What is the general solution for 90-70sin(x)-130cos(x)=0 ?

The general solution for 90-70sin(x)-130cos(x)=0 is x=1.40923…+2pin,x=-0.42135…+2pin

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90-70sin(x)-130cos(x)=0

Solution

90 - 70 sin (x) - 130 cos (x) = 0

Solution

x = 1.40923 \dots + 2 πn, x = - 0.42135 \dots + 2 πn

Degrees

x = 80.74328 \dots^{\circ} + 36 0^{\circ} n, x = - 24.14177 \dots^{\circ} + 36 0^{\circ} n

Solution steps

90 - 70 sin (x) - 130 cos (x) = 0

Add

130 cos (x)

to both sides

90 - 70 sin (x) = 130 cos (x)

Square both sides

(90 - 70 sin (x))^{2} = (130 cos (x))^{2}

Subtract

(130 cos (x))^{2}

from both sides

(90 - 70 sin (x))^{2} - 16900 cos^{2} (x) = 0

Rewrite using trig identities

(90 - 70 sin (x))^{2} - 16900 cos^{2} (x)

Use the Pythagorean identity:

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

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

= (90 - 70 sin (x))^{2} - 16900 (1 - sin^{2} (x))

Simplify

(90 - 70 sin (x))^{2} - 16900 (1 - sin^{2} (x)) : 21800 sin^{2} (x) - 12600 sin (x) - 8800

(90 - 70 sin (x))^{2} - 16900 (1 - sin^{2} (x))

(90 - 70 sin (x))^{2} : 8100 - 12600 sin (x) + 4900 sin^{2} (x)

Apply Perfect Square Formula:

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

a = 90, b = 70 sin (x)

= 9 0^{2} - 2 \cdot 90 \cdot 70 sin (x) + (70 sin (x))^{2}

Simplify

9 0^{2} - 2 \cdot 90 \cdot 70 sin (x) + (70 sin (x))^{2} : 8100 - 12600 sin (x) + 4900 sin^{2} (x)

9 0^{2} - 2 \cdot 90 \cdot 70 sin (x) + (70 sin (x))^{2}

9 0^{2} = 8100

9 0^{2}

9 0^{2} = 8100

= 8100

2 \cdot 90 \cdot 70 sin (x) = 12600 sin (x)

2 \cdot 90 \cdot 70 sin (x)

Multiply the numbers:

2 \cdot 90 \cdot 70 = 12600

= 12600 sin (x)

(70 sin (x))^{2} = 4900 sin^{2} (x)

(70 sin (x))^{2}

Apply exponent rule:

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

= 7 0^{2} sin^{2} (x)

7 0^{2} = 4900

= 4900 sin^{2} (x)

= 8100 - 12600 sin (x) + 4900 sin^{2} (x)

= 8100 - 12600 sin (x) + 4900 sin^{2} (x)

= 8100 - 12600 sin (x) + 4900 sin^{2} (x) - 16900 (1 - sin^{2} (x))

Expand

- 16900 (1 - sin^{2} (x)) : - 16900 + 16900 sin^{2} (x)

- 16900 (1 - sin^{2} (x))

Apply the distributive law:

a (b - c) = ab - a c

a = - 16900, b = 1, c = sin^{2} (x)

= - 16900 \cdot 1 - (- 16900) sin^{2} (x)

Apply minus-plus rules

- (- a) = a

= - 16900 \cdot 1 + 16900 sin^{2} (x)

Multiply the numbers:

16900 \cdot 1 = 16900

= - 16900 + 16900 sin^{2} (x)

= 8100 - 12600 sin (x) + 4900 sin^{2} (x) - 16900 + 16900 sin^{2} (x)

Simplify

8100 - 12600 sin (x) + 4900 sin^{2} (x) - 16900 + 16900 sin^{2} (x) : 21800 sin^{2} (x) - 12600 sin (x) - 8800

8100 - 12600 sin (x) + 4900 sin^{2} (x) - 16900 + 16900 sin^{2} (x)

Group like terms

= - 12600 sin (x) + 4900 sin^{2} (x) + 16900 sin^{2} (x) + 8100 - 16900

Add similar elements:

4900 sin^{2} (x) + 16900 sin^{2} (x) = 21800 sin^{2} (x)

= - 12600 sin (x) + 21800 sin^{2} (x) + 8100 - 16900

Add/Subtract the numbers:

