What is the general solution for cos^2(x)=tan(x) ?

The general solution for cos^2(x)=tan(x) is x=0.59876…+2pin,x=-2.54282…+2pin

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Popular Trigonometry >

cos^2(x)=tan(x)

Solution

cos^{2} (x) = tan (x)

Solution

x = 0.59876 \dots + 2 πn, x = - 2.54282 \dots + 2 πn

Degrees

x = 34.30680 \dots^{\circ} + 36 0^{\circ} n, x = - 145.69319 \dots^{\circ} + 36 0^{\circ} n

Solution steps

cos^{2} (x) = tan (x)

Subtract

tan (x)

from both sides

cos^{2} (x) - tan (x) = 0

Express with sin, cos

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

Simplify

cos^{2} (x) - \frac{sin ( x )}{cos ( x )} : \frac{cos ^{3} ( x ) - sin ( x )}{cos ( x )}

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

cos^{2} (x) cos (x) - sin (x) = cos^{3} (x) - sin (x)

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

cos^{2} (x) cos (x) = cos^{3} (x)

cos^{2} (x) cos (x)

Apply exponent rule:

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

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

= cos^{2 + 1} (x)

Add the numbers:

2 + 1 = 3

= cos^{3} (x)

= cos^{3} (x) - sin (x)

= \frac{cos ^{3} ( x ) - sin ( x )}{cos ( x )}

\frac{cos ^{3} ( x ) - sin ( x )}{cos ( x )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

cos^{3} (x) - sin (x) = 0

Add

sin (x)

to both sides

cos^{3} (x) = sin (x)

Square both sides

(cos^{3} (x))^{2} = sin^{2} (x)

Subtract

sin^{2} (x)

from both sides

cos^{6} (x) - sin^{2} (x) = 0

Rewrite using trig identities

cos^{6} (x) - sin^{2} (x)

Use the Pythagorean identity:

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

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

= cos^{6} (x) - (1 - cos^{2} (x))

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

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

Distribute parentheses

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

Apply minus-plus rules

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

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

= cos^{6} (x) - 1 + cos^{2} (x)

- 1 + cos^{2} (x) + cos^{6} (x) = 0

Solve by substitution

- 1 + cos^{2} (x) + cos^{6} (x) = 0

Let:

cos (x) = u

- 1 + u^{2} + u^{6} = 0

- 1 + u^{2} + u^{6} = 0 : u = 0.68232 \dots, u = - 0.68232 \dots

- 1 + u^{2} + u^{6} = 0

Write in the standard form

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

u^{6} + u^{2} - 1 = 0

Rewrite the equation with

v = u^{2}

and

v^{3} = u^{6}

v^{3} + v - 1 = 0

Solve

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

v^{3} + v - 1 = 0

Find one solution for

v^{3} + v - 1 = 0

using Newton-Raphson:

v \approx 0.68232 \dots

v^{3} + v - 1 = 0

Newton-Raphson Approximation Definition

f (v) = v^{3} + v - 1

Find

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

\frac{d}{d v} (v^{3} + v - 1)

Apply the Sum/Difference Rule:

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

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

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

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

Apply the Power Rule:

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

= 3 v^{3 - 1}

Simplify

= 3 v^{2}

\frac{d v}{d v} = 1

\frac{d v}{d v}

Apply the common derivative:

\frac{d v}{d v} = 1

= 1

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

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

Derivative of a constant:

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

= 0

= 3 v^{2} + 1 - 0

Simplify

= 3 v^{2} + 1

Let

v_{0} = 1

Compute

v_{n + 1}

until

Δ v_{n + 1} < 0.000001

v_{1} = 0.75 : Δ v_{1} = 0.25

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

f^{^{'}} (v_{0}) = 3 \cdot 1^{2} + 1 = 4

v_{1} = 0.75

Δ v_{1} = ∣ 0.75 - 1 ∣ = 0.25

Δ v_{1} = 0.25

v_{2} = 0.68604 \dots : Δ v_{2} = 0.06395 \dots

f (v_{1}) = 0.7 5^{3} + 0.75 - 1 = 0.171875

f^{^{'}} (v_{1}) = 3 \cdot 0.7 5^{2} + 1 = 2.6875

v_{2} = 0.68604 \dots

Δ v_{2} = ∣ 0.68604 \dots - 0.75 ∣ = 0.06395 \dots

Δ v_{2} = 0.06395 \dots

v_{3} = 0.68233 \dots : Δ v_{3} = 0.00370 \dots

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

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

v_{3} = 0.68233 \dots

Δ v_{3} = ∣ 0.68233 \dots - 0.68604 \dots ∣ = 0.00370 \dots

Δ v_{3} = 0.00370 \dots

v_{4} = 0.68232 \dots : Δ v_{4} = 0.00001 \dots

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

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

v_{4} = 0.68232 \dots

Δ v_{4} = ∣ 0.68232 \dots - 0.68233 \dots ∣ = 0.00001 \dots

Δ v_{4} = 0.00001 \dots

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

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

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

v_{5} = 0.68232 \dots

Δ v_{5} = ∣ 0.68232 \dots - 0.68232 \dots ∣ = 1.18493 E - 10

Δ v_{5} = 1.18493 E - 10

v \approx 0.68232 \dots

Apply long division:

