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Popular Trigonometry >

3sin(x)=2sec(x)tan(x)

Solution

3 sin (x) = 2 sec (x) tan (x)

Solution

x = 2 πn, x = π + 2 πn, x = 2.52611 \dots + 2 πn, x = - 2.52611 \dots + 2 πn, x = 0.61547 \dots + 2 πn, x = 2 π - 0.61547 \dots + 2 πn

Degrees

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = 144.73561 \dots^{\circ} + 36 0^{\circ} n, x = - 144.73561 \dots^{\circ} + 36 0^{\circ} n, x = 35.26438 \dots^{\circ} + 36 0^{\circ} n, x = 324.73561 \dots^{\circ} + 36 0^{\circ} n

Solution steps

3 sin (x) = 2 sec (x) tan (x)

Subtract

2 sec (x) tan (x)

from both sides

3 sin (x) - 2 sec (x) tan (x) = 0

Express with sin, cos

3 sin (x) - 2 sec (x) tan (x)

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

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

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

Simplify

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

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

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

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

Multiply fractions:

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

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

Multiply the numbers:

1 \cdot 2 = 2

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

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

cos (x) cos (x)

Apply exponent rule:

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

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

= cos^{1 + 1} (x)

Add the numbers:

1 + 1 = 2

= cos^{2} (x)

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

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

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

\frac{- 2 sin ( x ) + 3 cos ^{2} ( x ) sin ( x )}{cos ^{2} ( x )} = 0

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

- 2 sin (x) + 3 cos^{2} (x) sin (x) = 0

Factor

- 2 sin (x) + 3 cos^{2} (x) sin (x) : sin (x) (3 cos (x) + 2) (3 cos (x) - 2)

- 2 sin (x) + 3 cos^{2} (x) sin (x)

Factor out common term

sin (x)

= sin (x) (- 2 + 3 cos^{2} (x))

Factor

3 cos^{2} (x) - 2 : (3 cos (x) + 2) (3 cos (x) - 2)

3 cos^{2} (x) - 2

Rewrite

3 cos^{2} (x) - 2

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

3 cos^{2} (x) - 2

Apply radical rule:

a = (a)^{2}

3 = (3)^{2}

= (3)^{2} cos^{2} (x) - 2

Apply radical rule:

a = (a)^{2}

2 = (2)^{2}

= (3)^{2} cos^{2} (x) - (2)^{2}

Apply exponent rule:

a^{m} b^{m} = (ab)^{m}

(3)^{2} cos^{2} (x) = (3 cos (x))^{2}

= (3 cos (x))^{2} - (2)^{2}

= (3 cos (x))^{2} - (2)^{2}

Apply Difference of Two Squares Formula:

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

(3 cos (x))^{2} - (2)^{2} = (3 cos (x) + 2) (3 cos (x) - 2)

= (3 cos (x) + 2) (3 cos (x) - 2)

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

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

Solving each part separately

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

3 cos (x) + 2 = 0 : x = arccos (- \frac{2}{3}) + 2 πn, x = - arccos (- \frac{2}{3}) + 2 πn

3 cos (x) + 2 = 0

Move

2

to the right side

3 cos (x) + 2 = 0

Subtract

2

from both sides

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

Simplify

3 cos (x) = - 2

3 cos (x) = - 2

Divide both sides by

3

3 cos (x) = - 2

Divide both sides by

3

\frac{3 cos ( x )}{3} = \frac{- 2}{3}

Simplify

\frac{3 cos ( x )}{3} = \frac{- 2}{3}

Simplify

\frac{3 cos ( x )}{3} : cos (x)

\frac{3 cos ( x )}{3}

Cancel the common factor:

3

= cos (x)

Simplify

\frac{- 2}{3} : - \frac{2}{3}

\frac{- 2}{3}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{2}{3}

Combine same powers :

\frac{x}{y} = \frac{x}{y}

= - \frac{2}{3}

cos (x) = - \frac{2}{3}

cos (x) = - \frac{2}{3}

cos (x) = - \frac{2}{3}

Apply trig inverse properties

cos (x) = - \frac{2}{3}

General solutions for

cos (x) = - \frac{2}{3}

cos (x) = - a \Rightarrow x = arccos (- a) + 2 πn, x = - arccos (- a) + 2 πn

x = arccos (- \frac{2}{3}) + 2 πn, x = - arccos (- \frac{2}{3}) + 2 πn

x = arccos (- \frac{2}{3}) + 2 πn, x = - arccos (- \frac{2}{3}) + 2 πn

3 cos (x) - 2 = 0 : x = arccos (\frac{2}{3}) + 2 πn, x = 2 π - arccos (\frac{2}{3}) + 2 πn

3 cos (x) - 2 = 0

Move

2

to the right side

3 cos (x) - 2 = 0

Add

2

to both sides

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

Simplify

3 cos (x) = 2

3 cos (x) = 2

Divide both sides by

3

3 cos (x) = 2

Divide both sides by

3

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

Simplify

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

Simplify

\frac{3 cos ( x )}{3} : cos (x)

\frac{3 cos ( x )}{3}

Cancel the common factor:

3

= cos (x)

Simplify

\frac{2}{3} : \frac{2}{3}

\frac{2}{3}

Combine same powers :

\frac{x}{y} = \frac{x}{y}

= \frac{2}{3}

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

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

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

Apply trig inverse properties

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

General solutions for

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

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

x = arccos (\frac{2}{3}) + 2 πn, x = 2 π - arccos (\frac{2}{3}) + 2 πn

x = arccos (\frac{2}{3}) + 2 πn, x = 2 π - arccos (\frac{2}{3}) + 2 πn

Combine all the solutions

x = 2 πn, x = π + 2 πn, x = arccos (- \frac{2}{3}) + 2 πn, x = - arccos (- \frac{2}{3}) + 2 πn, x = arccos (\frac{2}{3}) + 2 πn, x = 2 π - arccos (\frac{2}{3}) + 2 πn

Show solutions in decimal form

x = 2 πn, x = π + 2 πn, x = 2.52611 \dots + 2 πn, x = - 2.52611 \dots + 2 πn, x = 0.61547 \dots + 2 πn, x = 2 π - 0.61547 \dots + 2 πn

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