8100 - 16900 = - 8800

= 21800 sin^{2} (x) - 12600 sin (x) - 8800

= 21800 sin^{2} (x) - 12600 sin (x) - 8800

= 21800 sin^{2} (x) - 12600 sin (x) - 8800

- 8800 - 12600 sin (x) + 21800 sin^{2} (x) = 0

Solve by substitution

- 8800 - 12600 sin (x) + 21800 sin^{2} (x) = 0

Let:

sin (x) = u

- 8800 - 12600 u + 21800 u^{2} = 0

- 8800 - 12600 u + 21800 u^{2} = 0 : u = \frac{63 + 13 137}{218}, u = \frac{63 - 13 137}{218}

- 8800 - 12600 u + 21800 u^{2} = 0

Divide both sides by

21800

- \frac{8800}{21800} - \frac{12600 u}{21800} + \frac{21800 u ^{2}}{21800} = \frac{0}{21800}

Write in the standard form

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

u^{2} - \frac{63 u}{109} - \frac{44}{109} = 0

Solve with the quadratic formula

u^{2} - \frac{63 u}{109} - \frac{44}{109} = 0

Quadratic Equation Formula:

For

a = 1, b = - \frac{63}{109}, c = - \frac{44}{109}

u_{1, 2} = \frac{- ( - \frac{63}{109} ) \pm ( - \frac{63}{109} ) ^{2} - 4 \cdot 1 \cdot ( - \frac{44}{109} )}{2 \cdot 1}

u_{1, 2} = \frac{- ( - \frac{63}{109} ) \pm ( - \frac{63}{109} ) ^{2} - 4 \cdot 1 \cdot ( - \frac{44}{109} )}{2 \cdot 1}

(- \frac{63}{109})^{2} - 4 \cdot 1 \cdot (- \frac{44}{109}) = \frac{13 137}{109}

(- \frac{63}{109})^{2} - 4 \cdot 1 \cdot (- \frac{44}{109})

Apply rule

- (- a) = a

= (- \frac{63}{109})^{2} + 4 \cdot 1 \cdot \frac{44}{109}

(- \frac{63}{109})^{2} = \frac{6 3 ^{2}}{10 9 ^{2}}

(- \frac{63}{109})^{2}

Apply exponent rule:

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

n

is even

(- \frac{63}{109})^{2} = (\frac{63}{109})^{2}

= (\frac{63}{109})^{2}

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{6 3 ^{2}}{10 9 ^{2}}

4 \cdot 1 \cdot \frac{44}{109} = \frac{176}{109}

4 \cdot 1 \cdot \frac{44}{109}

Multiply fractions:

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

= 1 \cdot \frac{44 \cdot 4}{109}

Multiply the numbers:

44 \cdot 4 = 176

= 1 \cdot \frac{176}{109}

Multiply:

1 \cdot \frac{176}{109} = \frac{176}{109}

= \frac{176}{109}

= \frac{6 3 ^{2}}{10 9 ^{2}} + \frac{176}{109}

\frac{6 3 ^{2}}{10 9 ^{2}} = \frac{3969}{11881}

\frac{6 3 ^{2}}{10 9 ^{2}}

6 3^{2} = 3969

= \frac{3969}{10 9 ^{2}}

10 9^{2} = 11881

= \frac{3969}{11881}

= \frac{3969}{11881} + \frac{176}{109}

Join

\frac{3969}{11881} + \frac{176}{109} : \frac{23153}{11881}

\frac{3969}{11881} + \frac{176}{109}

Least Common Multiplier of

11881, 109 : 11881

11881, 109

Least Common Multiplier (LCM)

Prime factorization of

11881 : 109 \cdot 109

11881

11881

divides by

10911881 = 109 \cdot 109

= 109 \cdot 109

Prime factorization of

109 : 109

109

109

is a prime number, therefore no factorization is possible

= 109

Multiply each factor the greatest number of times it occurs in either

11881

109

= 109 \cdot 109

Multiply the numbers:

109 \cdot 109 = 11881

= 11881

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

11881

For

\frac{176}{109} :

multiply the denominator and numerator by

109

\frac{176}{109} = \frac{176 \cdot 109}{109 \cdot 109} = \frac{19184}{11881}

= \frac{3969}{11881} + \frac{19184}{11881}

Since the denominators are equal, combine the fractions:

