What is the general solution for sin(x)+cos(x)= 3/2 ?

The general solution for sin(x)+cos(x)= 3/2 is No Solution for x\in\mathbb{R}

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sin(x)+cos(x)= 3/2

sin (x) + cos (x) = \frac{3}{2}

No Solution for x \in R

Solution steps

sin (x) + cos (x) = \frac{3}{2}

Rewrite using trig identities

sin (x) + cos (x)

sin (x) + cos (x) = 2 sin (x + \frac{π}{4})

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 (\frac{π}{4}) sin (x) + sin (\frac{π}{4}) cos (x))

Use the Angle Sum identity:

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

= 2 sin (x + \frac{π}{4})

= 2 sin (x + \frac{π}{4})

2 sin (x + \frac{π}{4}) = \frac{3}{2}

Divide both sides by

2

2 sin (x + \frac{π}{4}) = \frac{3}{2}

Divide both sides by

2

\frac{2 sin ( x + \frac{π}{4} )}{2} = \frac{\frac{3}{2}}{2}

Simplify

\frac{2 sin ( x + \frac{π}{4} )}{2} = \frac{\frac{3}{2}}{2}

Simplify

\frac{2 sin ( x + \frac{π}{4} )}{2} : sin (x + \frac{π}{4})

\frac{2 sin ( x + \frac{π}{4} )}{2}

Cancel the common factor:

2

= sin (x + \frac{π}{4})

Simplify

\frac{\frac{3}{2}}{2} : \frac{3 2}{4}

\frac{\frac{3}{2}}{2}

Apply the fraction rule:

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

= \frac{3}{2 2}

Rationalize

\frac{3}{2 2} : \frac{3 2}{4}

\frac{3}{2 2}

Multiply by the conjugate

\frac{2}{2}

= \frac{3 2}{2 2 2}

222 = 4

222

Apply exponent rule:

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

222 = 2 \cdot 2^{\frac{1}{2}} \cdot 2^{\frac{1}{2}} = 2^{1 + \frac{1}{2} + \frac{1}{2}}

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

Add similar elements:

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

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

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

2 \cdot \frac{1}{2}

Multiply fractions:

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

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

Cancel the common factor:

2

= 1

= 2^{1 + 1}

Add the numbers:

1 + 1 = 2

= 2^{2}

2^{2} = 4

= 4

= \frac{3 2}{4}

= \frac{3 2}{4}

sin (x + \frac{π}{4}) = \frac{3 2}{4}

sin (x + \frac{π}{4}) = \frac{3 2}{4}

sin (x + \frac{π}{4}) = \frac{3 2}{4}

- 1 \leq sin (x) \leq 1

No Solution for x \in R

cos^{2} (x) - sin^{2} (x) - 5 sin (x) = 3

sin (θ) = - \frac{21}{5}

t, cos (2 t) - sin (t) = o

cot (x) = - \frac{15}{8}

tan (x^{\circ}) = \frac{30}{73}

What is the general solution for sin(x)+cos(x)= 3/2 ?
The general solution for sin(x)+cos(x)= 3/2 is No Solution for x\in\mathbb{R}