What is the general solution for 7cos(x)-24sin(x)=10 ?

The general solution for 7cos(x)-24sin(x)=10 is x=pi+0.69531…+2pin,x=-0.12772…+2pin

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7cos(x)-24sin(x)=10

Solution

7 cos (x) - 24 sin (x) = 10

Solution

x = π + 0.69531 \dots + 2 πn, x = - 0.12772 \dots + 2 πn

Degrees

x = 219.83838 \dots^{\circ} + 36 0^{\circ} n, x = - 7.31797 \dots^{\circ} + 36 0^{\circ} n

Solution steps

7 cos (x) - 24 sin (x) = 10

Add

24 sin (x)

to both sides

7 cos (x) = 10 + 24 sin (x)

Square both sides

(7 cos (x))^{2} = (10 + 24 sin (x))^{2}

Subtract

(10 + 24 sin (x))^{2}

from both sides

49 cos^{2} (x) - 100 - 480 sin (x) - 576 sin^{2} (x) = 0

Rewrite using trig identities

- 100 - 480 sin (x) + 49 cos^{2} (x) - 576 sin^{2} (x)

Use the Pythagorean identity:

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

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

= - 100 - 480 sin (x) + 49 (1 - sin^{2} (x)) - 576 sin^{2} (x)

Simplify

- 100 - 480 sin (x) + 49 (1 - sin^{2} (x)) - 576 sin^{2} (x) : - 625 sin^{2} (x) - 480 sin (x) - 51

- 100 - 480 sin (x) + 49 (1 - sin^{2} (x)) - 576 sin^{2} (x)

Expand

49 (1 - sin^{2} (x)) : 49 - 49 sin^{2} (x)

49 (1 - sin^{2} (x))

Apply the distributive law:

a (b - c) = ab - a c

a = 49, b = 1, c = sin^{2} (x)

= 49 \cdot 1 - 49 sin^{2} (x)

Multiply the numbers:

49 \cdot 1 = 49

= 49 - 49 sin^{2} (x)

= - 100 - 480 sin (x) + 49 - 49 sin^{2} (x) - 576 sin^{2} (x)

Simplify

- 100 - 480 sin (x) + 49 - 49 sin^{2} (x) - 576 sin^{2} (x) : - 625 sin^{2} (x) - 480 sin (x) - 51

- 100 - 480 sin (x) + 49 - 49 sin^{2} (x) - 576 sin^{2} (x)

Add similar elements:

- 49 sin^{2} (x) - 576 sin^{2} (x) = - 625 sin^{2} (x)

= - 100 - 480 sin (x) + 49 - 625 sin^{2} (x)

Group like terms

= - 480 sin (x) - 625 sin^{2} (x) - 100 + 49

Add/Subtract the numbers:

- 100 + 49 = - 51

= - 625 sin^{2} (x) - 480 sin (x) - 51

= - 625 sin^{2} (x) - 480 sin (x) - 51

= - 625 sin^{2} (x) - 480 sin (x) - 51

- 51 - 480 sin (x) - 625 sin^{2} (x) = 0

Solve by substitution

- 51 - 480 sin (x) - 625 sin^{2} (x) = 0

Let:

sin (x) = u

- 51 - 480 u - 625 u^{2} = 0

- 51 - 480 u - 625 u^{2} = 0 : u = - \frac{48 + 7 21}{125}, u = - \frac{48 - 7 21}{125}

- 51 - 480 u - 625 u^{2} = 0

Write in the standard form

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

- 625 u^{2} - 480 u - 51 = 0

Solve with the quadratic formula

- 625 u^{2} - 480 u - 51 = 0

Quadratic Equation Formula:

For

a = - 625, b = - 480, c = - 51

u_{1, 2} = \frac{- ( - 480 ) \pm ( - 480 ) ^{2} - 4 ( - 625 ) ( - 51 )}{2 ( - 625 )}

u_{1, 2} = \frac{- ( - 480 ) \pm ( - 480 ) ^{2} - 4 ( - 625 ) ( - 51 )}{2 ( - 625 )}

(- 480)^{2} - 4 (- 625) (- 51) = 7021

(- 480)^{2} - 4 (- 625) (- 51)

Apply rule

- (- a) = a

= (- 480)^{2} - 4 \cdot 625 \cdot 51

Apply exponent rule:

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

n

is even

(- 480)^{2} = 48 0^{2}

= 48 0^{2} - 4 \cdot 625 \cdot 51

Multiply the numbers:

