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Trigonometry – Problem 2: Cosine Sum Formula Evaluate

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Question

Cosine Sum Formula

Evaluate cos⁡147°cos⁡333°+cos⁡57°cos⁡63°\cos147°\cos333° + \cos57°\cos63°.

Step-by-step solution

Using angle transformations:

cos⁡147°cos⁡333°+cos⁡57°cos⁡63°\cos147°\cos333° + \cos57°\cos63°

=cos⁡(180°−33°)cos⁡(360°−27°)+cos⁡(90°−33°)cos⁡(90°−27°)= \cos(180°-33°)\cos(360°-27°) + \cos(90°-33°)\cos(90°-27°)

=−cos⁡33°cos⁡27°+sin⁡33°sin⁡27°= -\cos33°\cos27° + \sin33°\sin27°

=−(cos⁡33°cos⁡27°−sin⁡33°sin⁡27°)= -(\cos33°\cos27° - \sin33°\sin27°)

=−cos⁡(33°+27°)=−cos⁡60°=−12= -\cos(33°+27°) = -\cos60° = -\frac{1}{2}

Final answer

−12-\frac{1}{2}

Marking scheme

1. Checkpoints (max 7 pts total)

  • Correct identity setup (2 pts): choose an appropriate sum/difference, double-angle, or auxiliary-angle idea and set up the key equation(s).
  • Correct algebra / trig simplification (2 pts): transform expressions without sign mistakes.
  • Solve for target quantity (2 pts): isolate the requested value and handle any constraints if needed.
  • Final answer (1 pt): clearly state the result in the required form.

2. Zero-credit items

  • Only writing the final answer with no supporting steps.
  • Using unrelated identities without reaching a valid equation.

3. Deductions

  • Algebra/sign error (-1)
  • Missing condition check (-1)

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