What is the general solution for cos(90-x)+csc(x)= 5/2 ?

The general solution for cos(90-x)+csc(x)= 5/2 is x=30+360n,x=150+360n

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Popular Trigonometry >

cos(90-x)+csc(x)= 5/2

Solution

cos (9 0^{\circ} - x) + csc (x) = \frac{5}{2}

Solution

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

Radians

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

Solution steps

cos (9 0^{\circ} - x) + csc (x) = \frac{5}{2}

Rewrite using trig identities

cos (9 0^{\circ} - x) + csc (x) = \frac{5}{2}

Rewrite using trig identities

cos (9 0^{\circ} - x)

Use the Angle Difference identity:

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

= cos (9 0^{\circ}) cos (x) + sin (9 0^{\circ}) sin (x)

Simplify

cos (9 0^{\circ}) cos (x) + sin (9 0^{\circ}) sin (x) : sin (x)

cos (9 0^{\circ}) cos (x) + sin (9 0^{\circ}) sin (x)

cos (9 0^{\circ}) cos (x) = 0

cos (9 0^{\circ}) cos (x)

Simplify

cos (9 0^{\circ}) : 0

cos (9 0^{\circ})

Use the following trivial identity:

cos (9 0^{\circ}) = 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}

= 0

= 0 \cdot cos (x)

Apply rule

0 \cdot a = 0

= 0

sin (9 0^{\circ}) sin (x) = sin (x)

sin (9 0^{\circ}) sin (x)

Simplify

sin (9 0^{\circ}) : 1

sin (9 0^{\circ})

Use the following trivial identity:

sin (9 0^{\circ}) = 1

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} 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 sin (x)

Multiply:

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

= sin (x)

= 0 + sin (x)

0 + sin (x) = sin (x)

= sin (x)

= sin (x)

sin (x) + csc (x) = \frac{5}{2}

sin (x) + csc (x) = \frac{5}{2}

Subtract

\frac{5}{2}

from both sides

sin (x) + csc (x) - \frac{5}{2} = 0

Simplify

sin (x) + csc (x) - \frac{5}{2} : \frac{2 sin ( x ) + 2 csc ( x ) - 5}{2}

sin (x) + csc (x) - \frac{5}{2}

Convert element to fraction:

sin (x) = \frac{sin ( x ) 2}{2}, csc (x) = \frac{csc ( x ) 2}{2}

= \frac{sin ( x ) \cdot 2}{2} + \frac{csc ( x ) \cdot 2}{2} - \frac{5}{2}

Since the denominators are equal, combine the fractions:

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

= \frac{sin ( x ) \cdot 2 + csc ( x ) \cdot 2 - 5}{2}

\frac{2 sin ( x ) + 2 csc ( x ) - 5}{2} = 0

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

2 sin (x) + 2 csc (x) - 5 = 0

Rewrite using trig identities

- 5 + 2 csc (x) + 2 sin (x)

Use the basic trigonometric identity:

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

= - 5 + 2 csc (x) + 2 \cdot \frac{1}{csc ( x )}

2 \cdot \frac{1}{csc ( x )} = \frac{2}{csc ( x )}

2 \cdot \frac{1}{csc ( x )}

Multiply fractions:

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

= \frac{1 \cdot 2}{csc ( x )}

Multiply the numbers:

1 \cdot 2 = 2

= \frac{2}{csc ( x )}

= - 5 + 2 csc (x) + \frac{2}{csc ( x )}

- 5 + \frac{2}{csc ( x )} + 2 csc (x) = 0

Solve by substitution

- 5 + \frac{2}{csc ( x )} + 2 csc (x) = 0

Let:

csc (x) = u

- 5 + \frac{2}{u} + 2 u = 0

- 5 + \frac{2}{u} + 2 u = 0 : u = 2, u = \frac{1}{2}

- 5 + \frac{2}{u} + 2 u = 0

Multiply both sides by

u

- 5 + \frac{2}{u} + 2 u = 0

Multiply both sides by

u

- 5 u + \frac{2}{u} u + 2 uu = 0 \cdot u

Simplify

- 5 u + \frac{2}{u} u + 2 uu = 0 \cdot u

Simplify

\frac{2}{u} u : 2

\frac{2}{u} u

Multiply fractions:

