Simplify the expression (1 + cos 30°) / (1 - sin 60°).

Question

ScanSolve AI Tutor · Accepted Answer

Step 1: Recall the values of cos 30° and sin 60°. cos 30° = (√(3))/(2); sin 60° = (√(3))/(2) Step 2: Substitute these values into the given expression. (1 + cos 30°)/(1 − sin 60°) = (1 + (√(3))/(2))/(1 − (√(3))/(2)) Step 3: Simplify the numerator and the denominator by finding a common denominator. (2/2 + (√(3))/(2))/(2/2 − (√(3))/(2)) = ((2 + √(3))/(2))/((2 − √(3))/(2)) Step 4: Simplify the complex fraction. (2 + √(3))/(2 − √(3)) Step 5: Rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator, which is 2 + √(3). (2 + √(3))/(2 − √(3)) × (2 + √(3))/(2 + √(3)) Step 6: Expand the numerator and the denominator. Numerator: (2 + √(3))(2 + √(3)) = 2² + 2√(3) + 2√(3) + (√(3))² = 4 + 4√(3) + 3 = 7 + 4√(3); Denominator: (2 − √(3))(2 + √(3)) = 2² − (√(3))² = 4 − 3 = 1 Step 7: Write the final simplified expression. (7 + 4√(3))/(1) = 7 + 4√(3) The rationalized expression is 7 + 4√(3).

Simplify the expression (1 + cos 30°) / (1 - sin 60°).

Answer

Step-by-step solution

Recall the values of $\cos 30^\circ$ and $\sin 60^\circ$ .

Substitute these values into the given expression.

Simplify the numerator and the denominator by finding a common denominator.

Simplify the complex fraction.

Rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator, which is $2 + \sqrt{3}$ .

Expand the numerator and the denominator.

Write the final simplified expression.

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Simplify the expression (1 + cos 30°) / (1 - sin 60°).

Answer

Step-by-step solution

Recall the values of cos⁡30∘\cos 30^\circcos30∘ and sin⁡60∘\sin 60^\circsin60∘.

Substitute these values into the given expression.

Simplify the numerator and the denominator by finding a common denominator.

Simplify the complex fraction.

Rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator, which is 2+32 + \sqrt{3}2+3​.

Expand the numerator and the denominator.

Write the final simplified expression.

Try one yourself

Still stuck on this one?

More Mathematics Questions

Recall the values of $\cos 30^\circ$ and $\sin 60^\circ$ .

Rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator, which is $2 + \sqrt{3}$ .