Find the Value Using the Unit Circle cos(75)

Problem

cos(75)

Solution

1. Identify two angles from the unit circle that sum to 75 We can use 45 and 30 because both have known coordinates on the unit circle.

2. Apply the cosine sum formula, which states cos(A+B)=cos(A)*cos(B)−sin(A)*sin(B)

3. Substitute A=45 and B=30 into the formula:

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

4. Evaluate the trigonometric functions using the unit circle values: cos(45)=√(,2)/2 cos(30)=√(,3)/2 sin(45)=√(,2)/2 and sin(30)=1/2

cos(75)=(√(,2)/2)*(√(,3)/2)−(√(,2)/2)*(1/2)

5. Simplify the expression by multiplying the fractions and combining the terms over a common denominator.

cos(75)=√(,6)/4−√(,2)/4

cos(75)=(√(,6)−√(,2))/4

Final Answer

cos(75)=(√(,6)−√(,2))/4

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