What is the general solution for 3sec^2(x)-5tan(x)=3 ?

The general solution for 3sec^2(x)-5tan(x)=3 is x=1.03037…+pin,x=pin

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Popular >

3sec^2(x)-5tan(x)=3

Solution

3 sec^{2} (x) - 5 tan (x) = 3

Solution

x = 1.03037 \dots + πn, x = πn

Degrees

x = 59.03624 \dots^{\circ} + 18 0^{\circ} n, x = 0^{\circ} + 18 0^{\circ} n

Solution steps

3 sec^{2} (x) - 5 tan (x) = 3

Subtract

3

from both sides

3 sec^{2} (x) - 5 tan (x) - 3 = 0

Rewrite using trig identities

- 3 + 3 sec^{2} (x) - 5 tan (x)

Use the Pythagorean identity:

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

= - 3 + 3 (tan^{2} (x) + 1) - 5 tan (x)

Simplify

- 3 + 3 (tan^{2} (x) + 1) - 5 tan (x) : 3 tan^{2} (x) - 5 tan (x)

- 3 + 3 (tan^{2} (x) + 1) - 5 tan (x)

Expand

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

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

Apply the distributive law:

a (b + c) = ab + a c

a = 3, b = tan^{2} (x), c = 1

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

Multiply the numbers:

3 \cdot 1 = 3

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

= - 3 + 3 tan^{2} (x) + 3 - 5 tan (x)

Simplify

- 3 + 3 tan^{2} (x) + 3 - 5 tan (x) : 3 tan^{2} (x) - 5 tan (x)

- 3 + 3 tan^{2} (x) + 3 - 5 tan (x)

Group like terms

= 3 tan^{2} (x) - 5 tan (x) - 3 + 3

- 3 + 3 = 0

= 3 tan^{2} (x) - 5 tan (x)

= 3 tan^{2} (x) - 5 tan (x)

= 3 tan^{2} (x) - 5 tan (x)

3 tan^{2} (x) - 5 tan (x) = 0

Solve by substitution

3 tan^{2} (x) - 5 tan (x) = 0

Let:

tan (x) = u

3 u^{2} - 5 u = 0

3 u^{2} - 5 u = 0 : u = \frac{5}{3}, u = 0

3 u^{2} - 5 u = 0

Solve with the quadratic formula

3 u^{2} - 5 u = 0

Quadratic Equation Formula:

For

a = 3, b = - 5, c = 0

u_{1, 2} = \frac{- ( - 5 ) \pm ( - 5 ) ^{2} - 4 \cdot 3 \cdot 0}{2 \cdot 3}

u_{1, 2} = \frac{- ( - 5 ) \pm ( - 5 ) ^{2} - 4 \cdot 3 \cdot 0}{2 \cdot 3}

(- 5)^{2} - 4 \cdot 3 \cdot 0 = 5

(- 5)^{2} - 4 \cdot 3 \cdot 0

Apply exponent rule:

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

n

is even

(- 5)^{2} = 5^{2}

= 5^{2} - 4 \cdot 3 \cdot 0

Apply rule

0 \cdot a = 0

= 5^{2} - 0

5^{2} - 0 = 5^{2}

= 5^{2}

Apply radical rule:

n a^{n} = a,

assuming

a \geq 0

= 5

u_{1, 2} = \frac{- ( - 5 ) \pm 5}{2 \cdot 3}

Separate the solutions

u_{1} = \frac{- ( - 5 ) + 5}{2 \cdot 3}, u_{2} = \frac{- ( - 5 ) - 5}{2 \cdot 3}

u = \frac{- ( - 5 ) + 5}{2 \cdot 3} : \frac{5}{3}

\frac{- ( - 5 ) + 5}{2 \cdot 3}

Apply rule

- (- a) = a

= \frac{5 + 5}{2 \cdot 3}

Add the numbers:

5 + 5 = 10

= \frac{10}{2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{10}{6}

Cancel the common factor:

2

= \frac{5}{3}

u = \frac{- ( - 5 ) - 5}{2 \cdot 3} : 0

\frac{- ( - 5 ) - 5}{2 \cdot 3}

Apply rule

- (- a) = a

= \frac{5 - 5}{2 \cdot 3}

Subtract the numbers:

5 - 5 = 0

= \frac{0}{2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{0}{6}

Apply rule

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

= 0

The solutions to the quadratic equation are:

u = \frac{5}{3}, u = 0

Substitute back

u = tan (x)

tan (x) = \frac{5}{3}, tan (x) = 0

tan (x) = \frac{5}{3}, tan (x) = 0

tan (x) = \frac{5}{3} : x = arctan (\frac{5}{3}) + πn

tan (x) = \frac{5}{3}

Apply trig inverse properties

tan (x) = \frac{5}{3}

General solutions for

tan (x) = \frac{5}{3}

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

x = arctan (\frac{5}{3}) + πn

x = arctan (\frac{5}{3}) + 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{5}{3}) + πn, x = πn

Show solutions in decimal form

x = 1.03037 \dots + πn, x = πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for 3sec^2(x)-5tan(x)=3 ?
The general solution for 3sec^2(x)-5tan(x)=3 is x=1.03037…+pin,x=pin