What is the general solution for 7sin(x)cos(x)+cos^2(x)=1 ?

The general solution for 7sin(x)cos(x)+cos^2(x)=1 is x=2pin,x=pi+2pin,x=1.42889…+pin

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7sin(x)cos(x)+cos^2(x)=1

Solution

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

Solution

x = 2 πn, x = π + 2 πn, x = 1.42889 \dots + πn

Degrees

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = 81.86989 \dots^{\circ} + 18 0^{\circ} n

Solution steps

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

Subtract

1

from both sides

7 sin (x) cos (x) + cos^{2} (x) - 1 = 0

Rewrite using trig identities

- 1 + cos^{2} (x) + 7 cos (x) sin (x)

Use the Pythagorean identity:

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

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

= 7 cos (x) sin (x) - sin^{2} (x)

- sin^{2} (x) + 7 cos (x) sin (x) = 0

Factor

- sin^{2} (x) + 7 cos (x) sin (x) : sin (x) (- sin (x) + 7 cos (x))

- sin^{2} (x) + 7 cos (x) sin (x)

Apply exponent rule:

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

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

= - sin (x) sin (x) + 7 sin (x) cos (x)

Factor out common term

sin (x)

= sin (x) (- sin (x) + 7 cos (x))

sin (x) (- sin (x) + 7 cos (x)) = 0

Solving each part separately

sin (x) = 0 or - sin (x) + 7 cos (x) = 0

sin (x) = 0 : x = 2 πn, x = π + 2 πn

sin (x) = 0

General solutions for

sin (x) = 0

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 2 πn

Solve

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn, x = π + 2 πn

- sin (x) + 7 cos (x) = 0 : x = arctan (7) + πn

- sin (x) + 7 cos (x) = 0

Rewrite using trig identities

- sin (x) + 7 cos (x) = 0

Divide both sides by

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

\frac{- sin ( x ) + 7 cos ( x )}{cos ( x )} = \frac{0}{cos ( x )}

Simplify

- \frac{sin ( x )}{cos ( x )} + 7 = 0

Use the basic trigonometric identity:

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

- tan (x) + 7 = 0

- tan (x) + 7 = 0

Move

7

to the right side

- tan (x) + 7 = 0

Subtract

7

from both sides

- tan (x) + 7 - 7 = 0 - 7

Simplify

- tan (x) = - 7

- tan (x) = - 7

Divide both sides by

- 1

- tan (x) = - 7

Divide both sides by

- 1

\frac{- tan ( x )}{- 1} = \frac{- 7}{- 1}

Simplify

tan (x) = 7

tan (x) = 7

Apply trig inverse properties

tan (x) = 7

General solutions for

tan (x) = 7

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

x = arctan (7) + πn

x = arctan (7) + πn

Combine all the solutions

x = 2 πn, x = π + 2 πn, x = arctan (7) + πn

Show solutions in decimal form

x = 2 πn, x = π + 2 πn, x = 1.42889 \dots + πn

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Frequently Asked Questions (FAQ)

What is the general solution for 7sin(x)cos(x)+cos^2(x)=1 ?
The general solution for 7sin(x)cos(x)+cos^2(x)=1 is x=2pin,x=pi+2pin,x=1.42889…+pin