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

= \frac{2 u}{u}

Cancel the common factor:

u

= 2

Simplify

2 uu : 2 u^{2}

2 uu

Apply exponent rule:

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

uu = u^{1 + 1}

= 2 u^{1 + 1}

Add the numbers:

1 + 1 = 2

= 2 u^{2}

Simplify

0 \cdot u : 0

0 \cdot u

Apply rule

0 \cdot a = 0

= 0

- 5 u + 2 + 2 u^{2} = 0

- 5 u + 2 + 2 u^{2} = 0

- 5 u + 2 + 2 u^{2} = 0

Solve

- 5 u + 2 + 2 u^{2} = 0 : u = 2, u = \frac{1}{2}

- 5 u + 2 + 2 u^{2} = 0

Write in the standard form

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

2 u^{2} - 5 u + 2 = 0

Solve with the quadratic formula

2 u^{2} - 5 u + 2 = 0

Quadratic Equation Formula:

For

a = 2, b = - 5, c = 2

u_{1, 2} = \frac{- ( - 5 ) \pm ( - 5 ) ^{2} - 4 \cdot 2 \cdot 2}{2 \cdot 2}

u_{1, 2} = \frac{- ( - 5 ) \pm ( - 5 ) ^{2} - 4 \cdot 2 \cdot 2}{2 \cdot 2}

(- 5)^{2} - 4 \cdot 2 \cdot 2 = 3

(- 5)^{2} - 4 \cdot 2 \cdot 2

Apply exponent rule:

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

n

is even

(- 5)^{2} = 5^{2}

= 5^{2} - 4 \cdot 2 \cdot 2

Multiply the numbers:

4 \cdot 2 \cdot 2 = 16

= 5^{2} - 16

5^{2} = 25

= 25 - 16

Subtract the numbers:

25 - 16 = 9

= 9

Factor the number:

9 = 3^{2}

= 3^{2}

Apply radical rule:

3^{2} = 3

= 3

u_{1, 2} = \frac{- ( - 5 ) \pm 3}{2 \cdot 2}

Separate the solutions

u_{1} = \frac{- ( - 5 ) + 3}{2 \cdot 2}, u_{2} = \frac{- ( - 5 ) - 3}{2 \cdot 2}

u = \frac{- ( - 5 ) + 3}{2 \cdot 2} : 2

\frac{- ( - 5 ) + 3}{2 \cdot 2}

Apply rule

- (- a) = a

= \frac{5 + 3}{2 \cdot 2}

Add the numbers:

5 + 3 = 8

= \frac{8}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{8}{4}

Divide the numbers:

\frac{8}{4} = 2

= 2

u = \frac{- ( - 5 ) - 3}{2 \cdot 2} : \frac{1}{2}

\frac{- ( - 5 ) - 3}{2 \cdot 2}

Apply rule

- (- a) = a

= \frac{5 - 3}{2 \cdot 2}

Subtract the numbers:

5 - 3 = 2

= \frac{2}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{2}{4}

Cancel the common factor:

2

= \frac{1}{2}

The solutions to the quadratic equation are:

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

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

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

- 5 + \frac{2}{u} + 2 u

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

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

Substitute back

u = csc (x)

csc (x) = 2, csc (x) = \frac{1}{2}

csc (x) = 2, csc (x) = \frac{1}{2}

csc (x) = 2 : x = 3 0^{\circ} + 36 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n

csc (x) = 2

General solutions for

csc (x) = 2

csc (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} csc (x) Undefiend 22 \frac{2 3}{3} 1 \frac{2 3}{3} 22 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} csc (x) Undefiend - 2 - 2 - \frac{2 3}{3} - 1 - \frac{2 3}{3} - 2 - 2

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

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

csc (x) = \frac{1}{2} :

No Solution

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

csc (x) \leq - 1 or csc (x) \geq 1

No Solution

Combine all the solutions

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

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for cos(90-x)+csc(x)= 5/2 ?
The general solution for cos(90-x)+csc(x)= 5/2 is x=30+360n,x=150+360n