What is the general solution for 10sec(2x)+5tan(2x)-15=0 ?

The general solution for 10sec(2x)+5tan(2x)-15=0 is x=(0.56432…)/2+pin,x=-(1.20782…)/2+pin

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Popular Trigonometry >

10sec(2x)+5tan(2x)-15=0

Solution

10 sec (2 x) + 5 tan (2 x) - 15 = 0

Solution

x = \frac{0.56432 \dots}{2} + πn, x = - \frac{1.20782 \dots}{2} + πn

Degrees

x = 16.16676 \dots^{\circ} + 18 0^{\circ} n, x = - 34.60171 \dots^{\circ} + 18 0^{\circ} n

Solution steps

10 sec (2 x) + 5 tan (2 x) - 15 = 0

Express with sin, cos

10 \cdot \frac{1}{cos ( 2 x )} + 5 \cdot \frac{sin ( 2 x )}{cos ( 2 x )} - 15 = 0

Simplify

10 \cdot \frac{1}{cos ( 2 x )} + 5 \cdot \frac{sin ( 2 x )}{cos ( 2 x )} - 15 : \frac{10 + 5 sin ( 2 x ) - 15 cos ( 2 x )}{cos ( 2 x )}

10 \cdot \frac{1}{cos ( 2 x )} + 5 \cdot \frac{sin ( 2 x )}{cos ( 2 x )} - 15

10 \cdot \frac{1}{cos ( 2 x )} = \frac{10}{cos ( 2 x )}

10 \cdot \frac{1}{cos ( 2 x )}

Multiply fractions:

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

= \frac{1 \cdot 10}{cos ( 2 x )}

Multiply the numbers:

1 \cdot 10 = 10

= \frac{10}{cos ( 2 x )}

5 \cdot \frac{sin ( 2 x )}{cos ( 2 x )} = \frac{5 sin ( 2 x )}{cos ( 2 x )}

5 \cdot \frac{sin ( 2 x )}{cos ( 2 x )}

Multiply fractions:

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

= \frac{sin ( 2 x ) \cdot 5}{cos ( 2 x )}

= \frac{10}{cos ( 2 x )} + \frac{5 sin ( 2 x )}{cos ( 2 x )} - 15

Combine the fractions

\frac{10}{cos ( 2 x )} + \frac{5 sin ( 2 x )}{cos ( 2 x )} : \frac{10 + 5 sin ( 2 x )}{cos ( 2 x )}

Apply rule

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

= \frac{10 + 5 sin ( 2 x )}{cos ( 2 x )}

= \frac{5 sin ( 2 x ) + 10}{cos ( 2 x )} - 15

Convert element to fraction:

15 = \frac{15 cos ( 2 x )}{cos ( 2 x )}

= \frac{10 + sin ( 2 x ) \cdot 5}{cos ( 2 x )} - \frac{15 cos ( 2 x )}{cos ( 2 x )}

Since the denominators are equal, combine the fractions:

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

= \frac{10 + sin ( 2 x ) \cdot 5 - 15 cos ( 2 x )}{cos ( 2 x )}

\frac{10 + 5 sin ( 2 x ) - 15 cos ( 2 x )}{cos ( 2 x )} = 0

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

10 + 5 sin (2 x) - 15 cos (2 x) = 0

Add

15 cos (2 x)

to both sides

10 + 5 sin (2 x) = 15 cos (2 x)

Square both sides

(10 + 5 sin (2 x))^{2} = (15 cos (2 x))^{2}

Subtract

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

from both sides

(10 + 5 sin (2 x))^{2} - 225 cos^{2} (2 x) = 0

Rewrite using trig identities

(10 + 5 sin (2 x))^{2} - 225 cos^{2} (2 x)

Use the Pythagorean identity:

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

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

= (10 + 5 sin (2 x))^{2} - 225 (1 - sin^{2} (2 x))

Simplify

(10 + 5 sin (2 x))^{2} - 225 (1 - sin^{2} (2 x)) : 250 sin^{2} (2 x) + 100 sin (2 x) - 125

(10 + 5 sin (2 x))^{2} - 225 (1 - sin^{2} (2 x))

(10 + 5 sin (2 x))^{2} : 100 + 100 sin (2 x) + 25 sin^{2} (2 x)

Apply Perfect Square Formula:

(a + b)^{2} = a^{2} + 2 ab + b^{2}

a = 10, b = 5 sin (2 x)

