Find the Exact Value cos(78)

Problem

cos(78^∘)

Solution

1. Identify the expression as the cosine of an angle in degrees. Since 78^∘ is not a standard reference angle, we look for a way to express it using sum or difference identities.

2. Rewrite the angle using the sum of two angles that have known trigonometric values. We can use 78^∘=45^∘+33^∘ or 78^∘=60^∘+18^∘ Using 60^∘ and 18^∘ is effective because the values for 18^∘ are known constants.

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

4. Substitute A=60^∘ and B=18^∘ into the formula.

cos(60^∘+18^∘)=cos(60^∘)*cos(18^∘)−sin(60^∘)*sin(18^∘)

5. Recall the exact values for the required trigonometric functions.

cos(60^∘)=1/2

sin(60^∘)=√(,3)/2

cos(18^∘)=√(,10+2√(,5))/4

sin(18^∘)=(√(,5)−1)/4

6. Plug these values into the expanded expression.

cos(78^∘)=(1/2)*(√(,10+2√(,5))/4)−(√(,3)/2)*((√(,5)−1)/4)

7. Simplify the expression by multiplying the fractions and combining them over a common denominator of 8

cos(78^∘)=(√(,10+2√(,5))−√(,3)*(√(,5)−1))/8

8. Distribute the √(,3) in the second term.

cos(78^∘)=(√(,10+2√(,5))−√(,15)+√(,3))/8

Final Answer

cos(78^∘)=(√(,10+2√(,5))−√(,15)+√(,3))/8

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