What is the general solution for 14sin(x+pi/2)+21tan(pi-x)=0 ?

The general solution for 14sin(x+pi/2)+21tan(pi-x)=0 is x= pi/6+2pin,x=(5pi)/6+2pin

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Popular Trigonometry >

14sin(x+pi/2)+21tan(pi-x)=0

Solution

14 sin (x + \frac{π}{2}) + 21 tan (π - x) = 0

Solution

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

Degrees

x = 3 0^{\circ} + 36 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n

Solution steps

14 sin (x + \frac{π}{2}) + 21 tan (π - x) = 0

Rewrite using trig identities

14 sin (x + \frac{π}{2}) + 21 tan (π - x) = 0

Rewrite using trig identities

sin (x + \frac{π}{2})

Use the Angle Sum identity:

sin (s + t) = sin (s) cos (t) + cos (s) sin (t)

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

Simplify

sin (x) cos (\frac{π}{2}) + cos (x) sin (\frac{π}{2}) : cos (x)

sin (x) cos (\frac{π}{2}) + cos (x) sin (\frac{π}{2})

sin (x) cos (\frac{π}{2}) = 0

sin (x) cos (\frac{π}{2})

Simplify

cos (\frac{π}{2}) : 0

cos (\frac{π}{2})

Use the following trivial identity:

cos (\frac{π}{2}) = 0

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= 0

= 0 \cdot sin (x)

Apply rule

0 \cdot a = 0

= 0

cos (x) sin (\frac{π}{2}) = cos (x)

cos (x) sin (\frac{π}{2})

Simplify

sin (\frac{π}{2}) : 1

sin (\frac{π}{2})

Use the following trivial identity:

sin (\frac{π}{2}) = 1

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}

= 1

= 1 \cdot cos (x)

Multiply:

cos (x) \cdot 1 = cos (x)

= cos (x)

= 0 + cos (x)

0 + cos (x) = cos (x)

= cos (x)

= cos (x)

Use the basic trigonometric identity:

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

= \frac{sin ( π - x )}{cos ( π - x )}

Use the Angle Difference identity:

sin (s - t) = sin (s) cos (t) - cos (s) sin (t)

= \frac{sin ( π ) cos ( x ) - cos ( π ) sin ( x )}{cos ( π - x )}

Use the Angle Difference identity:

cos (s - t) = cos (s) cos (t) + sin (s) sin (t)

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

Simplify

\frac{sin ( π ) cos ( x ) - cos ( π ) sin ( x )}{cos ( π ) cos ( x ) + sin ( π ) sin ( x )} : - \frac{sin ( x )}{cos ( x )}

\frac{sin ( π ) cos ( x ) - cos ( π ) sin ( x )}{cos ( π ) cos ( x ) + sin ( π ) sin ( x )}

sin (π) cos (x) - cos (π) sin (x) = sin (x)

sin (π) cos (x) - cos (π) sin (x)

sin (π) cos (x) = 0

sin (π) cos (x)

Simplify

sin (π) : 0

sin (π)

Use the following trivial identity:

sin (π) = 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}

= 0

= 0 \cdot cos (x)

Apply rule

0 \cdot a = 0

= 0

cos (π) sin (x) = - sin (x)

cos (π) sin (x)

Simplify

cos (π) : - 1

cos (π)

Use the following trivial identity:

cos (π) = (- 1)

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= - 1

= - 1 \cdot sin (x)

Multiply:

1 \cdot sin (x) = sin (x)

= - sin (x)

= 0 - (- sin (x))

Refine

= sin (x)

= \frac{sin ( x )}{cos ( π ) cos ( x ) + sin ( π ) sin ( x )}

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

cos (π) cos (x) + sin (π) sin (x)

cos (π) cos (x) = - cos (x)

cos (π) cos (x)

Simplify

cos (π) : - 1

cos (π)

Use the following trivial identity:

cos (π) = (- 1)

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= - 1

= - 1 \cdot cos (x)