= 1 0^{2} + 2 \cdot 10 \cdot 5 sin (2 x) + (5 sin (2 x))^{2}

Simplify

1 0^{2} + 2 \cdot 10 \cdot 5 sin (2 x) + (5 sin (2 x))^{2} : 100 + 100 sin (2 x) + 25 sin^{2} (2 x)

1 0^{2} + 2 \cdot 10 \cdot 5 sin (2 x) + (5 sin (2 x))^{2}

1 0^{2} = 100

1 0^{2}

1 0^{2} = 100

= 100

2 \cdot 10 \cdot 5 sin (2 x) = 100 sin (2 x)

2 \cdot 10 \cdot 5 sin (2 x)

Multiply the numbers:

2 \cdot 10 \cdot 5 = 100

= 100 sin (2 x)

(5 sin (2 x))^{2} = 25 sin^{2} (2 x)

(5 sin (2 x))^{2}

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

= 5^{2} sin^{2} (2 x)

5^{2} = 25

= 25 sin^{2} (2 x)

= 100 + 100 sin (2 x) + 25 sin^{2} (2 x)

= 100 + 100 sin (2 x) + 25 sin^{2} (2 x)

= 100 + 100 sin (2 x) + 25 sin^{2} (2 x) - 225 (1 - sin^{2} (2 x))

Expand

- 225 (1 - sin^{2} (2 x)) : - 225 + 225 sin^{2} (2 x)

- 225 (1 - sin^{2} (2 x))

Apply the distributive law:

a (b - c) = ab - a c

a = - 225, b = 1, c = sin^{2} (2 x)

= - 225 \cdot 1 - (- 225) sin^{2} (2 x)

Apply minus-plus rules

- (- a) = a

= - 225 \cdot 1 + 225 sin^{2} (2 x)

Multiply the numbers:

225 \cdot 1 = 225

= - 225 + 225 sin^{2} (2 x)

= 100 + 100 sin (2 x) + 25 sin^{2} (2 x) - 225 + 225 sin^{2} (2 x)

Simplify

100 + 100 sin (2 x) + 25 sin^{2} (2 x) - 225 + 225 sin^{2} (2 x) : 250 sin^{2} (2 x) + 100 sin (2 x) - 125

100 + 100 sin (2 x) + 25 sin^{2} (2 x) - 225 + 225 sin^{2} (2 x)

Group like terms

= 100 sin (2 x) + 25 sin^{2} (2 x) + 225 sin^{2} (2 x) + 100 - 225

Add similar elements:

25 sin^{2} (2 x) + 225 sin^{2} (2 x) = 250 sin^{2} (2 x)

= 100 sin (2 x) + 250 sin^{2} (2 x) + 100 - 225

Add/Subtract the numbers:

100 - 225 = - 125

= 250 sin^{2} (2 x) + 100 sin (2 x) - 125

= 250 sin^{2} (2 x) + 100 sin (2 x) - 125

= 250 sin^{2} (2 x) + 100 sin (2 x) - 125

- 125 + 100 sin (2 x) + 250 sin^{2} (2 x) = 0

Solve by substitution

- 125 + 100 sin (2 x) + 250 sin^{2} (2 x) = 0

Let:

sin (2 x) = u

- 125 + 100 u + 250 u^{2} = 0

- 125 + 100 u + 250 u^{2} = 0 : u = \frac{- 2 + 3 6}{10}, u = - \frac{2 + 3 6}{10}

- 125 + 100 u + 250 u^{2} = 0

Write in the standard form

a x^{2} + b x + c = 0

250 u^{2} + 100 u - 125 = 0

Solve with the quadratic formula

250 u^{2} + 100 u - 125 = 0

Quadratic Equation Formula:

For

a = 250, b = 100, c = - 125

u_{1, 2} = \frac{- 100 \pm 10 0 ^{2} - 4 \cdot 250 ( - 125 )}{2 \cdot 250}

u_{1, 2} = \frac{- 100 \pm 10 0 ^{2} - 4 \cdot 250 ( - 125 )}{2 \cdot 250}

10 0^{2} - 4 \cdot 250 (- 125) = 1506

10 0^{2} - 4 \cdot 250 (- 125)

Apply rule

- (- a) = a

= 10 0^{2} + 4 \cdot 250 \cdot 125

Multiply the numbers:

4 \cdot 250 \cdot 125 = 125000

= 10 0^{2} + 125000

10 0^{2} = 10000

= 10000 + 125000

Add the numbers:

10000 + 125000 = 135000

= 135000

Prime factorization of

135000 : 2^{3} \cdot 3^{3} \cdot 5^{4}

135000

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

Apply exponent rule:

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

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

Apply radical rule:

n ab = n a n b

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

Apply radical rule:

n a^{n} = a

2^{2} = 2

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

Apply radical rule:

n a^{n} = a

3^{2} = 3

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

Apply radical rule:

n a^{m} = a^{\frac{m}{n}}

5^{4} = 5^{\frac{4}{2}} = 5^{2}

= 5^{2} \cdot 2 \cdot 3 2 \cdot 3

Refine

= 1506

u_{1, 2} = \frac{- 100 \pm 150 6}{2 \cdot 250}

Separate the solutions

u_{1} = \frac{- 100 + 150 6}{2 \cdot 250}, u_{2} = \frac{- 100 - 150 6}{2 \cdot 250}

u = \frac{- 100 + 150 6}{2 \cdot 250} : \frac{- 2 + 3 6}{10}

\frac{- 100 + 150 6}{2 \cdot 250}

Multiply the numbers:

2 \cdot 250 = 500

= \frac{- 100 + 150 6}{500}

Factor

- 100 + 1506 : 50 (- 2 + 36)

- 100 + 1506

Rewrite as

= - 50 \cdot 2 + 50 \cdot 36

Factor out common term

50

= 50 (- 2 + 36)

= \frac{50 ( - 2 + 3 6 )}{500}

Cancel the common factor:

50

= \frac{- 2 + 3 6}{10}

u = \frac{- 100 - 150 6}{2 \cdot 250} : - \frac{2 + 3 6}{10}

\frac{- 100 - 150 6}{2 \cdot 250}

Multiply the numbers:

2 \cdot 250 = 500

= \frac{- 100 - 150 6}{500}

Factor

- 100 - 1506 : - 50 (2 + 36)

- 100 - 1506

Rewrite as

= - 50 \cdot 2 - 50 \cdot 36

Factor out common term

50

= - 50 (2 + 36)

= - \frac{50 ( 2 + 3 6 )}{500}

Cancel the common factor:

50

= - \frac{2 + 3 6}{10}

The solutions to the quadratic equation are:

u = \frac{- 2 + 3 6}{10}, u = - \frac{2 + 3 6}{10}

Substitute back

u = sin (2 x)

sin (2 x) = \frac{- 2 + 3 6}{10}, sin (2 x) = - \frac{2 + 3 6}{10}

sin (2 x) = \frac{- 2 + 3 6}{10}, sin (2 x) = - \frac{2 + 3 6}{10}

sin (2 x) = \frac{- 2 + 3 6}{10} : x = \frac{arcsin ( \frac{- 2 + 3 6}{10} )}{2} + πn, x = \frac{π}{2} - \frac{arcsin ( \frac{- 2 + 3 6}{10} )}{2} + πn

sin (2 x) = \frac{- 2 + 3 6}{10}

Apply trig inverse properties

sin (2 x) = \frac{- 2 + 3 6}{10}

General solutions for

sin (2 x) = \frac{- 2 + 3 6}{10}

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

2 x = arcsin (\frac{- 2 + 3 6}{10}) + 2 πn, 2 x = π - arcsin (\frac{- 2 + 3 6}{10}) + 2 πn

2 x = arcsin (\frac{- 2 + 3 6}{10}) + 2 πn, 2 x = π - arcsin (\frac{- 2 + 3 6}{10}) + 2 πn

Solve

2 x = arcsin (\frac{- 2 + 3 6}{10}) + 2 πn : x = \frac{arcsin ( \frac{- 2 + 3 6}{10} )}{2} + πn

2 x = arcsin (\frac{- 2 + 3 6}{10}) + 2 πn

Divide both sides by

2

2 x = arcsin (\frac{- 2 + 3 6}{10}) + 2 πn

Divide both sides by

2

\frac{2 x}{2} = \frac{arcsin ( \frac{- 2 + 3 6}{10} )}{2} + \frac{2 πn}{2}

Simplify

x = \frac{arcsin ( \frac{- 2 + 3 6}{10} )}{2} + πn

x = \frac{arcsin ( \frac{- 2 + 3 6}{10} )}{2} + πn

Solve

2 x = π - arcsin (\frac{- 2 + 3 6}{10}) + 2 πn : x = \frac{π}{2} - \frac{arcsin ( \frac{- 2 + 3 6}{10} )}{2} + πn

