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Simplify ( 30^ - 30^)/( 45^ - 60^) and leave the answer in the form a - bsqrt(c).

Mathematics

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Answer

12\frac{1}{2}

Step-by-step solution

7 steps

To simplify the expression sin⁡30∘−cos⁡30∘tan⁡45∘−tan⁡60∘\frac{\sin 30^\circ - \cos 30^\circ}{\tan 45^\circ - \tan 60^\circ} and leave the answer in the form a−bca - b\sqrt{c}, we will use the exact values of the trigonometric functions for special angles.

  1. Recall the values of the trigonometric functions.

    sin⁡30∘=12\sin 30^\circ = \frac{1}{2} cos⁡30∘=32\cos 30^\circ = \frac{\sqrt{3}}{2} tan⁡45∘=1\tan 45^\circ = 1 tan⁡60∘=3\tan 60^\circ = \sqrt{3}

  2. Substitute these values into the given expression.

    sin⁡30∘−cos⁡30∘tan⁡45∘−tan⁡60∘=12−321−3\frac{\sin 30^\circ - \cos 30^\circ}{\tan 45^\circ - \tan 60^\circ} = \frac{\frac{1}{2} - \frac{\sqrt{3}}{2}}{1 - \sqrt{3}}

  3. Simplify the numerator.

    12−32=1−32\frac{1}{2} - \frac{\sqrt{3}}{2} = \frac{1 - \sqrt{3}}{2}

  4. Rewrite the expression with the simplified numerator.

    1−321−3\frac{\frac{1 - \sqrt{3}}{2}}{1 - \sqrt{3}}

  5. Perform the division.

    1−32×11−3\frac{1 - \sqrt{3}}{2} \times \frac{1}{1 - \sqrt{3}}

  6. Cancel out the common term (1−3)(1 - \sqrt{3}) from the numerator and the denominator.

    12\frac{1}{2}

  7. Express the answer in the form a−bca - b\sqrt{c}.

    Since the result is 12\frac{1}{2}, we can write this as 12−03\frac{1}{2} - 0\sqrt{3} (or any other c\sqrt{c}, as b=0b=0).
    Here, a=12a = \frac{1}{2}, b=0b = 0, and c=3c = 3.

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