What is the general solution for 3sin(x)+4=5cos(x),[0.2pi] ?

The general solution for 3sin(x)+4=5cos(x),[0.2pi] is No Solution for x\in\mathbb{R}

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Popular Trigonometry >

3sin(x)+4=5cos(x),[0.2pi]

Solution

3 sin (x) + 4 = 5 cos (x), [0.2 π]

Solution

No Solution for x \in R

Solution steps

3 sin (x) + 4 = 5 cos (x), [0.2 π]

Square both sides

(3 sin (x) + 4)^{2} = (5 cos (x))^{2}

Subtract

(5 cos (x))^{2}

from both sides

(3 sin (x) + 4)^{2} - 25 cos^{2} (x) = 0

Rewrite using trig identities

(4 + 3 sin (x))^{2} - 25 cos^{2} (x)

Use the Pythagorean identity:

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

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

= (4 + 3 sin (x))^{2} - 25 (1 - sin^{2} (x))

Simplify

(4 + 3 sin (x))^{2} - 25 (1 - sin^{2} (x)) : 34 sin^{2} (x) + 24 sin (x) - 9

(4 + 3 sin (x))^{2} - 25 (1 - sin^{2} (x))

(4 + 3 sin (x))^{2} : 16 + 24 sin (x) + 9 sin^{2} (x)

Apply Perfect Square Formula:

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

a = 4, b = 3 sin (x)

= 4^{2} + 2 \cdot 4 \cdot 3 sin (x) + (3 sin (x))^{2}

Simplify

4^{2} + 2 \cdot 4 \cdot 3 sin (x) + (3 sin (x))^{2} : 16 + 24 sin (x) + 9 sin^{2} (x)

4^{2} + 2 \cdot 4 \cdot 3 sin (x) + (3 sin (x))^{2}

4^{2} = 16

4^{2}

4^{2} = 16

= 16

2 \cdot 4 \cdot 3 sin (x) = 24 sin (x)

2 \cdot 4 \cdot 3 sin (x)

Multiply the numbers:

2 \cdot 4 \cdot 3 = 24

= 24 sin (x)

(3 sin (x))^{2} = 9 sin^{2} (x)

(3 sin (x))^{2}

Apply exponent rule:

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

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

3^{2} = 9

= 9 sin^{2} (x)

= 16 + 24 sin (x) + 9 sin^{2} (x)

= 16 + 24 sin (x) + 9 sin^{2} (x)

= 16 + 24 sin (x) + 9 sin^{2} (x) - 25 (1 - sin^{2} (x))

Expand

- 25 (1 - sin^{2} (x)) : - 25 + 25 sin^{2} (x)

- 25 (1 - sin^{2} (x))

Apply the distributive law:

a (b - c) = ab - a c

a = - 25, b = 1, c = sin^{2} (x)

= - 25 \cdot 1 - (- 25) sin^{2} (x)

Apply minus-plus rules

- (- a) = a

= - 25 \cdot 1 + 25 sin^{2} (x)

Multiply the numbers:

25 \cdot 1 = 25

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

= 16 + 24 sin (x) + 9 sin^{2} (x) - 25 + 25 sin^{2} (x)

Simplify

16 + 24 sin (x) + 9 sin^{2} (x) - 25 + 25 sin^{2} (x) : 34 sin^{2} (x) + 24 sin (x) - 9

16 + 24 sin (x) + 9 sin^{2} (x) - 25 + 25 sin^{2} (x)

Group like terms

= 24 sin (x) + 9 sin^{2} (x) + 25 sin^{2} (x) + 16 - 25

Add similar elements:

9 sin^{2} (x) + 25 sin^{2} (x) = 34 sin^{2} (x)

= 24 sin (x) + 34 sin^{2} (x) + 16 - 25

Add/Subtract the numbers:

16 - 25 = - 9

= 34 sin^{2} (x) + 24 sin (x) - 9

= 34 sin^{2} (x) + 24 sin (x) - 9

= 34 sin^{2} (x) + 24 sin (x) - 9

- 9 + 24 sin (x) + 34 sin^{2} (x) = 0

Solve by substitution

- 9 + 24 sin (x) + 34 sin^{2} (x) = 0

Let:

sin (x) = u

- 9 + 24 u + 34 u^{2} = 0

- 9 + 24 u + 34 u^{2} = 0 : u = \frac{3 ( 5 2 - 4 )}{34}, u = - \frac{3 ( 4 + 5 2 )}{34}

- 9 + 24 u + 34 u^{2} = 0

Write in the standard form

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

34 u^{2} + 24 u - 9 = 0

Solve with the quadratic formula

34 u^{2} + 24 u - 9 = 0

Quadratic Equation Formula:

For

a = 34, b = 24, c = - 9

u_{1, 2} = \frac{- 24 \pm 2 4 ^{2} - 4 \cdot 34 ( - 9 )}{2 \cdot 34}

u_{1, 2} = \frac{- 24 \pm 2 4 ^{2} - 4 \cdot 34 ( - 9 )}{2 \cdot 34}

2 4^{2} - 4 \cdot 34 (- 9) = 302

2 4^{2} - 4 \cdot 34 (- 9)

Apply rule

- (- a) = a

= 2 4^{2} + 4 \cdot 34 \cdot 9

Multiply the numbers:

4 \cdot 34 \cdot 9 = 1224

= 2 4^{2} + 1224

2 4^{2} = 576

= 576 + 1224

Add the numbers:

