What is the value of cos(345) ?

The value of cos(345) is (sqrt(2)+sqrt(6))/4

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cos(345)

cos (34 5^{\circ})

\frac{2 + 6}{4}

Solution steps

cos (34 5^{\circ})

Rewrite using trig identities:

cos (22 5^{\circ}) cos (12 0^{\circ}) - sin (22 5^{\circ}) sin (12 0^{\circ})

= cos (22 5^{\circ}) cos (12 0^{\circ}) - sin (22 5^{\circ}) sin (12 0^{\circ})

Rewrite using trig identities:

cos (22 5^{\circ}) = - \frac{2}{2}

Use the following trivial identity:

cos (12 0^{\circ}) = - \frac{1}{2}

Rewrite using trig identities:

sin (22 5^{\circ}) = - \frac{2}{2}

Use the following trivial identity:

sin (12 0^{\circ}) = \frac{3}{2}

= (- \frac{2}{2}) (- \frac{1}{2}) - (- \frac{2}{2}) \frac{3}{2}

Simplify

(- \frac{2}{2}) (- \frac{1}{2}) - (- \frac{2}{2}) \frac{3}{2} : \frac{2 + 6}{4}

= \frac{2 + 6}{4}

sec (2)

tan (arcsin (\frac{2}{2}))

sin (7 2^{\circ})

tan (- \frac{7 π}{12})

arcsec (- \frac{2 3}{3})