en

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular Trigonometry >

25sin(x)cos(x)-5sin(x)+20cos(x)=4

Solution

25 sin (x) cos (x) - 5 sin (x) + 20 cos (x) = 4

Solution

x = 1.36943 \dots + 2 πn, x = 2 π - 1.36943 \dots + 2 πn, x = - 0.92729 \dots + 2 πn, x = π + 0.92729 \dots + 2 πn

+1

Degrees

x = 78.46304 \dots^{\circ} + 36 0^{\circ} n, x = 281.53695 \dots^{\circ} + 36 0^{\circ} n, x = - 53.13010 \dots^{\circ} + 36 0^{\circ} n, x = 233.13010 \dots^{\circ} + 36 0^{\circ} n

Solution steps

25 sin (x) cos (x) - 5 sin (x) + 20 cos (x) = 4

Subtract

4

from both sides

25 sin (x) cos (x) - 5 sin (x) + 20 cos (x) - 4 = 0

Factor

25 sin (x) cos (x) - 5 sin (x) + 20 cos (x) - 4 : (5 cos (x) - 1) (5 sin (x) + 4)

25 sin (x) cos (x) - 5 sin (x) + 20 cos (x) - 4

= (25 sin (x) cos (x) - 5 sin (x)) + (20 cos (x) - 4)

Factor out

4

from

20 cos (x) - 4 : 4 (5 cos (x) - 1)

20 cos (x) - 4

Rewrite

20

as

5 \cdot 4

= 5 \cdot 4 cos (x) - 4

Factor out common term

4

= 4 (5 cos (x) - 1)

Factor out

5 sin (x)

from

25 sin (x) cos (x) - 5 sin (x) : 5 sin (x) (5 cos (x) - 1)

25 sin (x) cos (x) - 5 sin (x)

Rewrite

25

as

5 \cdot 5

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

Factor out common term

5 sin (x)

= 5 sin (x) (5 cos (x) - 1)

= 4 (5 cos (x) - 1) + 5 sin (x) (5 cos (x) - 1)

Factor out common term

5 cos (x) - 1

= (5 cos (x) - 1) (5 sin (x) + 4)

(5 cos (x) - 1) (5 sin (x) + 4) = 0

Solving each part separately

5 cos (x) - 1 = 0 or 5 sin (x) + 4 = 0

5 cos (x) - 1 = 0 : x = arccos (\frac{1}{5}) + 2 πn, x = 2 π - arccos (\frac{1}{5}) + 2 πn

5 cos (x) - 1 = 0

Move

1

to the right side

5 cos (x) - 1 = 0

Add

1

to both sides

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

Simplify

5 cos (x) = 1

5 cos (x) = 1

Divide both sides by

5

5 cos (x) = 1

Divide both sides by

5

\frac{5 cos ( x )}{5} = \frac{1}{5}

Simplify

cos (x) = \frac{1}{5}

cos (x) = \frac{1}{5}

Apply trig inverse properties

cos (x) = \frac{1}{5}

General solutions for

cos (x) = \frac{1}{5}

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

x = arccos (\frac{1}{5}) + 2 πn, x = 2 π - arccos (\frac{1}{5}) + 2 πn

x = arccos (\frac{1}{5}) + 2 πn, x = 2 π - arccos (\frac{1}{5}) + 2 πn

5 sin (x) + 4 = 0 : x = arcsin (- \frac{4}{5}) + 2 πn, x = π + arcsin (\frac{4}{5}) + 2 πn

5 sin (x) + 4 = 0

Move

4

to the right side

5 sin (x) + 4 = 0

Subtract

4

from both sides

5 sin (x) + 4 - 4 = 0 - 4

Simplify

5 sin (x) = - 4

5 sin (x) = - 4

Divide both sides by

5

5 sin (x) = - 4

Divide both sides by

5

\frac{5 sin ( x )}{5} = \frac{- 4}{5}

Simplify

sin (x) = - \frac{4}{5}

sin (x) = - \frac{4}{5}

Apply trig inverse properties

sin (x) = - \frac{4}{5}

General solutions for

sin (x) = - \frac{4}{5}

sin (x) = - a \Rightarrow x = arcsin (- a) + 2 πn, x = π + arcsin (a) + 2 πn

x = arcsin (- \frac{4}{5}) + 2 πn, x = π + arcsin (\frac{4}{5}) + 2 πn

x = arcsin (- \frac{4}{5}) + 2 πn, x = π + arcsin (\frac{4}{5}) + 2 πn

Combine all the solutions

x = arccos (\frac{1}{5}) + 2 πn, x = 2 π - arccos (\frac{1}{5}) + 2 πn, x = arcsin (- \frac{4}{5}) + 2 πn, x = π + arcsin (\frac{4}{5}) + 2 πn

Show solutions in decimal form

x = 1.36943 \dots + 2 πn, x = 2 π - 1.36943 \dots + 2 πn, x = - 0.92729 \dots + 2 πn, x = π + 0.92729 \dots + 2 πn

Graph

Popular Examples

cos(sqrt(2)T)=1,0<= T<= 2pi

2-2cos^2(x/2)=2cos^2(x)

sec^2(θ)-2=tan^2(θ)

cos(x)=sin(x)+1