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QuestionEvaluate the expression using special right triangles: csc⁡245∘+tan⁡260∘cot⁡230∘\frac{\csc ^{2} 45^{\circ}+\tan ^{2} 60^{\circ}}{\cot ^{2} 30^{\circ}} Simplify your answer with integers or fractions.

Studdy Solution

STEP 1

Assumptions1. We are working with special right triangles, which include30-60-90 and45-45-90 triangles. . We know the values of the trigonometric functions for these special angles.
3. The trigonometric functions are defined as follows - csc⁡θ=1sin⁡θ\csc \theta = \frac{1}{\sin \theta} - tan⁡θ=sin⁡θcos⁡θ\tan \theta = \frac{\sin \theta}{\cos \theta} - cot⁡θ=1tan⁡θ=cos⁡θsin⁡θ\cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta}

STEP 2

First, we need to find the values of the trigonometric functions at the given angles.For a45-45-90 triangle, the sides are in the ratio11:2\sqrt{2}, so sin⁡45∘=cos⁡45∘=12\sin45^{\circ} = \cos45^{\circ} = \frac{1}{\sqrt{2}}.
For a30-60-90 triangle, the sides are in the ratio1:\sqrt{}2, so sin⁡30∘=12\sin30^{\circ} = \frac{1}{2}, cos⁡30∘=2\cos30^{\circ} = \frac{\sqrt{}}{2}, sin⁡60∘=2\sin60^{\circ} = \frac{\sqrt{}}{2}, and cos⁡60∘=12\cos60^{\circ} = \frac{1}{2}.

STEP 3

Now, we can substitute the values of the trigonometric functions into the given expression.
csc⁡245∘+tan⁡260∘cot⁡230∘=(1sin⁡45∘)2+(sin⁡60∘cos⁡60∘)2(cos⁡30∘sin⁡30∘)2\frac{\csc ^{2}45^{\circ}+\tan ^{2}60^{\circ}}{\cot ^{2}30^{\circ}} = \frac{\left(\frac{1}{\sin45^{\circ}}\right)^{2} + \left(\frac{\sin60^{\circ}}{\cos60^{\circ}}\right)^{2}}{\left(\frac{\cos30^{\circ}}{\sin30^{\circ}}\right)^{2}}

STEP 4

Substitute the values of the trigonometric functions into the expression.
(112)2+(3212)2(3212)2\frac{\left(\frac{1}{\frac{1}{\sqrt{2}}}\right)^{2} + \left(\frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}}\right)^{2}}{\left(\frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}}\right)^{2}}

STEP 5

implify the expression.
22+3232=2+33=53\frac{\sqrt{2}^{2} + \sqrt{3}^{2}}{\sqrt{3}^{2}} = \frac{2 +3}{3} = \frac{5}{3} So, the simplified expression is 53\frac{5}{3}.

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