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Chapter 5: Problem 100

Use the sum-to-product formulas to find the exact value of the expression. $$\cos 120^{\circ}+\cos 60^{\circ}$$

Short Answer

Expert verified
The exact value of the expression \(\cos 120^{\circ}+\cos 60^{\circ}\) using the sum-to-product formulas is 0.

Step by step solution

01

Identify correct formula

The sum to product formula for cosine is \[2\cos(\frac{1}{2}(a+b))\cos(\frac{1}{2}(a-b))\]. Since we are dealing with the angles 120 and 60, our a and b values would be 120 and 60 respectively.
02

Substitute values into the equation

Make the substitutions with a=120 and b=60 into the sum-to-product formula to get \[2\cos(\frac{1}{2}(120+60))\cos(\frac{1}{2}(120-60)).\]
03

Simplify the equation

The above equation then simplifies to \[2\cos(90^{\circ})\cos(30^{\circ}).\]
04

Calculate the cosine values

Now we can use the unit circle to look up these values. We can see that \(\cos(90^{\circ})=0\) and \(\cos(30^{\circ})=\frac{\sqrt{3}}{2}\).
05

Final Calculation

Substitute these values into the equation from step 3, \[2*0*\frac{\sqrt{3}}{2}\] to get a final value of 0.
06

Confirm the result

The result makes sense because the cosine of 90 degrees is always 0, and anything multiplied by 0 is 0.

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