576 + 1224 = 1800

= 1800

Prime factorization of

1800 : 2^{3} \cdot 3^{2} \cdot 5^{2}

1800

1800

divides by

21800 = 900 \cdot 2

= 2 \cdot 900

900

divides by

2900 = 450 \cdot 2

= 2 \cdot 2 \cdot 450

450

divides by

2450 = 225 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 225

225

divides by

3225 = 75 \cdot 3

= 2 \cdot 2 \cdot 2 \cdot 3 \cdot 75

75

divides by

375 = 25 \cdot 3

= 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 \cdot 25

25

divides by

525 = 5 \cdot 5

= 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 \cdot 5 \cdot 5

2, 3, 5

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 \cdot 5 \cdot 5

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

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

Apply exponent rule:

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

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

Apply radical rule:

n ab = n a n b

= 2 2^{2} 3^{2} 5^{2}

Apply radical rule:

n a^{n} = a

2^{2} = 2

= 22 3^{2} 5^{2}

Apply radical rule:

n a^{n} = a

3^{2} = 3

= 2 \cdot 32 5^{2}

Apply radical rule:

n a^{n} = a

5^{2} = 5

= 2 \cdot 3 \cdot 52

Refine

= 302

u_{1, 2} = \frac{- 24 \pm 30 2}{2 \cdot 34}

Separate the solutions

u_{1} = \frac{- 24 + 30 2}{2 \cdot 34}, u_{2} = \frac{- 24 - 30 2}{2 \cdot 34}

u = \frac{- 24 + 30 2}{2 \cdot 34} : \frac{3 ( 5 2 - 4 )}{34}

\frac{- 24 + 30 2}{2 \cdot 34}

Multiply the numbers:

2 \cdot 34 = 68

= \frac{- 24 + 30 2}{68}

Factor

- 24 + 302 : 6 (- 4 + 52)

- 24 + 302

Rewrite as

= - 6 \cdot 4 + 6 \cdot 52

Factor out common term

6

= 6 (- 4 + 52)

= \frac{6 ( - 4 + 5 2 )}{68}

Cancel the common factor:

2

= \frac{3 ( 5 2 - 4 )}{34}

u = \frac{- 24 - 30 2}{2 \cdot 34} : - \frac{3 ( 4 + 5 2 )}{34}

\frac{- 24 - 30 2}{2 \cdot 34}

Multiply the numbers:

2 \cdot 34 = 68

= \frac{- 24 - 30 2}{68}

Factor

- 24 - 302 : - 6 (4 + 52)

- 24 - 302

Rewrite as

= - 6 \cdot 4 - 6 \cdot 52

Factor out common term

6

= - 6 (4 + 52)

= - \frac{6 ( 4 + 5 2 )}{68}

Cancel the common factor:

2

= - \frac{3 ( 4 + 5 2 )}{34}

The solutions to the quadratic equation are:

u = \frac{3 ( 5 2 - 4 )}{34}, u = - \frac{3 ( 4 + 5 2 )}{34}

Substitute back

u = sin (x)

sin (x) = \frac{3 ( 5 2 - 4 )}{34}, sin (x) = - \frac{3 ( 4 + 5 2 )}{34}

sin (x) = \frac{3 ( 5 2 - 4 )}{34}, sin (x) = - \frac{3 ( 4 + 5 2 )}{34}

sin (x) = \frac{3 ( 5 2 - 4 )}{34}, [0.2 π] :

No Solution

sin (x) = \frac{3 ( 5 2 - 4 )}{34}, [0.2 π]

Apply trig inverse properties

sin (x) = \frac{3 ( 5 2 - 4 )}{34}

General solutions for

sin (x) = \frac{3 ( 5 2 - 4 )}{34}

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

x = arcsin (\frac{3 ( 5 2 - 4 )}{34}) + 2 πn, x = π - arcsin (\frac{3 ( 5 2 - 4 )}{34}) + 2 πn

x = arcsin (\frac{3 ( 5 2 - 4 )}{34}) + 2 πn, x = π - arcsin (\frac{3 ( 5 2 - 4 )}{34}) + 2 πn

Solutions for the range

[0.2 π]

No Solution

sin (x) = - \frac{3 ( 4 + 5 2 )}{34}, [0.2 π] :

No Solution

sin (x) = - \frac{3 ( 4 + 5 2 )}{34}, [0.2 π]

Apply trig inverse properties

sin (x) = - \frac{3 ( 4 + 5 2 )}{34}

General solutions for

sin (x) = - \frac{3 ( 4 + 5 2 )}{34}

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

x = arcsin (- \frac{3 ( 4 + 5 2 )}{34}) + 2 πn, x = π + arcsin (\frac{3 ( 4 + 5 2 )}{34}) + 2 πn

x = arcsin (- \frac{3 ( 4 + 5 2 )}{34}) + 2 πn, x = π + arcsin (\frac{3 ( 4 + 5 2 )}{34}) + 2 πn

Solutions for the range

[0.2 π]

No Solution

Combine all the solutions

No Solution

Verify solutions by plugging them into the original equation

Check the solutions by plugging them into

3 sin (x) + 4 = 5 cos (x)

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

No Solution for x \in R

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for 3sin(x)+4=5cos(x),[0.2pi] ?
The general solution for 3sin(x)+4=5cos(x),[0.2pi] is No Solution for x\in\mathbb{R}