Math

QuestionEvaluate the expression using special right triangles:
sec2π3+tan2π4cot2π6= \frac{\sec ^{2} \frac{\pi}{3}+\tan ^{2} \frac{\pi}{4}}{\cot ^{2} \frac{\pi}{6}}=
Simplify your answer, including radicals.

Studdy Solution

STEP 1

Assumptions1. The given expression is secπ3+tanπ4cotπ6\frac{\sec ^{} \frac{\pi}{3}+\tan ^{} \frac{\pi}{4}}{\cot ^{} \frac{\pi}{6}} . We are to use the properties of special right triangles to evaluate this expression.

STEP 2

We need to recall the values of the trigonometric functions at the given angles. These are standard angles that correspond to special right triangles.
1. secπ=2\sec \frac{\pi}{} =2
2. tanπ4=1\tan \frac{\pi}{4} =1 . cotπ6=\cot \frac{\pi}{6} = \sqrt{}

STEP 3

Substitute the values of the trigonometric functions into the given expression.
sec2π3+tan2πcot2π6=22+12(3)2\frac{\sec ^{2} \frac{\pi}{3}+\tan ^{2} \frac{\pi}{}}{\cot ^{2} \frac{\pi}{6}} = \frac{2^{2} +1^{2}}{(\sqrt{3})^{2}}

STEP 4

implify the expression in the numerator and the denominator.
22+12(3)2=4+13\frac{2^{2} +1^{2}}{(\sqrt{3})^{2}} = \frac{4 +1}{3}

STEP 5

Further simplify the expression.
4+13=53\frac{4 +1}{3} = \frac{5}{3}So, the given expression simplifies to 53\frac{5}{3}.

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