What is the value of (cos(pi/4)+sin(pi/6))^2 ?

The value of (cos(pi/4)+sin(pi/6))^2 is (3+2sqrt(2))/4

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(cos(pi/4)+sin(pi/6))^2

(cos (\frac{π}{4}) + sin (\frac{π}{6}))^{2}

\frac{3 + 2 2}{4}

Decimal Notation

1.45710 \dots

Solution steps

(cos (\frac{π}{4}) + sin (\frac{π}{6}))^{2}

Use the following trivial identity:

cos (\frac{π}{4}) = \frac{2}{2}

cos (\frac{π}{4})

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}

= \frac{2}{2}

Use the following trivial identity:

sin (\frac{π}{6}) = \frac{1}{2}

sin (\frac{π}{6})

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}

= \frac{1}{2}

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

Simplify

(\frac{2}{2} + \frac{1}{2})^{2} : \frac{3 + 2 2}{4}

(\frac{2}{2} + \frac{1}{2})^{2}

Combine the fractions

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

Apply rule

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

= \frac{2 + 1}{2}

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

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

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

(2 + 1)^{2} = 3 + 22

(2 + 1)^{2}

Apply Perfect Square Formula:

(a + b)^{2} = a^{2} + 2 ab + b^{2}

a = 2, b = 1

= (2)^{2} + 22 \cdot 1 + 1^{2}

Simplify

(2)^{2} + 22 \cdot 1 + 1^{2} : 3 + 22

(2)^{2} + 22 \cdot 1 + 1^{2}

Apply rule

1^{a} = 1

1^{2} = 1

= (2)^{2} + 2 \cdot 1 \cdot 2 + 1

(2)^{2} = 2

(2)^{2}

Apply radical rule:

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

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

Apply exponent rule:

(a^{b})^{c} = a^{b c}

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

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

\frac{1}{2} \cdot 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

22 \cdot 1 = 22

22 \cdot 1

Multiply the numbers:

2 \cdot 1 = 2

= 22

= 2 + 22 + 1

Add the numbers:

2 + 1 = 3

= 3 + 22

= 3 + 22

= \frac{3 + 2 2}{2 ^{2}}

2^{2} = 4

= \frac{3 + 2 2}{4}

= \frac{3 + 2 2}{4}

cos (94 5^{\circ})

cot (1 3^{\circ})

cos (arctan (- 5))

e^{c o s (π)} cos (π) - e^{c o s (π)} sin^{2} (π)

cos (6 0^{\circ} + 22 5^{\circ})

What is the value of (cos(pi/4)+sin(pi/6))^2 ?
The value of (cos(pi/4)+sin(pi/6))^2 is (3+2sqrt(2))/4