Find the exact value of cos 240 degrees without a calculator. show your work!

it equals (−12) because cos(60∘)=12

The reference angle for 240∘ is 60∘ (since 240∘=180∘+60∘)

60∘ is an angle of one of the standard triangles with 
cos(60∘)=12

240∘ is in the 3rd quadrant so (either by CAST or noting that the "x-side" of the associate triangle is negative)
cos(240∘)=−cos(60∘)

cos(240∘)=−12

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