4 \cdot 625 \cdot 51 = 127500

= 48 0^{2} - 127500

48 0^{2} = 230400

= 230400 - 127500

Subtract the numbers:

230400 - 127500 = 102900

= 102900

Prime factorization of

102900 : 2^{2} \cdot 3 \cdot 5^{2} \cdot 7^{3}

102900

= 7^{3} \cdot 2^{2} \cdot 5^{2} \cdot 3

Apply exponent rule:

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

= 2^{2} \cdot 5^{2} \cdot 7^{2} \cdot 3 \cdot 7

Apply radical rule:

= 2^{2} 5^{2} 7^{2} 3 \cdot 7

Apply radical rule:

2^{2} = 2

= 2 5^{2} 7^{2} 3 \cdot 7

Apply radical rule:

5^{2} = 5

= 2 \cdot 5 7^{2} 3 \cdot 7

Apply radical rule:

7^{2} = 7

= 2 \cdot 5 \cdot 7 3 \cdot 7

Refine

= 7021

u_{1, 2} = \frac{- ( - 480 ) \pm 70 21}{2 ( - 625 )}

Separate the solutions

u_{1} = \frac{- ( - 480 ) + 70 21}{2 ( - 625 )}, u_{2} = \frac{- ( - 480 ) - 70 21}{2 ( - 625 )}

u = \frac{- ( - 480 ) + 70 21}{2 ( - 625 )} : - \frac{48 + 7 21}{125}

\frac{- ( - 480 ) + 70 21}{2 ( - 625 )}

Remove parentheses:

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

= \frac{480 + 70 21}{- 2 \cdot 625}

Multiply the numbers:

2 \cdot 625 = 1250

= \frac{480 + 70 21}{- 1250}

Apply the fraction rule:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{480 + 70 21}{1250}

Cancel

\frac{480 + 70 21}{1250} : \frac{48 + 7 21}{125}

\frac{480 + 70 21}{1250}

Factor

480 + 7021 : 10 (48 + 721)

480 + 7021

Rewrite as

= 10 \cdot 48 + 10 \cdot 721

Factor out common term

10

= 10 (48 + 721)

= \frac{10 ( 48 + 7 21 )}{1250}

Cancel the common factor:

10

= \frac{48 + 7 21}{125}

= - \frac{48 + 7 21}{125}

u = \frac{- ( - 480 ) - 70 21}{2 ( - 625 )} : - \frac{48 - 7 21}{125}

\frac{- ( - 480 ) - 70 21}{2 ( - 625 )}

Remove parentheses:

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

= \frac{480 - 70 21}{- 2 \cdot 625}

Multiply the numbers:

2 \cdot 625 = 1250

= \frac{480 - 70 21}{- 1250}

Apply the fraction rule:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{480 - 70 21}{1250}

Cancel

\frac{480 - 70 21}{1250} : \frac{48 - 7 21}{125}

\frac{480 - 70 21}{1250}

Factor

480 - 7021 : 10 (48 - 721)

480 - 7021

Rewrite as

= 10 \cdot 48 - 10 \cdot 721

Factor out common term

10

= 10 (48 - 721)

= \frac{10 ( 48 - 7 21 )}{1250}

Cancel the common factor:

10

= \frac{48 - 7 21}{125}

= - \frac{48 - 7 21}{125}

The solutions to the quadratic equation are:

u = - \frac{48 + 7 21}{125}, u = - \frac{48 - 7 21}{125}

Substitute back

u = sin (x)

sin (x) = - \frac{48 + 7 21}{125}, sin (x) = - \frac{48 - 7 21}{125}

sin (x) = - \frac{48 + 7 21}{125}, sin (x) = - \frac{48 - 7 21}{125}

sin (x) = - \frac{48 + 7 21}{125} : x = arcsin (- \frac{48 + 7 21}{125}) + 2 πn, x = π + arcsin (\frac{48 + 7 21}{125}) + 2 πn

sin (x) = - \frac{48 + 7 21}{125}

Apply trig inverse properties

sin (x) = - \frac{48 + 7 21}{125}

General solutions for

sin (x) = - \frac{48 + 7 21}{125}

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

x = arcsin (- \frac{48 + 7 21}{125}) + 2 πn, x = π + arcsin (\frac{48 + 7 21}{125}) + 2 πn

x = arcsin (- \frac{48 + 7 21}{125}) + 2 πn, x = π + arcsin (\frac{48 + 7 21}{125}) + 2 πn

