What is the general solution for sin(x)+cos(x)=1,-360<= x<= 360 ?

Q: What is the general solution for sin(x)+cos(x)=1,-360<= x<= 360 ?

The general solution for sin(x)+cos(x)=1,-360<= x<= 360 is x=0,x=90,x=360,x=-360,x=-270

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sin(x)+cos(x)=1,-360<= x<= 360

Solution

sin (x) + cos (x) = 1, - 36 0^{\circ} \leq x \leq 36 0^{\circ}

Solution

x = 0, x = 9 0^{\circ}, x = 36 0^{\circ}, x = - 36 0^{\circ}, x = - 27 0^{\circ}

Radians

x = 0, x = \frac{π}{2}, x = 2 π, x = - 2 π, x = - \frac{3 π}{2}

Solution steps

sin (x) + cos (x) = 1, - 36 0^{\circ} \leq x \leq 36 0^{\circ}

Rewrite using trig identities

sin (x) + cos (x)

sin (x) + cos (x) = 2 sin (x + 4 5^{\circ})

sin (x) + cos (x)

Rewrite as

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

Use the following trivial identity:

cos (\frac{π}{4}) = \frac{1}{2}

Use the following trivial identity:

sin (\frac{π}{4}) = \frac{1}{2}

= 2 (cos (4 5^{\circ}) sin (x) + sin (4 5^{\circ}) cos (x))

Use the Angle Sum identity:

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

= 2 sin (x + 4 5^{\circ})

= 2 sin (x + 4 5^{\circ})

2 sin (x + 4 5^{\circ}) = 1

Divide both sides by

2

2 sin (x + 4 5^{\circ}) = 1

Divide both sides by

2

\frac{2 sin ( x + 4 5 ^{\circ} )}{2} = \frac{1}{2}

Simplify

\frac{2 sin ( x + 4 5 ^{\circ} )}{2} = \frac{1}{2}

Simplify

\frac{2 sin ( x + 4 5 ^{\circ} )}{2} : sin (x + 4 5^{\circ})

\frac{2 sin ( x + 4 5 ^{\circ} )}{2}

Cancel the common factor:

2

= sin (x + 4 5^{\circ})

Simplify

\frac{1}{2} : \frac{2}{2}

\frac{1}{2}

Multiply by the conjugate

\frac{2}{2}

= \frac{1 \cdot 2}{2 2}

1 \cdot 2 = 2

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= \frac{2}{2}

sin (x + 4 5^{\circ}) = \frac{2}{2}

sin (x + 4 5^{\circ}) = \frac{2}{2}

sin (x + 4 5^{\circ}) = \frac{2}{2}

General solutions for

sin (x + 4 5^{\circ}) = \frac{2}{2}

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}

x + 4 5^{\circ} = 4 5^{\circ} + 36 0^{\circ} n, x + 4 5^{\circ} = 13 5^{\circ} + 36 0^{\circ} n

x + 4 5^{\circ} = 4 5^{\circ} + 36 0^{\circ} n, x + 4 5^{\circ} = 13 5^{\circ} + 36 0^{\circ} n

Solve

x + 4 5^{\circ} = 4 5^{\circ} + 36 0^{\circ} n : x = 36 0^{\circ} n

x + 4 5^{\circ} = 4 5^{\circ} + 36 0^{\circ} n

Subtract

4 5^{\circ}

from both sides

x + 4 5^{\circ} - 4 5^{\circ} = 4 5^{\circ} + 36 0^{\circ} n - 4 5^{\circ}

Simplify

x = 36 0^{\circ} n

Solve

x + 4 5^{\circ} = 13 5^{\circ} + 36 0^{\circ} n : x = 36 0^{\circ} n + 9 0^{\circ}

x + 4 5^{\circ} = 13 5^{\circ} + 36 0^{\circ} n

Move

4 5^{\circ}

to the right side

x + 4 5^{\circ} = 13 5^{\circ} + 36 0^{\circ} n

Subtract

4 5^{\circ}

from both sides

x + 4 5^{\circ} - 4 5^{\circ} = 13 5^{\circ} + 36 0^{\circ} n - 4 5^{\circ}

Simplify

x + 4 5^{\circ} - 4 5^{\circ} = 13 5^{\circ} + 36 0^{\circ} n - 4 5^{\circ}

Simplify

x + 4 5^{\circ} - 4 5^{\circ} : x

x + 4 5^{\circ} - 4 5^{\circ}

Add similar elements:

4 5^{\circ} - 4 5^{\circ} = 0

= x

Simplify

13 5^{\circ} + 36 0^{\circ} n - 4 5^{\circ} : 36 0^{\circ} n + 9 0^{\circ}

13 5^{\circ} + 36 0^{\circ} n - 4 5^{\circ}

Group like terms

= 36 0^{\circ} n - 4 5^{\circ} + 13 5^{\circ}

Combine the fractions

- 4 5^{\circ} + 13 5^{\circ} : 9 0^{\circ}

Apply rule

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

= \frac{- 18 0 ^{\circ} + 54 0 ^{\circ}}{4}

Add similar elements:

- 18 0^{\circ} + 54 0^{\circ} = 36 0^{\circ}

= 9 0^{\circ}

Cancel the common factor:

2

= 9 0^{\circ}

= 36 0^{\circ} n + 9 0^{\circ}

x = 36 0^{\circ} n + 9 0^{\circ}

x = 36 0^{\circ} n + 9 0^{\circ}

x = 36 0^{\circ} n + 9 0^{\circ}

x = 36 0^{\circ} n, x = 36 0^{\circ} n + 9 0^{\circ}

Solutions for the range

- 36 0^{\circ} \leq x \leq 36 0^{\circ}

x = 0, x = 9 0^{\circ}, x = 36 0^{\circ}, x = - 36 0^{\circ}, x = - 27 0^{\circ}

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Frequently Asked Questions (FAQ)

What is the general solution for sin(x)+cos(x)=1,-360<= x<= 360 ?
The general solution for sin(x)+cos(x)=1,-360<= x<= 360 is x=0,x=90,x=360,x=-360,x=-270