What is the value of 2/(2+cos(270)) ?

The value of 2/(2+cos(270)) is 1

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2/(2+cos(270))

\frac{2}{2 + cos ( 27 0 ^{\circ} )}

1

Solution steps

\frac{2}{2 + cos ( 27 0 ^{\circ} )}

Rewrite using trig identities:

cos (27 0^{\circ}) = cos^{2} (13 5^{\circ}) - sin^{2} (13 5^{\circ})

cos (27 0^{\circ})

Write

cos (27 0^{\circ})

cos (2 \cdot 13 5^{\circ})

= cos (2 \cdot 13 5^{\circ})

Use the Double Angle identity:

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

= cos^{2} (13 5^{\circ}) - sin^{2} (13 5^{\circ})

= \frac{2}{2 + cos ^{2} ( 13 5 ^{\circ} ) - sin ^{2} ( 13 5 ^{\circ} )}

Use the following trivial identity:

cos (13 5^{\circ}) = - \frac{2}{2}

cos (13 5^{\circ})

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}

Use the following trivial identity:

sin (13 5^{\circ}) = \frac{2}{2}

sin (13 5^{\circ})

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

Simplify

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

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

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

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

Apply exponent rule:

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

n

is even

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

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

Add similar elements:

(\frac{2}{2})^{2} - (\frac{2}{2})^{2} = 0

= 2

= \frac{2}{2}

Apply rule

\frac{a}{a} = 1

= 1

= 1

sin (\frac{7}{12})

arccot (cot (- 1 0^{\circ}))

2 sin (- 7 0^{\circ})

cos (6 0^{\circ}) cos (6 5^{\circ}) - sin (6 0^{\circ}) sin (6 5^{\circ})

arcsin (\frac{0}{3})