sin (x) = - \frac{48 - 7 21}{125} : x = arcsin (- \frac{48 - 7 21}{125}) + 2 πn, x = π + arcsin (\frac{48 - 7 21}{125}) + 2 πn

sin (x) = - \frac{48 - 7 21}{125}

Apply trig inverse properties

sin (x) = - \frac{48 - 7 21}{125}

General solutions for

sin (x) = - \frac{48 - 7 21}{125}

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

x = arcsin (- \frac{48 - 7 21}{125}) + 2 πn, x = π + arcsin (\frac{48 - 7 21}{125}) + 2 πn

x = arcsin (- \frac{48 - 7 21}{125}) + 2 πn, x = π + arcsin (\frac{48 - 7 21}{125}) + 2 πn

Combine all the solutions

x = arcsin (- \frac{48 + 7 21}{125}) + 2 πn, x = π + arcsin (\frac{48 + 7 21}{125}) + 2 πn, x = arcsin (- \frac{48 - 7 21}{125}) + 2 πn, x = π + arcsin (\frac{48 - 7 21}{125}) + 2 πn

Verify solutions by plugging them into the original equation

Check the solutions by plugging them into

7 cos (x) - 24 sin (x) = 10

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

Check the solution

arcsin (- \frac{48 + 7 21}{125}) + 2 πn :

False

arcsin (- \frac{48 + 7 21}{125}) + 2 πn

Plug in

n = 1

arcsin (- \frac{48 + 7 21}{125}) + 2 π 1

For

7 cos (x) - 24 sin (x) = 10

plug in

x = arcsin (- \frac{48 + 7 21}{125}) + 2 π 1

7 cos (arcsin (- \frac{48 + 7 21}{125}) + 2 π 1) - 24 sin (arcsin (- \frac{48 + 7 21}{125}) + 2 π 1) = 10

Refine

20.74996 \dots = 10

\Rightarrow False

Check the solution

π + arcsin (\frac{48 + 7 21}{125}) + 2 πn :

True

π + arcsin (\frac{48 + 7 21}{125}) + 2 πn

Plug in

n = 1

π + arcsin (\frac{48 + 7 21}{125}) + 2 π 1

For

7 cos (x) - 24 sin (x) = 10

plug in

x = π + arcsin (\frac{48 + 7 21}{125}) + 2 π 1

7 cos (π + arcsin (\frac{48 + 7 21}{125}) + 2 π 1) - 24 sin (π + arcsin (\frac{48 + 7 21}{125}) + 2 π 1) = 10

Refine

10 = 10

\Rightarrow True

Check the solution

arcsin (- \frac{48 - 7 21}{125}) + 2 πn :

True

arcsin (- \frac{48 - 7 21}{125}) + 2 πn

Plug in

n = 1

arcsin (- \frac{48 - 7 21}{125}) + 2 π 1

For

7 cos (x) - 24 sin (x) = 10

plug in

x = arcsin (- \frac{48 - 7 21}{125}) + 2 π 1

7 cos (arcsin (- \frac{48 - 7 21}{125}) + 2 π 1) - 24 sin (arcsin (- \frac{48 - 7 21}{125}) + 2 π 1) = 10

Refine

10 = 10

\Rightarrow True

Check the solution

π + arcsin (\frac{48 - 7 21}{125}) + 2 πn :

False

π + arcsin (\frac{48 - 7 21}{125}) + 2 πn

Plug in

n = 1

π + arcsin (\frac{48 - 7 21}{125}) + 2 π 1

For

7 cos (x) - 24 sin (x) = 10

plug in

x = π + arcsin (\frac{48 - 7 21}{125}) + 2 π 1

7 cos (π + arcsin (\frac{48 - 7 21}{125}) + 2 π 1) - 24 sin (π + arcsin (\frac{48 - 7 21}{125}) + 2 π 1) = 10

Refine

- 3.88596 \dots = 10

\Rightarrow False

x = π + arcsin (\frac{48 + 7 21}{125}) + 2 πn, x = arcsin (- \frac{48 - 7 21}{125}) + 2 πn

Show solutions in decimal form

x = π + 0.69531 \dots + 2 πn, x = - 0.12772 \dots + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for 7cos(x)-24sin(x)=10 ?
The general solution for 7cos(x)-24sin(x)=10 is x=pi+0.69531…+2pin,x=-0.12772…+2pin