Find the Value of: cos(24°) + cos(55°) + cos(125°) + cos(204°)

To solve this, we must shrink the large, ugly angles (> 90°) down into small, acute angles using the supplementary angle rules (180 - θ) and (180 + θ).

Rule 1: In the 2nd Quadrant, Cosine is completely negative.

• cos(180° - θ) = -cos(θ)

Rule 2: In the 3rd Quadrant, Cosine is also completely negative.

• cos(180° + θ) = -cos(θ)

Let us break down the two massive angles (125° and 204°) and rewrite them using the number 180:

Step 1: Simplify cos(125°)

• We can rewrite 125 as (180 - 55).
• So, cos(125°) = cos(180° - 55°)
• Using Rule 1, cos(180° - 55°) magically becomes -cos(55°).

Step 2: Simplify cos(204°)

• We can rewrite 204 as (180 + 24).
• So, cos(204°) = cos(180° + 24°)
• Using Rule 2, cos(180° + 24°) magically becomes -cos(24°).

Step 3: Put it all together in the main equation Now, replace the large angles in the original problem with our new negative values: = cos(24°) + cos(55°) + [ -cos(55°) ] + [ -cos(24°) ]

Step 4: Cancel them out

• The positive cos(24°) perfectly cancels the negative -cos(24°).
• The positive cos(55°) perfectly cancels the negative -cos(55°).

Result = 0 + 0 = 0. The proof is complete!