What is the general solution for sin^2(x)sec(x)cot(x)=(sqrt(3))/2 ?

The general solution for sin^2(x)sec(x)cot(x)=(sqrt(3))/2 is x= pi/3+2pin,x=(2pi)/3+2pin

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Popular Trigonometry >

sin^2(x)sec(x)cot(x)=(sqrt(3))/2

Solution

sin^{2} (x) sec (x) cot (x) = \frac{3}{2}

Solution

x = \frac{π}{3} + 2 πn, x = \frac{2 π}{3} + 2 πn

Degrees

x = 6 0^{\circ} + 36 0^{\circ} n, x = 12 0^{\circ} + 36 0^{\circ} n

Solution steps

sin^{2} (x) sec (x) cot (x) = \frac{3}{2}

Subtract

\frac{3}{2}

from both sides

sin^{2} (x) sec (x) cot (x) - \frac{3}{2} = 0

Simplify

sin^{2} (x) sec (x) cot (x) - \frac{3}{2} : \frac{2 sin ^{2} ( x ) sec ( x ) cot ( x ) - 3}{2}

sin^{2} (x) sec (x) cot (x) - \frac{3}{2}

Convert element to fraction:

sin^{2} (x) sec (x) cot (x) = \frac{sin ^{2} ( x ) sec ( x ) cot ( x ) 2}{2}

= \frac{sin ^{2} ( x ) sec ( x ) cot ( x ) \cdot 2}{2} - \frac{3}{2}

Since the denominators are equal, combine the fractions:

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

= \frac{sin ^{2} ( x ) sec ( x ) cot ( x ) \cdot 2 - 3}{2}

\frac{2 sin ^{2} ( x ) sec ( x ) cot ( x ) - 3}{2} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

2 sin^{2} (x) sec (x) cot (x) - 3 = 0

Express with sin, cos

- 3 + 2 cot (x) sec (x) sin^{2} (x)

Use the basic trigonometric identity:

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

= - 3 + 2 \cdot \frac{cos ( x )}{sin ( x )} sec (x) sin^{2} (x)

Use the basic trigonometric identity:

sec (x) = \frac{1}{cos ( x )}

= - 3 + 2 \cdot \frac{cos ( x )}{sin ( x )} \cdot \frac{1}{cos ( x )} sin^{2} (x)

Simplify

- 3 + 2 \cdot \frac{cos ( x )}{sin ( x )} \cdot \frac{1}{cos ( x )} sin^{2} (x) : - 3 + 2 sin (x)

- 3 + 2 \cdot \frac{cos ( x )}{sin ( x )} \cdot \frac{1}{cos ( x )} sin^{2} (x)

2 \cdot \frac{cos ( x )}{sin ( x )} \cdot \frac{1}{cos ( x )} sin^{2} (x) = 2 sin (x)

2 \cdot \frac{cos ( x )}{sin ( x )} \cdot \frac{1}{cos ( x )} sin^{2} (x)

Multiply fractions:

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

= \frac{cos ( x ) \cdot 1 \cdot 2 sin ^{2} ( x )}{sin ( x ) cos ( x )}

Cancel the common factor:

cos (x)

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

Multiply the numbers:

1 \cdot 2 = 2

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

Cancel the common factor:

sin (x)

= 2 sin (x)

= - 3 + 2 sin (x)

= - 3 + 2 sin (x)

- 3 + 2 sin (x) = 0

Move

3

to the right side

- 3 + 2 sin (x) = 0

Add

3

to both sides

- 3 + 2 sin (x) + 3 = 0 + 3

Simplify

2 sin (x) = 3

2 sin (x) = 3

Divide both sides by

2

2 sin (x) = 3

Divide both sides by

2

\frac{2 sin ( x )}{2} = \frac{3}{2}

Simplify

sin (x) = \frac{3}{2}

sin (x) = \frac{3}{2}

General solutions for

sin (x) = \frac{3}{2}

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 = \frac{π}{3} + 2 πn, x = \frac{2 π}{3} + 2 πn

x = \frac{π}{3} + 2 πn, x = \frac{2 π}{3} + 2 πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for sin^2(x)sec(x)cot(x)=(sqrt(3))/2 ?
The general solution for sin^2(x)sec(x)cot(x)=(sqrt(3))/2 is x= pi/3+2pin,x=(2pi)/3+2pin