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

= \frac{3969 + 19184}{11881}

Add the numbers:

3969 + 19184 = 23153

= \frac{23153}{11881}

= \frac{23153}{11881}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{23153}{11881}

11881 = 109

11881

Factor the number:

11881 = 10 9^{2}

= 10 9^{2}

Apply radical rule:

10 9^{2} = 109

= 109

= \frac{23153}{109}

23153 = 13137

23153

Prime factorization of

23153 : 1 3^{2} \cdot 137

23153

23153

divides by

1323153 = 1781 \cdot 13

= 13 \cdot 1781

1781

divides by

131781 = 137 \cdot 13

= 13 \cdot 13 \cdot 137

13, 137

are all prime numbers, therefore no further factorization is possible

= 13 \cdot 13 \cdot 137

= 1 3^{2} \cdot 137

= 1 3^{2} \cdot 137

Apply radical rule:

= 137 1 3^{2}

Apply radical rule:

1 3^{2} = 13

= 13137

= \frac{13 137}{109}

u_{1, 2} = \frac{- ( - \frac{63}{109} ) \pm \frac{13 137}{109}}{2 \cdot 1}

Separate the solutions

u_{1} = \frac{- ( - \frac{63}{109} ) + \frac{13 137}{109}}{2 \cdot 1}, u_{2} = \frac{- ( - \frac{63}{109} ) - \frac{13 137}{109}}{2 \cdot 1}

u = \frac{- ( - \frac{63}{109} ) + \frac{13 137}{109}}{2 \cdot 1} : \frac{63 + 13 137}{218}

\frac{- ( - \frac{63}{109} ) + \frac{13 137}{109}}{2 \cdot 1}

Apply rule

- (- a) = a

= \frac{\frac{63}{109} + \frac{13 137}{109}}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{\frac{63}{109} + \frac{13 137}{109}}{2}

Combine the fractions

\frac{63}{109} + \frac{13 137}{109} : \frac{63 + 13 137}{109}

Apply rule

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

= \frac{63 + 13 137}{109}

= \frac{\frac{63 + 13 137}{109}}{2}

Apply the fraction rule:

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

= \frac{63 + 13 137}{109 \cdot 2}

Multiply the numbers:

109 \cdot 2 = 218

= \frac{63 + 13 137}{218}

u = \frac{- ( - \frac{63}{109} ) - \frac{13 137}{109}}{2 \cdot 1} : \frac{63 - 13 137}{218}

\frac{- ( - \frac{63}{109} ) - \frac{13 137}{109}}{2 \cdot 1}

Apply rule

- (- a) = a

= \frac{\frac{63}{109} - \frac{13 137}{109}}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{\frac{63}{109} - \frac{13 137}{109}}{2}

Combine the fractions

\frac{63}{109} - \frac{13 137}{109} : \frac{63 - 13 137}{109}

Apply rule

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

= \frac{63 - 13 137}{109}

= \frac{\frac{63 - 13 137}{109}}{2}

Apply the fraction rule:

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

= \frac{63 - 13 137}{109 \cdot 2}

Multiply the numbers:

109 \cdot 2 = 218

= \frac{63 - 13 137}{218}

The solutions to the quadratic equation are:

u = \frac{63 + 13 137}{218}, u = \frac{63 - 13 137}{218}

Substitute back

u = sin (x)

sin (x) = \frac{63 + 13 137}{218}, sin (x) = \frac{63 - 13 137}{218}

sin (x) = \frac{63 + 13 137}{218}, sin (x) = \frac{63 - 13 137}{218}

sin (x) = \frac{63 + 13 137}{218} : x = arcsin (\frac{63 + 13 137}{218}) + 2 πn, x = π - arcsin (\frac{63 + 13 137}{218}) + 2 πn

sin (x) = \frac{63 + 13 137}{218}

Apply trig inverse properties

sin (x) = \frac{63 + 13 137}{218}

General solutions for

sin (x) = \frac{63 + 13 137}{218}

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

x = arcsin (\frac{63 + 13 137}{218}) + 2 πn, x = π - arcsin (\frac{63 + 13 137}{218}) + 2 πn

x = arcsin (\frac{63 + 13 137}{218}) + 2 πn, x = π - arcsin (\frac{63 + 13 137}{218}) + 2 πn