Multiply:

1 \cdot cos (x) = cos (x)

= - cos (x)

= - cos (x) + sin (π) sin (x)

sin (π) sin (x) = 0

sin (π) sin (x)

Simplify

sin (π) : 0

sin (π)

Use the following trivial identity:

sin (π) = 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}

= 0

= 0 \cdot sin (x)

Apply rule

0 \cdot a = 0

= 0

= - cos (x) + 0

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

= - cos (x)

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

Apply the fraction rule:

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

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

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

14 cos (x) + 21 (- \frac{sin ( x )}{cos ( x )}) = 0

Simplify

14 cos (x) + 21 (- \frac{sin ( x )}{cos ( x )}) : 14 cos (x) - \frac{21 sin ( x )}{cos ( x )}

14 cos (x) + 21 (- \frac{sin ( x )}{cos ( x )})

Remove parentheses:

(- a) = - a

= 14 cos (x) - 21 \cdot \frac{sin ( x )}{cos ( x )}

Multiply

21 \cdot \frac{sin ( x )}{cos ( x )} : \frac{21 sin ( x )}{cos ( x )}

21 \cdot \frac{sin ( x )}{cos ( x )}

Multiply fractions:

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

= \frac{sin ( x ) \cdot 21}{cos ( x )}

= 14 cos (x) - \frac{21 sin ( x )}{cos ( x )}

14 cos (x) - \frac{21 sin ( x )}{cos ( x )} = 0

14 cos (x) - \frac{21 sin ( x )}{cos ( x )} = 0

Simplify

14 cos (x) - \frac{21 sin ( x )}{cos ( x )} : \frac{14 cos ^{2} ( x ) - 21 sin ( x )}{cos ( x )}

14 cos (x) - \frac{21 sin ( x )}{cos ( x )}

Convert element to fraction:

14 cos (x) = \frac{14 cos ( x ) cos ( x )}{cos ( x )}

= \frac{14 cos ( x ) cos ( x )}{cos ( x )} - \frac{21 sin ( x )}{cos ( x )}

Since the denominators are equal, combine the fractions:

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

= \frac{14 cos ( x ) cos ( x ) - 21 sin ( x )}{cos ( x )}

14 cos (x) cos (x) - 21 sin (x) = 14 cos^{2} (x) - 21 sin (x)

14 cos (x) cos (x) - 21 sin (x)

14 cos (x) cos (x) = 14 cos^{2} (x)

14 cos (x) cos (x)

Apply exponent rule:

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

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

= 14 cos^{1 + 1} (x)

Add the numbers:

1 + 1 = 2

= 14 cos^{2} (x)

= 14 cos^{2} (x) - 21 sin (x)

= \frac{14 cos ^{2} ( x ) - 21 sin ( x )}{cos ( x )}

\frac{14 cos ^{2} ( x ) - 21 sin ( x )}{cos ( x )} = 0

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

14 cos^{2} (x) - 21 sin (x) = 0

Rewrite using trig identities

14 cos^{2} (x) - 21 sin (x)

Use the Pythagorean identity:

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

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

= 14 (1 - sin^{2} (x)) - 21 sin (x)

(1 - sin^{2} (x)) \cdot 14 - 21 sin (x) = 0

Solve by substitution

(1 - sin^{2} (x)) \cdot 14 - 21 sin (x) = 0

Let:

sin (x) = u

(1 - u^{2}) \cdot 14 - 21 u = 0

(1 - u^{2}) \cdot 14 - 21 u = 0 : u = - 2, u = \frac{1}{2}

(1 - u^{2}) \cdot 14 - 21 u = 0

Expand

(1 - u^{2}) \cdot 14 - 21 u : 14 - 14 u^{2} - 21 u

(1 - u^{2}) \cdot 14 - 21 u

= 14 (1 - u^{2}) - 21 u

Expand

14 (1 - u^{2}) : 14 - 14 u^{2}

14 (1 - u^{2})