2 x = π - arcsin (\frac{- 2 + 3 6}{10}) + 2 πn

Divide both sides by

2

2 x = π - arcsin (\frac{- 2 + 3 6}{10}) + 2 πn

Divide both sides by

2

\frac{2 x}{2} = \frac{π}{2} - \frac{arcsin ( \frac{- 2 + 3 6}{10} )}{2} + \frac{2 πn}{2}

Simplify

x = \frac{π}{2} - \frac{arcsin ( \frac{- 2 + 3 6}{10} )}{2} + πn

x = \frac{π}{2} - \frac{arcsin ( \frac{- 2 + 3 6}{10} )}{2} + πn

x = \frac{arcsin ( \frac{- 2 + 3 6}{10} )}{2} + πn, x = \frac{π}{2} - \frac{arcsin ( \frac{- 2 + 3 6}{10} )}{2} + πn

sin (2 x) = - \frac{2 + 3 6}{10} : x = - \frac{arcsin ( \frac{2 + 3 6}{10} )}{2} + πn, x = \frac{π}{2} + \frac{arcsin ( \frac{2 + 3 6}{10} )}{2} + πn

sin (2 x) = - \frac{2 + 3 6}{10}

Apply trig inverse properties

sin (2 x) = - \frac{2 + 3 6}{10}

General solutions for

sin (2 x) = - \frac{2 + 3 6}{10}

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

2 x = arcsin (- \frac{2 + 3 6}{10}) + 2 πn, 2 x = π + arcsin (\frac{2 + 3 6}{10}) + 2 πn

2 x = arcsin (- \frac{2 + 3 6}{10}) + 2 πn, 2 x = π + arcsin (\frac{2 + 3 6}{10}) + 2 πn

Solve

2 x = arcsin (- \frac{2 + 3 6}{10}) + 2 πn : x = - \frac{arcsin ( \frac{2 + 3 6}{10} )}{2} + πn

2 x = arcsin (- \frac{2 + 3 6}{10}) + 2 πn

Simplify

arcsin (- \frac{2 + 3 6}{10}) + 2 πn : - arcsin (\frac{2 + 3 6}{10}) + 2 πn

arcsin (- \frac{2 + 3 6}{10}) + 2 πn

Use the following property:

arcsin (- x) = - arcsin (x)

arcsin (- \frac{2 + 3 6}{10}) = - arcsin (\frac{2 + 3 6}{10})

= - arcsin (\frac{2 + 3 6}{10}) + 2 πn

2 x = - arcsin (\frac{2 + 3 6}{10}) + 2 πn

Divide both sides by

2

2 x = - arcsin (\frac{2 + 3 6}{10}) + 2 πn

Divide both sides by

2

\frac{2 x}{2} = - \frac{arcsin ( \frac{2 + 3 6}{10} )}{2} + \frac{2 πn}{2}

Simplify

x = - \frac{arcsin ( \frac{2 + 3 6}{10} )}{2} + πn

x = - \frac{arcsin ( \frac{2 + 3 6}{10} )}{2} + πn

Solve

2 x = π + arcsin (\frac{2 + 3 6}{10}) + 2 πn : x = \frac{π}{2} + \frac{arcsin ( \frac{2 + 3 6}{10} )}{2} + πn

2 x = π + arcsin (\frac{2 + 3 6}{10}) + 2 πn

Divide both sides by

2

2 x = π + arcsin (\frac{2 + 3 6}{10}) + 2 πn

Divide both sides by

2

\frac{2 x}{2} = \frac{π}{2} + \frac{arcsin ( \frac{2 + 3 6}{10} )}{2} + \frac{2 πn}{2}

Simplify

x = \frac{π}{2} + \frac{arcsin ( \frac{2 + 3 6}{10} )}{2} + πn

x = \frac{π}{2} + \frac{arcsin ( \frac{2 + 3 6}{10} )}{2} + πn

x = - \frac{arcsin ( \frac{2 + 3 6}{10} )}{2} + πn, x = \frac{π}{2} + \frac{arcsin ( \frac{2 + 3 6}{10} )}{2} + πn

