Find the Exact Value tan(67)

Problem

tan(67.5^∘)

Solution

1. Identify the angle as a half-angle of a known value from the unit circle.

θ=67.5^∘=135^∘/2

2. Apply the formula for the half-angle of tangent.

tan(α/2)=(1−cos(α))/sin(α)

3. Substitute α=135^∘ into the formula.

tan(67.5^∘)=(1−cos(135^∘))/sin(135^∘)

4. Determine the exact values for the sine and cosine of 135^∘

cos(135^∘)=−√(,2)/2

sin(135^∘)=√(,2)/2

5. Substitute these values back into the expression.

tan(67.5^∘)=(1−(−√(,2)/2))/√(,2)/2

6. Simplify the numerator by finding a common denominator.

tan(67.5^∘)=(2+√(,2))/2/√(,2)/2

7. Divide the fractions by multiplying by the reciprocal.

tan(67.5^∘)=(2+√(,2))/√(,2)

8. Rationalize the denominator by multiplying the numerator and denominator by √(,2)

tan(67.5^∘)=(2√(,2)+2)/2

9. Simplify the final expression by dividing each term by 2.

tan(67.5^∘)=√(,2)+1

Final Answer

tan(67.5^∘)=√(,2)+1

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