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Algebra Difficulty 3.1 AMC 10/12 Find the answer

Calculate: tan60+2sin452cos30\tan 60^{\circ}+2\sin 45^{\circ}-2\cos 30^{\circ}

A number or a short expression. Spacing and $ signs are ignored.

Solution

To solve the given problem, we start by breaking down the trigonometric functions into their exact values:

1. tan60=3\tan 60^{\circ} = \sqrt{3}
2. sin45=22\sin 45^{\circ} = \dfrac{\sqrt{2}}{2}
3. cos30=32\cos 30^{\circ} = \dfrac{\sqrt{3}}{2}

Now, substituting these values into the original expression:

tan60+2sin452cos30=3+2×222×32\tan 60^{\circ}+2\sin 45^{\circ}-2\cos 30^{\circ} = \sqrt{3}+2\times \dfrac{\sqrt{2}}{2}-2\times \dfrac{\sqrt{3}}{2}

Simplifying the expression step by step:

=3+23= \sqrt{3} + \sqrt{2} - \sqrt{3}

Notice that 3\sqrt{3} and 3-\sqrt{3} cancel each other out, leaving us with:

=2= \sqrt{2}

Therefore, the final answer is 2\boxed{\sqrt{2}}.

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.