (1.46557 \dots)

Derivative of a constant:

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

= 0

= 2 u - 0.68232 \dots + 0

Simplify

= 2 u - 0.68232 \dots

Let

u_{0} = 2

Compute

u_{n + 1}

until

Δ u_{n + 1} < 0.000001

u_{1} = 0.76391 \dots : Δ u_{1} = 1.23608 \dots

f (u_{0}) = 2^{2} - 0.68232 \dots \cdot 2 + 1.46557 \dots = 4.10091 \dots

f^{^{'}} (u_{0}) = 2 \cdot 2 - 0.68232 \dots = 3.31767 \dots

u_{1} = 0.76391 \dots

Δ u_{1} = ∣ 0.76391 \dots - 2 ∣ = 1.23608 \dots

Δ u_{1} = 1.23608 \dots

u_{2} = - 1.04316 \dots : Δ u_{2} = 1.80707 \dots

f (u_{1}) = 0.76391 \dots^{2} - 0.68232 \dots \cdot 0.76391 \dots + 1.46557 \dots = 1.52789 \dots

f^{^{'}} (u_{1}) = 2 \cdot 0.76391 \dots - 0.68232 \dots = 0.84550 \dots

u_{2} = - 1.04316 \dots

Δ u_{2} = ∣ - 1.04316 \dots - 0.76391 \dots ∣ = 1.80707 \dots

Δ u_{2} = 1.80707 \dots

u_{3} = 0.13630 \dots : Δ u_{3} = 1.17946 \dots

f (u_{2}) = (- 1.04316 \dots)^{2} - 0.68232 \dots (- 1.04316 \dots) + 1.46557 \dots = 3.26553 \dots

f^{^{'}} (u_{2}) = 2 (- 1.04316 \dots) - 0.68232 \dots = - 2.76865 \dots

u_{3} = 0.13630 \dots

Δ u_{3} = ∣ 0.13630 \dots - (- 1.04316 \dots) ∣ = 1.17946 \dots

Δ u_{3} = 1.17946 \dots

u_{4} = 3.53171 \dots : Δ u_{4} = 3.39540 \dots

f (u_{3}) = 0.13630 \dots^{2} - 0.68232 \dots \cdot 0.13630 \dots + 1.46557 \dots = 1.39114 \dots

f^{^{'}} (u_{3}) = 2 \cdot 0.13630 \dots - 0.68232 \dots = - 0.40971 \dots

u_{4} = 3.53171 \dots

Δ u_{4} = ∣ 3.53171 \dots - 0.13630 \dots ∣ = 3.39540 \dots

Δ u_{4} = 3.39540 \dots

u_{5} = 1.72500 \dots : Δ u_{5} = 1.80670 \dots

f (u_{4}) = 3.53171 \dots^{2} - 0.68232 \dots \cdot 3.53171 \dots + 1.46557 \dots = 11.52876 \dots

f^{^{'}} (u_{4}) = 2 \cdot 3.53171 \dots - 0.68232 \dots = 6.38109 \dots

u_{5} = 1.72500 \dots

Δ u_{5} = ∣ 1.72500 \dots - 3.53171 \dots ∣ = 1.80670 \dots

Δ u_{5} = 1.80670 \dots

u_{6} = 0.54560 \dots : Δ u_{6} = 1.17939 \dots

f (u_{5}) = 1.72500 \dots^{2} - 0.68232 \dots \cdot 1.72500 \dots + 1.46557 \dots = 3.26419 \dots

f^{^{'}} (u_{5}) = 2 \cdot 1.72500 \dots - 0.68232 \dots = 2.76767 \dots

u_{6} = 0.54560 \dots

Δ u_{6} = ∣ 0.54560 \dots - 1.72500 \dots ∣ = 1.17939 \dots

Δ u_{6} = 1.17939 \dots

u_{7} = - 2.85625 \dots : Δ u_{7} = 3.40185 \dots

f (u_{6}) = 0.54560 \dots^{2} - 0.68232 \dots \cdot 0.54560 \dots + 1.46557 \dots = 1.39097 \dots

f^{^{'}} (u_{6}) = 2 \cdot 0.54560 \dots - 0.68232 \dots = 0.40888 \dots

u_{7} = - 2.85625 \dots

Δ u_{7} = ∣ - 2.85625 \dots - 0.54560 \dots ∣ = 3.40185 \dots

Δ u_{7} = 3.40185 \dots

u_{8} = - 1.04656 \dots : Δ u_{8} = 1.80968 \dots

f (u_{7}) = (- 2.85625 \dots)^{2} - 0.68232 \dots (- 2.85625 \dots) + 1.46557 \dots = 11.57264 \dots

f^{^{'}} (u_{7}) = 2 (- 2.85625 \dots) - 0.68232 \dots = - 6.39483 \dots

u_{8} = - 1.04656 \dots

Δ u_{8} = ∣ - 1.04656 \dots - (- 2.85625 \dots) ∣ = 1.80968 \dots

Δ u_{8} = 1.80968 \dots

u_{9} = 0.13340 \dots : Δ u_{9} = 1.17997 \dots

f (u_{8}) = (- 1.04656 \dots)^{2} - 0.68232 \dots (- 1.04656 \dots) + 1.46557 \dots = 3.27496 \dots

f^{^{'}} (u_{8}) = 2 (- 1.04656 \dots) - 0.68232 \dots = - 2.77545 \dots

u_{9} = 0.13340 \dots

Δ u_{9} = ∣ 0.13340 \dots - (- 1.04656 \dots) ∣ = 1.17997 \dots

Δ u_{9} = 1.17997 \dots

u_{10} = 3.48434 \dots : Δ u_{10} = 3.35093 \dots

f (u_{9}) = 0.13340 \dots^{2} - 0.68232 \dots \cdot 0.13340 \dots + 1.46557 \dots = 1.39234 \dots

f^{^{'}} (u_{9}) = 2 \cdot 0.13340 \dots - 0.68232 \dots = - 0.41550 \dots

u_{10} = 3.48434 \dots

Δ u_{10} = ∣ 3.48434 \dots - 0.13340 \dots ∣ = 3.35093 \dots

Δ u_{10} = 3.35093 \dots

Cannot find solution

The solution is

u \approx - 0.68232 \dots

u \approx - 0.68232 \dots

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

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

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u \approx - 0.68232 \dots

Substitute back

u = sec (x)

sec (x) \approx - 0.68232 \dots

sec (x) \approx - 0.68232 \dots

sec (x) = - 0.68232 \dots :

No Solution

sec (x) = - 0.68232 \dots

sec (x) \leq - 1 or sec (x) \geq 1

No Solution

Combine all the solutions

No Solution for x \in R

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Frequently Asked Questions (FAQ)

What is the general solution for sec^2(x)+cos(x)=-1 ?
The general solution for sec^2(x)+cos(x)=-1 is No Solution for x\in\mathbb{R}