Apply the distributive law:

a (b - c) = ab - a c

a = 14, b = 1, c = u^{2}

= 14 \cdot 1 - 14 u^{2}

Multiply the numbers:

14 \cdot 1 = 14

= 14 - 14 u^{2}

= 14 - 14 u^{2} - 21 u

14 - 14 u^{2} - 21 u = 0

Write in the standard form

a x^{2} + b x + c = 0

- 14 u^{2} - 21 u + 14 = 0

Solve with the quadratic formula

- 14 u^{2} - 21 u + 14 = 0

Quadratic Equation Formula:

For

a = - 14, b = - 21, c = 14

u_{1, 2} = \frac{- ( - 21 ) \pm ( - 21 ) ^{2} - 4 ( - 14 ) \cdot 14}{2 ( - 14 )}

u_{1, 2} = \frac{- ( - 21 ) \pm ( - 21 ) ^{2} - 4 ( - 14 ) \cdot 14}{2 ( - 14 )}

(- 21)^{2} - 4 (- 14) \cdot 14 = 35

(- 21)^{2} - 4 (- 14) \cdot 14

Apply rule

- (- a) = a

= (- 21)^{2} + 4 \cdot 14 \cdot 14

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

(- 21)^{2} = 2 1^{2}

= 2 1^{2} + 4 \cdot 14 \cdot 14

Multiply the numbers:

4 \cdot 14 \cdot 14 = 784

= 2 1^{2} + 784

2 1^{2} = 441

= 441 + 784

Add the numbers:

441 + 784 = 1225

= 1225

Factor the number:

1225 = 3 5^{2}

= 3 5^{2}

Apply radical rule:

3 5^{2} = 35

= 35

u_{1, 2} = \frac{- ( - 21 ) \pm 35}{2 ( - 14 )}

Separate the solutions

u_{1} = \frac{- ( - 21 ) + 35}{2 ( - 14 )}, u_{2} = \frac{- ( - 21 ) - 35}{2 ( - 14 )}

u = \frac{- ( - 21 ) + 35}{2 ( - 14 )} : - 2

\frac{- ( - 21 ) + 35}{2 ( - 14 )}

Remove parentheses:

(- a) = - a, - (- a) = a

= \frac{21 + 35}{- 2 \cdot 14}

Add the numbers:

21 + 35 = 56

= \frac{56}{- 2 \cdot 14}

Multiply the numbers:

2 \cdot 14 = 28

= \frac{56}{- 28}

Apply the fraction rule:

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

= - \frac{56}{28}

Divide the numbers:

\frac{56}{28} = 2

= - 2

u = \frac{- ( - 21 ) - 35}{2 ( - 14 )} : \frac{1}{2}

\frac{- ( - 21 ) - 35}{2 ( - 14 )}

Remove parentheses:

(- a) = - a, - (- a) = a

= \frac{21 - 35}{- 2 \cdot 14}

Subtract the numbers:

21 - 35 = - 14

= \frac{- 14}{- 2 \cdot 14}

Multiply the numbers:

2 \cdot 14 = 28

= \frac{- 14}{- 28}

Apply the fraction rule:

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

= \frac{14}{28}

Cancel the common factor:

14

= \frac{1}{2}

The solutions to the quadratic equation are:

u = - 2, u = \frac{1}{2}

Substitute back

u = sin (x)

sin (x) = - 2, sin (x) = \frac{1}{2}

sin (x) = - 2, sin (x) = \frac{1}{2}

sin (x) = - 2 :

No Solution

sin (x) = - 2

- 1 \leq sin (x) \leq 1

No Solution

sin (x) = \frac{1}{2} : x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

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

General solutions for

sin (x) = \frac{1}{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}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

Combine all the solutions

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for 14sin(x+pi/2)+21tan(pi-x)=0 ?
The general solution for 14sin(x+pi/2)+21tan(pi-x)=0 is x= pi/6+2pin,x=(5pi)/6+2pin