Combine all the solutions

x = \frac{arcsin ( \frac{- 2 + 3 6}{10} )}{2} + πn, x = \frac{π}{2} - \frac{arcsin ( \frac{- 2 + 3 6}{10} )}{2} + πn, x = - \frac{arcsin ( \frac{2 + 3 6}{10} )}{2} + πn, x = \frac{π}{2} + \frac{arcsin ( \frac{2 + 3 6}{10} )}{2} + πn

Verify solutions by plugging them into the original equation

Check the solutions by plugging them into

10 sec (2 x) + 5 tan (2 x) - 15 = 0

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

Check the solution

\frac{arcsin ( \frac{- 2 + 3 6}{10} )}{2} + πn :

True

\frac{arcsin ( \frac{- 2 + 3 6}{10} )}{2} + πn

Plug in

n = 1

\frac{arcsin ( \frac{- 2 + 3 6}{10} )}{2} + π 1

For

10 sec (2 x) + 5 tan (2 x) - 15 = 0

plug in

x = \frac{arcsin ( \frac{- 2 + 3 6}{10} )}{2} + π 1

10 sec 2 \frac{arcsin ( \frac{- 2 + 3 6}{10} )}{2} + π 1 + 5 tan 2 \frac{arcsin ( \frac{- 2 + 3 6}{10} )}{2} + π 1 - 15 = 0

Refine

0 = 0

\Rightarrow True

Check the solution

\frac{π}{2} - \frac{arcsin ( \frac{- 2 + 3 6}{10} )}{2} + πn :

False

\frac{π}{2} - \frac{arcsin ( \frac{- 2 + 3 6}{10} )}{2} + πn

Plug in

n = 1

\frac{π}{2} - \frac{arcsin ( \frac{- 2 + 3 6}{10} )}{2} + π 1

For

10 sec (2 x) + 5 tan (2 x) - 15 = 0

plug in

x = \frac{π}{2} - \frac{arcsin ( \frac{- 2 + 3 6}{10} )}{2} + π 1

10 sec 2 \frac{π}{2} - \frac{arcsin ( \frac{- 2 + 3 6}{10} )}{2} + π 1 + 5 tan 2 \frac{π}{2} - \frac{arcsin ( \frac{- 2 + 3 6}{10} )}{2} + π 1 - 15 = 0

Refine

- 30 = 0

\Rightarrow False

Check the solution

- \frac{arcsin ( \frac{2 + 3 6}{10} )}{2} + πn :

True

- \frac{arcsin ( \frac{2 + 3 6}{10} )}{2} + πn

Plug in

n = 1

- \frac{arcsin ( \frac{2 + 3 6}{10} )}{2} + π 1

For

10 sec (2 x) + 5 tan (2 x) - 15 = 0

plug in

x = - \frac{arcsin ( \frac{2 + 3 6}{10} )}{2} + π 1

10 sec 2 - \frac{arcsin ( \frac{2 + 3 6}{10} )}{2} + π 1 + 5 tan 2 - \frac{arcsin ( \frac{2 + 3 6}{10} )}{2} + π 1 - 15 = 0

Refine

0 = 0

\Rightarrow True

Check the solution

\frac{π}{2} + \frac{arcsin ( \frac{2 + 3 6}{10} )}{2} + πn :

False

\frac{π}{2} + \frac{arcsin ( \frac{2 + 3 6}{10} )}{2} + πn

Plug in

n = 1

\frac{π}{2} + \frac{arcsin ( \frac{2 + 3 6}{10} )}{2} + π 1

For

10 sec (2 x) + 5 tan (2 x) - 15 = 0

plug in

x = \frac{π}{2} + \frac{arcsin ( \frac{2 + 3 6}{10} )}{2} + π 1

10 sec 2 \frac{π}{2} + \frac{arcsin ( \frac{2 + 3 6}{10} )}{2} + π 1 + 5 tan 2 \frac{π}{2} + \frac{arcsin ( \frac{2 + 3 6}{10} )}{2} + π 1 - 15 = 0

Refine

- 30 = 0

\Rightarrow False

x = \frac{arcsin ( \frac{- 2 + 3 6}{10} )}{2} + πn, x = - \frac{arcsin ( \frac{2 + 3 6}{10} )}{2} + πn

Show solutions in decimal form

x = \frac{0.56432 \dots}{2} + πn, x = - \frac{1.20782 \dots}{2} + πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for 10sec(2x)+5tan(2x)-15=0 ?
The general solution for 10sec(2x)+5tan(2x)-15=0 is x=(0.56432…)/2+pin,x=-(1.20782…)/2+pin