Is sin(θ)+cos(θ)=sqrt(2)sin(θ+45) ?

The answer to whether sin(θ)+cos(θ)=sqrt(2)sin(θ+45) is True

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular Trigonometry >

prove sin(θ)+cos(θ)=sqrt(2)sin(θ+45)

Solution

prove

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

Solution

True

Solution steps

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

Manipulating right side

2 sin (θ + 4 5^{\circ})

Rewrite using trig identities

sin (θ + 4 5^{\circ})

Use the Angle Sum identity:

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

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

Simplify

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

sin (θ) cos (4 5^{\circ}) + cos (θ) sin (4 5^{\circ})

sin (θ) cos (4 5^{\circ}) = \frac{2 sin ( θ )}{2}

sin (θ) cos (4 5^{\circ})

Simplify

cos (4 5^{\circ}) : \frac{2}{2}

cos (4 5^{\circ})

Use the following trivial identity:

cos (4 5^{\circ}) = \frac{2}{2}

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}

= \frac{2}{2}

= \frac{2}{2} sin (θ)

Multiply fractions:

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

= \frac{2 sin ( θ )}{2}

cos (θ) sin (4 5^{\circ}) = \frac{2 cos ( θ )}{2}

cos (θ) sin (4 5^{\circ})

Simplify

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

sin (4 5^{\circ})

Use the following trivial identity:

sin (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}

= \frac{2}{2}

= \frac{2}{2} cos (θ)

Multiply fractions:

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

= \frac{2 cos ( θ )}{2}

= \frac{2 sin ( θ )}{2} + \frac{2 cos ( θ )}{2}

Apply rule

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

= \frac{2 sin ( θ ) + 2 cos ( θ )}{2}

= \frac{2 sin ( θ ) + 2 cos ( θ )}{2}

= 2 \frac{2 sin ( θ ) + 2 cos ( θ )}{2}

Simplify

2 \frac{2 sin ( θ ) + 2 cos ( θ )}{2} : sin (θ) + cos (θ)

2 \frac{2 sin ( θ ) + 2 cos ( θ )}{2}

Multiply fractions:

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

= \frac{( 2 sin ( θ ) + 2 cos ( θ ) ) 2}{2}

Cancel

\frac{( 2 sin ( θ ) + 2 cos ( θ ) ) 2}{2} : \frac{2 sin ( θ ) + 2 cos ( θ )}{2}

\frac{( 2 sin ( θ ) + 2 cos ( θ ) ) 2}{2}

Apply radical rule:

2 = 2^{\frac{1}{2}}

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

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

\frac{2 ^{\frac{1}{2}}}{2 ^{1}} = \frac{1}{2 ^{1 - \frac{1}{2}}}

= \frac{2 sin ( θ ) + 2 cos ( θ )}{2 ^{1 - \frac{1}{2}}}

Subtract the numbers:

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

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

Apply radical rule:

2^{\frac{1}{2}} = 2

= \frac{2 sin ( θ ) + 2 cos ( θ )}{2}

= \frac{2 sin ( θ ) + 2 cos ( θ )}{2}

Factor out common term

2

= \frac{2 ( sin ( θ ) + cos ( θ ) )}{2}

Cancel the common factor:

2

= sin (θ) + cos (θ)

= sin (θ) + cos (θ)

We showed that the two sides could take the same form

\Rightarrow True

Popular Examples

prove (1-tan(θ))/(1+tan(θ))=(cot(θ)-1)/(cot(θ)+1)

prove cot(θ)sin(θ)cos(θ)=1-sin^2(θ)

prove csc(θ)= 1/(sin(θ))

prove 1/(sin(θ))-sin(θ)=cot(θ)cos(θ)

prove (sec(-x))/(tan(x)+cot(x))=-sin(-x)

Frequently Asked Questions (FAQ)

Is sin(θ)+cos(θ)=sqrt(2)sin(θ+45) ?
The answer to whether sin(θ)+cos(θ)=sqrt(2)sin(θ+45) is True