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

v^{2} + 0.68232 \dots v + 1.46557 \dots \approx 0

Find one solution for

v^{2} + 0.68232 \dots v + 1.46557 \dots = 0

using Newton-Raphson:

No Solution for

v \in R

v^{2} + 0.68232 \dots v + 1.46557 \dots = 0

Newton-Raphson Approximation Definition

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

Find

f^{^{'}} (v) : 2 v + 0.68232 \dots

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

Apply the Sum/Difference Rule:

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

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

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

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

Apply the Power Rule:

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

= 2 v^{2 - 1}

Simplify

= 2 v

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

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

Take the constant out:

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

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

Apply the common derivative:

\frac{d v}{d v} = 1

= 0.68232 \dots \cdot 1

Simplify

= 0.68232 \dots

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

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

Derivative of a constant:

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

= 0

= 2 v + 0.68232 \dots + 0

Simplify

= 2 v + 0.68232 \dots

Let

v_{0} = - 2

Compute

v_{n + 1}

until

Δ v_{n + 1} < 0.000001

v_{1} = - 0.76391 \dots : Δ v_{1} = 1.23608 \dots

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

f^{^{'}} (v_{0}) = 2 (- 2) + 0.68232 \dots = - 3.31767 \dots

v_{1} = - 0.76391 \dots

Δ v_{1} = ∣ - 0.76391 \dots - (- 2) ∣ = 1.23608 \dots

Δ v_{1} = 1.23608 \dots

v_{2} = 1.04316 \dots : Δ v_{2} = 1.80707 \dots

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

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

v_{2} = 1.04316 \dots

Δ v_{2} = ∣ 1.04316 \dots - (- 0.76391 \dots) ∣ = 1.80707 \dots

Δ v_{2} = 1.80707 \dots

v_{3} = - 0.13630 \dots : Δ v_{3} = 1.17946 \dots

f (v_{2}) = 1.04316 \dots^{2} + 0.68232 \dots \cdot 1.04316 \dots + 1.46557 \dots = 3.26553 \dots

f^{^{'}} (v_{2}) = 2 \cdot 1.04316 \dots + 0.68232 \dots = 2.76865 \dots

v_{3} = - 0.13630 \dots

Δ v_{3} = ∣ - 0.13630 \dots - 1.04316 \dots ∣ = 1.17946 \dots

Δ v_{3} = 1.17946 \dots

v_{4} = - 3.53171 \dots : Δ v_{4} = 3.39540 \dots

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

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

v_{4} = - 3.53171 \dots

Δ v_{4} = ∣ - 3.53171 \dots - (- 0.13630 \dots) ∣ = 3.39540 \dots

Δ v_{4} = 3.39540 \dots

v_{5} = - 1.72500 \dots : Δ v_{5} = 1.80670 \dots

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

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

v_{5} = - 1.72500 \dots

Δ v_{5} = ∣ - 1.72500 \dots - (- 3.53171 \dots) ∣ = 1.80670 \dots

Δ v_{5} = 1.80670 \dots

v_{6} = - 0.54560 \dots : Δ v_{6} = 1.17939 \dots

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

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

v_{6} = - 0.54560 \dots

Δ v_{6} = ∣ - 0.54560 \dots - (- 1.72500 \dots) ∣ = 1.17939 \dots

Δ v_{6} = 1.17939 \dots

v_{7} = 2.85625 \dots : Δ v_{7} = 3.40185 \dots

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

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

v_{7} = 2.85625 \dots

Δ v_{7} = ∣ 2.85625 \dots - (- 0.54560 \dots) ∣ = 3.40185 \dots

Δ v_{7} = 3.40185 \dots

v_{8} = 1.04656 \dots : Δ v_{8} = 1.80968 \dots

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

f^{^{'}} (v_{7}) = 2 \cdot 2.85625 \dots + 0.68232 \dots = 6.39483 \dots

v_{8} = 1.04656 \dots

Δ v_{8} = ∣ 1.04656 \dots - 2.85625 \dots ∣ = 1.80968 \dots

Δ v_{8} = 1.80968 \dots

v_{9} = - 0.13340 \dots : Δ v_{9} = 1.17997 \dots

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

f^{^{'}} (v_{8}) = 2 \cdot 1.04656 \dots + 0.68232 \dots = 2.77545 \dots

v_{9} = - 0.13340 \dots

Δ v_{9} = ∣ - 0.13340 \dots - 1.04656 \dots ∣ = 1.17997 \dots

Δ v_{9} = 1.17997 \dots

v_{10} = - 3.48434 \dots : Δ v_{10} = 3.35093 \dots

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

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

v_{10} = - 3.48434 \dots

Δ v_{10} = ∣ - 3.48434 \dots - (- 0.13340 \dots) ∣ = 3.35093 \dots

Δ v_{10} = 3.35093 \dots

Cannot find solution

The solution is

v \approx 0.68232 \dots

v \approx 0.68232 \dots

Substitute back

v = u^{2},

solve for

u

Solve

u^{2} = 0.68232 \dots : u = 0.68232 \dots, u = - 0.68232 \dots

u^{2} = 0.68232 \dots

For

x^{2} = f (a)