sin (x) = \frac{63 - 13 137}{218} : x = arcsin (\frac{63 - 13 137}{218}) + 2 πn, x = π + arcsin (- \frac{63 - 13 137}{218}) + 2 πn

sin (x) = \frac{63 - 13 137}{218}

Apply trig inverse properties

sin (x) = \frac{63 - 13 137}{218}

General solutions for

sin (x) = \frac{63 - 13 137}{218}

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

x = arcsin (\frac{63 - 13 137}{218}) + 2 πn, x = π + arcsin (- \frac{63 - 13 137}{218}) + 2 πn

x = arcsin (\frac{63 - 13 137}{218}) + 2 πn, x = π + arcsin (- \frac{63 - 13 137}{218}) + 2 πn

Combine all the solutions

x = arcsin (\frac{63 + 13 137}{218}) + 2 πn, x = π - arcsin (\frac{63 + 13 137}{218}) + 2 πn, x = arcsin (\frac{63 - 13 137}{218}) + 2 πn, x = π + arcsin (- \frac{63 - 13 137}{218}) + 2 πn

Verify solutions by plugging them into the original equation

Check the solutions by plugging them into

90 - 70 sin (x) - 130 cos (x) = 0

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

Check the solution

arcsin (\frac{63 + 13 137}{218}) + 2 πn :

True

arcsin (\frac{63 + 13 137}{218}) + 2 πn

Plug in

n = 1

arcsin (\frac{63 + 13 137}{218}) + 2 π 1

For

90 - 70 sin (x) - 130 cos (x) = 0

plug in

x = arcsin (\frac{63 + 13 137}{218}) + 2 π 1

90 - 70 sin (arcsin (\frac{63 + 13 137}{218}) + 2 π 1) - 130 cos (arcsin (\frac{63 + 13 137}{218}) + 2 π 1) = 0

Refine

0 = 0

\Rightarrow True

Check the solution

π - arcsin (\frac{63 + 13 137}{218}) + 2 πn :

False

π - arcsin (\frac{63 + 13 137}{218}) + 2 πn

Plug in

n = 1

π - arcsin (\frac{63 + 13 137}{218}) + 2 π 1

For

90 - 70 sin (x) - 130 cos (x) = 0

plug in

x = π - arcsin (\frac{63 + 13 137}{218}) + 2 π 1

90 - 70 sin (π - arcsin (\frac{63 + 13 137}{218}) + 2 π 1) - 130 cos (π - arcsin (\frac{63 + 13 137}{218}) + 2 π 1) = 0

Refine

41.82314 \dots = 0

\Rightarrow False

Check the solution

arcsin (\frac{63 - 13 137}{218}) + 2 πn :

True

arcsin (\frac{63 - 13 137}{218}) + 2 πn

Plug in

n = 1

arcsin (\frac{63 - 13 137}{218}) + 2 π 1

For

90 - 70 sin (x) - 130 cos (x) = 0

plug in

x = arcsin (\frac{63 - 13 137}{218}) + 2 π 1

90 - 70 sin (arcsin (\frac{63 - 13 137}{218}) + 2 π 1) - 130 cos (arcsin (\frac{63 - 13 137}{218}) + 2 π 1) = 0

Refine

0 = 0

\Rightarrow True

Check the solution

π + arcsin (- \frac{63 - 13 137}{218}) + 2 πn :

False

π + arcsin (- \frac{63 - 13 137}{218}) + 2 πn

Plug in

n = 1

π + arcsin (- \frac{63 - 13 137}{218}) + 2 π 1

For

90 - 70 sin (x) - 130 cos (x) = 0

plug in

x = π + arcsin (- \frac{63 - 13 137}{218}) + 2 π 1

90 - 70 sin (π + arcsin (- \frac{63 - 13 137}{218}) + 2 π 1) - 130 cos (π + arcsin (- \frac{63 - 13 137}{218}) + 2 π 1) = 0

Refine

237.25942 \dots = 0

\Rightarrow False

x = arcsin (\frac{63 + 13 137}{218}) + 2 πn, x = arcsin (\frac{63 - 13 137}{218}) + 2 πn

Show solutions in decimal form

x = 1.40923 \dots + 2 πn, x = - 0.42135 \dots + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for 90-70sin(x)-130cos(x)=0 ?
The general solution for 90-70sin(x)-130cos(x)=0 is x=1.40923…+2pin,x=-0.42135…+2pin