What is the general solution for 2sec^2(x)-3tan(x)=2 ?

The general solution for 2sec^2(x)-3tan(x)=2 is x=0.98279…+pin,x=pin

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Popular Trigonometry >

2sec^2(x)-3tan(x)=2

Solution

2 sec^{2} (x) - 3 tan (x) = 2

Solution

x = 0.98279 \dots + πn, x = πn

Degrees

x = 56.30993 \dots^{\circ} + 18 0^{\circ} n, x = 0^{\circ} + 18 0^{\circ} n

Solution steps

2 sec^{2} (x) - 3 tan (x) = 2

Subtract

2

from both sides

2 sec^{2} (x) - 3 tan (x) - 2 = 0

Rewrite using trig identities

- 2 + 2 sec^{2} (x) - 3 tan (x)

Use the Pythagorean identity:

sec^{2} (x) = tan^{2} (x) + 1

= - 2 + 2 (tan^{2} (x) + 1) - 3 tan (x)

Simplify

- 2 + 2 (tan^{2} (x) + 1) - 3 tan (x) : 2 tan^{2} (x) - 3 tan (x)

- 2 + 2 (tan^{2} (x) + 1) - 3 tan (x)

Expand

2 (tan^{2} (x) + 1) : 2 tan^{2} (x) + 2

2 (tan^{2} (x) + 1)

Apply the distributive law:

a (b + c) = ab + a c

a = 2, b = tan^{2} (x), c = 1

= 2 tan^{2} (x) + 2 \cdot 1

Multiply the numbers:

2 \cdot 1 = 2

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

= - 2 + 2 tan^{2} (x) + 2 - 3 tan (x)

Simplify

- 2 + 2 tan^{2} (x) + 2 - 3 tan (x) : 2 tan^{2} (x) - 3 tan (x)

- 2 + 2 tan^{2} (x) + 2 - 3 tan (x)

Group like terms

= 2 tan^{2} (x) - 3 tan (x) - 2 + 2

- 2 + 2 = 0

= 2 tan^{2} (x) - 3 tan (x)

= 2 tan^{2} (x) - 3 tan (x)

= 2 tan^{2} (x) - 3 tan (x)

2 tan^{2} (x) - 3 tan (x) = 0

Solve by substitution

2 tan^{2} (x) - 3 tan (x) = 0

Let:

tan (x) = u

2 u^{2} - 3 u = 0

2 u^{2} - 3 u = 0 : u = \frac{3}{2}, u = 0

2 u^{2} - 3 u = 0

Solve with the quadratic formula

2 u^{2} - 3 u = 0

Quadratic Equation Formula:

For

a = 2, b = - 3, c = 0

u_{1, 2} = \frac{- ( - 3 ) \pm ( - 3 ) ^{2} - 4 \cdot 2 \cdot 0}{2 \cdot 2}

u_{1, 2} = \frac{- ( - 3 ) \pm ( - 3 ) ^{2} - 4 \cdot 2 \cdot 0}{2 \cdot 2}

(- 3)^{2} - 4 \cdot 2 \cdot 0 = 3

(- 3)^{2} - 4 \cdot 2 \cdot 0

Apply exponent rule:

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

n

is even

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

= 3^{2} - 4 \cdot 2 \cdot 0

Apply rule

0 \cdot a = 0

= 3^{2} - 0

3^{2} - 0 = 3^{2}

= 3^{2}

Apply radical rule:

assuming

a \geq 0

= 3

u_{1, 2} = \frac{- ( - 3 ) \pm 3}{2 \cdot 2}

Separate the solutions

u_{1} = \frac{- ( - 3 ) + 3}{2 \cdot 2}, u_{2} = \frac{- ( - 3 ) - 3}{2 \cdot 2}

u = \frac{- ( - 3 ) + 3}{2 \cdot 2} : \frac{3}{2}

\frac{- ( - 3 ) + 3}{2 \cdot 2}

Apply rule

- (- a) = a

= \frac{3 + 3}{2 \cdot 2}

Add the numbers:

3 + 3 = 6

= \frac{6}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{6}{4}

Cancel the common factor:

2

= \frac{3}{2}

u = \frac{- ( - 3 ) - 3}{2 \cdot 2} : 0

\frac{- ( - 3 ) - 3}{2 \cdot 2}

Apply rule

- (- a) = a

= \frac{3 - 3}{2 \cdot 2}

Subtract the numbers:

3 - 3 = 0

= \frac{0}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{0}{4}

Apply rule

\frac{0}{a} = 0, a \neq = 0

= 0

The solutions to the quadratic equation are:

u = \frac{3}{2}, u = 0

Substitute back

u = tan (x)

tan (x) = \frac{3}{2}, tan (x) = 0

tan (x) = \frac{3}{2}, tan (x) = 0

tan (x) = \frac{3}{2} : x = arctan (\frac{3}{2}) + πn

tan (x) = \frac{3}{2}

Apply trig inverse properties

tan (x) = \frac{3}{2}

General solutions for

tan (x) = \frac{3}{2}

tan (x) = a \Rightarrow x = arctan (a) + πn

x = arctan (\frac{3}{2}) + πn

x = arctan (\frac{3}{2}) + πn

tan (x) = 0 : x = πn

tan (x) = 0

General solutions for

tan (x) = 0

tan (x)

periodicity table with

πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = 0 + πn

x = 0 + πn

Solve

x = 0 + πn : x = πn

x = 0 + πn

0 + πn = πn

x = πn

x = πn

Combine all the solutions

x = arctan (\frac{3}{2}) + πn, x = πn

Show solutions in decimal form

x = 0.98279 \dots + πn, x = πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for 2sec^2(x)-3tan(x)=2 ?
The general solution for 2sec^2(x)-3tan(x)=2 is x=0.98279…+pin,x=pin