What is the general solution for sec^2(x)cos(x)=cos(x),0<x<360 ?

Q: What is the general solution for sec^2(x)cos(x)=cos(x),0<x<360 ?

The general solution for sec^2(x)cos(x)=cos(x),0<x<360 is x=180

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sec^2(x)cos(x)=cos(x),0<x<360

Solution

sec^{2} (x) cos (x) = cos (x), 0^{\circ} < x < 36 0^{\circ}

Solution

x = 18 0^{\circ}

Radians

x = π

Solution steps

sec^{2} (x) cos (x) = cos (x), 0^{\circ} < x < 36 0^{\circ}

Subtract

cos (x)

from both sides

sec^{2} (x) cos (x) - cos (x) = 0

Factor

sec^{2} (x) cos (x) - cos (x) : cos (x) (sec (x) + 1) (sec (x) - 1)

sec^{2} (x) cos (x) - cos (x)

Factor out common term

cos (x)

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

Factor

sec^{2} (x) - 1 : (sec (x) + 1) (sec (x) - 1)

sec^{2} (x) - 1

Rewrite

1

1^{2}

= sec^{2} (x) - 1^{2}

Apply Difference of Two Squares Formula:

x^{2} - y^{2} = (x + y) (x - y)

sec^{2} (x) - 1^{2} = (sec (x) + 1) (sec (x) - 1)

= (sec (x) + 1) (sec (x) - 1)

= cos (x) (sec (x) + 1) (sec (x) - 1)

cos (x) (sec (x) + 1) (sec (x) - 1) = 0

Solving each part separately

cos (x) = 0 or sec (x) + 1 = 0 or sec (x) - 1 = 0

cos (x) = 0, 0 < x < 36 0^{\circ} : x = 9 0^{\circ}, x = 27 0^{\circ}

cos (x) = 0, 0 < x < 36 0^{\circ}

General solutions for

cos (x) = 0

cos (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n

x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n

Solutions for the range

0 < x < 36 0^{\circ}

x = 9 0^{\circ}, x = 27 0^{\circ}

sec (x) + 1 = 0, 0 < x < 36 0^{\circ} : x = 18 0^{\circ}

sec (x) + 1 = 0, 0 < x < 36 0^{\circ}

Move

1

to the right side

sec (x) + 1 = 0

Subtract

1

from both sides

sec (x) + 1 - 1 = 0 - 1

Simplify

sec (x) = - 1

sec (x) = - 1

General solutions for

sec (x) = - 1

sec (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

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

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

Solutions for the range

0 < x < 36 0^{\circ}

x = 18 0^{\circ}

sec (x) - 1 = 0, 0 < x < 36 0^{\circ} :

No Solution

sec (x) - 1 = 0, 0 < x < 36 0^{\circ}

Move

1

to the right side

sec (x) - 1 = 0

Add

1

to both sides

sec (x) - 1 + 1 = 0 + 1

Simplify

sec (x) = 1

sec (x) = 1

General solutions for

sec (x) = 1

sec (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

x = 0 + 36 0^{\circ} n

x = 0 + 36 0^{\circ} n

Solve

x = 0 + 36 0^{\circ} n : x = 36 0^{\circ} n

x = 0 + 36 0^{\circ} n

0 + 36 0^{\circ} n = 36 0^{\circ} n

x = 36 0^{\circ} n

x = 36 0^{\circ} n

Solutions for the range

0 < x < 36 0^{\circ}

No Solution

Combine all the solutions

x = 9 0^{\circ}, x = 27 0^{\circ}, x = 18 0^{\circ}

Since the equation is undefined for:

9 0^{\circ}, 27 0^{\circ}

x = 18 0^{\circ}

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Frequently Asked Questions (FAQ)

What is the general solution for sec^2(x)cos(x)=cos(x),0<x<360 ?
The general solution for sec^2(x)cos(x)=cos(x),0<x<360 is x=180