Expand Using Sum/Difference Formulas cos(75)

Problem

cos(75)

Solution

1. Identify two common angles from the unit circle that sum or subtract to 75 We can use 45 and 30 since 45 + 30 = 75$.

2. Apply the formula for the cosine of a sum, which is cos(α+β)=cos(α)*cos(β)−sin(α)*sin(β)

3. Substitute the values α=45 and β=30 into the formula.

cos(45+30)=cos(45)*cos(30)−sin(45)*sin(30)

4. Evaluate the trigonometric functions for the specific angles.

cos(45)=√(,2)/2

cos(30)=√(,3)/2

sin(45)=√(,2)/2

sin(30)=1/2

5. Multiply the terms together.

√(,2)/2⋅√(,3)/2=√(,6)/4

√(,2)/2⋅1/2=√(,2)/4

6. Subtract the resulting fractions to find the final value.

√(,6)/4−√(,2)/4=(√(,6)−√(,2))/4

Final Answer

cos(75)=(√(,6)−√(,2))/4

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