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

Want more problems? Check here!