the solutions are

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

u = 0.68232 \dots, u = - 0.68232 \dots

The solutions are

u = 0.68232 \dots, u = - 0.68232 \dots

Substitute back

u = cos (x)

cos (x) = 0.68232 \dots, cos (x) = - 0.68232 \dots

cos (x) = 0.68232 \dots, cos (x) = - 0.68232 \dots

cos (x) = 0.68232 \dots : x = arccos (0.68232 \dots) + 2 πn, x = 2 π - arccos (0.68232 \dots) + 2 πn

cos (x) = 0.68232 \dots

Apply trig inverse properties

cos (x) = 0.68232 \dots

General solutions for

cos (x) = 0.68232 \dots

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

x = arccos (0.68232 \dots) + 2 πn, x = 2 π - arccos (0.68232 \dots) + 2 πn

x = arccos (0.68232 \dots) + 2 πn, x = 2 π - arccos (0.68232 \dots) + 2 πn

cos (x) = - 0.68232 \dots : x = arccos (- 0.68232 \dots) + 2 πn, x = - arccos (- 0.68232 \dots) + 2 πn

cos (x) = - 0.68232 \dots

Apply trig inverse properties

cos (x) = - 0.68232 \dots

General solutions for

cos (x) = - 0.68232 \dots

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

x = arccos (- 0.68232 \dots) + 2 πn, x = - arccos (- 0.68232 \dots) + 2 πn

x = arccos (- 0.68232 \dots) + 2 πn, x = - arccos (- 0.68232 \dots) + 2 πn

Combine all the solutions

x = arccos (0.68232 \dots) + 2 πn, x = 2 π - arccos (0.68232 \dots) + 2 πn, x = arccos (- 0.68232 \dots) + 2 πn, x = - arccos (- 0.68232 \dots) + 2 πn

Verify solutions by plugging them into the original equation

Check the solutions by plugging them into

cos^{2} (x) = tan (x)

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

Check the solution

arccos (0.68232 \dots) + 2 πn :

True

arccos (0.68232 \dots) + 2 πn

Plug in

n = 1

arccos (0.68232 \dots) + 2 π 1

For

cos^{2} (x) = tan (x)

plug in

x = arccos (0.68232 \dots) + 2 π 1

cos^{2} (arccos (0.68232 \dots) + 2 π 1) = tan (arccos (0.68232 \dots) + 2 π 1)

Refine

0.68232 \dots = 0.68232 \dots

\Rightarrow True

Check the solution

2 π - arccos (0.68232 \dots) + 2 πn :

False

2 π - arccos (0.68232 \dots) + 2 πn

Plug in

n = 1

2 π - arccos (0.68232 \dots) + 2 π 1

For

cos^{2} (x) = tan (x)

plug in

x = 2 π - arccos (0.68232 \dots) + 2 π 1

cos^{2} (2 π - arccos (0.68232 \dots) + 2 π 1) = tan (2 π - arccos (0.68232 \dots) + 2 π 1)

Refine

0.68232 \dots = - 0.68232 \dots

\Rightarrow False

Check the solution

arccos (- 0.68232 \dots) + 2 πn :

False

arccos (- 0.68232 \dots) + 2 πn

Plug in

n = 1

arccos (- 0.68232 \dots) + 2 π 1

For

cos^{2} (x) = tan (x)

plug in

x = arccos (- 0.68232 \dots) + 2 π 1

cos^{2} (arccos (- 0.68232 \dots) + 2 π 1) = tan (arccos (- 0.68232 \dots) + 2 π 1)

Refine

0.68232 \dots = - 0.68232 \dots

\Rightarrow False

Check the solution

- arccos (- 0.68232 \dots) + 2 πn :

True

- arccos (- 0.68232 \dots) + 2 πn

Plug in

n = 1

- arccos (- 0.68232 \dots) + 2 π 1

For

cos^{2} (x) = tan (x)

plug in

x = - arccos (- 0.68232 \dots) + 2 π 1

cos^{2} (- arccos (- 0.68232 \dots) + 2 π 1) = tan (- arccos (- 0.68232 \dots) + 2 π 1)

Refine

0.68232 \dots = 0.68232 \dots

\Rightarrow True

x = arccos (0.68232 \dots) + 2 πn, x = - arccos (- 0.68232 \dots) + 2 πn

Show solutions in decimal form

x = 0.59876 \dots + 2 πn, x = - 2.54282 \dots + 2 πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for cos^2(x)=tan(x) ?
The general solution for cos^2(x)=tan(x) is x=0.59876…+2pin,x=-2.54282…+2pin