What is the general solution for sin^2(x)-15sin(x)cos(x)+50cos^2(x)=0 ?

The general solution for sin^2(x)-15sin(x)cos(x)+50cos^2(x)=0 is x=1.37340…+pin,x=1.47112…+pin

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sin^2(x)-15sin(x)cos(x)+50cos^2(x)=0

Solution

sin^{2} (x) - 15 sin (x) cos (x) + 50 cos^{2} (x) = 0

Solution

x = 1.37340 \dots + πn, x = 1.47112 \dots + πn

Degrees

x = 78.69006 \dots^{\circ} + 18 0^{\circ} n, x = 84.28940 \dots^{\circ} + 18 0^{\circ} n

Solution steps

sin^{2} (x) - 15 sin (x) cos (x) + 50 cos^{2} (x) = 0

Factor

sin^{2} (x) - 15 sin (x) cos (x) + 50 cos^{2} (x) : (sin (x) - 5 cos (x)) (sin (x) - 10 cos (x))

sin^{2} (x) - 15 sin (x) cos (x) + 50 cos^{2} (x)

Break the expression into groups

sin^{2} (x) - 15 sin (x) cos (x) + 50 cos^{2} (x)

Definition

Factors of

50 : 1, 2, 5, 10, 25, 50

50

Divisors (Factors)

Find the Prime factors of

50 : 2, 5, 5

50

50

divides by

250 = 25 \cdot 2

= 2 \cdot 25

25

divides by

525 = 5 \cdot 5

= 2 \cdot 5 \cdot 5

2, 5

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 5 \cdot 5

Multiply the prime factors of

50 : 10, 25

2 \cdot 5 = 10

5 \cdot 5 = 25

10, 25

10, 25

Add the prime factors:

2, 5

Add 1 and the number

50

itself

1, 50

The factors of

50

1, 2, 5, 10, 25, 50

Negative factors of

50 : - 1, - 2, - 5, - 10, - 25, - 50

Multiply the factors by

- 1

to get the negative factors

- 1, - 2, - 5, - 10, - 25, - 50

For every two factors such that

u * v = 50,

check if

u + v = - 15

Check

u = 1, v = 50 : u * v = 50, u + v = 51 \Rightarrow

FalseCheck

u = 2, v = 25 : u * v = 50, u + v = 27 \Rightarrow

False

u = - 5, v = - 10

Group into

(a x^{2} + ux y) + (vx y + c y^{2})

(sin^{2} (x) - 5 sin (x) cos (x)) + (- 10 sin (x) cos (x) + 50 cos^{2} (x))

= (sin^{2} (x) - 5 sin (x) cos (x)) + (- 10 sin (x) cos (x) + 50 cos^{2} (x))

Factor out

sin (x)

from

sin^{2} (x) - 5 sin (x) cos (x) : sin (x) (sin (x) - 5 cos (x))

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

Apply exponent rule:

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

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

= sin (x) sin (x) - 5 sin (x) cos (x)

Factor out common term

sin (x)

= sin (x) (sin (x) - 5 cos (x))

Factor out

- 10 cos (x)

from

- 10 sin (x) cos (x) + 50 cos^{2} (x) : - 10 cos (x) (sin (x) - 5 cos (x))

- 10 sin (x) cos (x) + 50 cos^{2} (x)

Apply exponent rule:

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

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

= - 10 sin (x) cos (x) + 50 cos (x) cos (x)

Rewrite

50

5 \cdot 10

= - 10 sin (x) cos (x) + 5 \cdot 10 cos (x) cos (x)

Factor out common term

- 10 cos (x)

= - 10 cos (x) (sin (x) - 5 cos (x))

= sin (x) (sin (x) - 5 cos (x)) - 10 cos (x) (sin (x) - 5 cos (x))

Factor out common term

sin (x) - 5 cos (x)

= (sin (x) - 5 cos (x)) (sin (x) - 10 cos (x))

(sin (x) - 5 cos (x)) (sin (x) - 10 cos (x)) = 0

Solving each part separately

sin (x) - 5 cos (x) = 0 or sin (x) - 10 cos (x) = 0

sin (x) - 5 cos (x) = 0 : x = arctan (5) + πn

sin (x) - 5 cos (x) = 0

Rewrite using trig identities

sin (x) - 5 cos (x) = 0

Divide both sides by

cos (x), cos (x) \neq = 0

\frac{sin ( x ) - 5 cos ( x )}{cos ( x )} = \frac{0}{cos ( x )}

Simplify

\frac{sin ( x )}{cos ( x )} - 5 = 0

Use the basic trigonometric identity:

\frac{sin ( x )}{cos ( x )} = tan (x)

tan (x) - 5 = 0

tan (x) - 5 = 0

Move

5

to the right side

tan (x) - 5 = 0

Add

5

to both sides

tan (x) - 5 + 5 = 0 + 5

Simplify

tan (x) = 5

tan (x) = 5

Apply trig inverse properties

tan (x) = 5

General solutions for

tan (x) = 5

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

x = arctan (5) + πn

x = arctan (5) + πn

sin (x) - 10 cos (x) = 0 : x = arctan (10) + πn

sin (x) - 10 cos (x) = 0

Rewrite using trig identities

sin (x) - 10 cos (x) = 0

Divide both sides by

cos (x), cos (x) \neq = 0

\frac{sin ( x ) - 10 cos ( x )}{cos ( x )} = \frac{0}{cos ( x )}

Simplify

\frac{sin ( x )}{cos ( x )} - 10 = 0

Use the basic trigonometric identity:

\frac{sin ( x )}{cos ( x )} = tan (x)

tan (x) - 10 = 0

tan (x) - 10 = 0

Move

10

to the right side

tan (x) - 10 = 0

Add

10

to both sides

tan (x) - 10 + 10 = 0 + 10

Simplify

tan (x) = 10

tan (x) = 10

Apply trig inverse properties

tan (x) = 10

General solutions for

tan (x) = 10

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

x = arctan (10) + πn

x = arctan (10) + πn

Combine all the solutions

x = arctan (5) + πn, x = arctan (10) + πn

Show solutions in decimal form

x = 1.37340 \dots + πn, x = 1.47112 \dots + πn

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Frequently Asked Questions (FAQ)

What is the general solution for sin^2(x)-15sin(x)cos(x)+50cos^2(x)=0 ?
The general solution for sin^2(x)-15sin(x)cos(x)+50cos^2(x)=0 is x=1.37340…+pin,x=1.47112…+pin