Select the answer which is equivalent to the given expression using your calculator. tan(arccos ((8)/(9))) (sqrt17)/(8) (9)/(8) (sqrt17)/(9) (9)/(sqrt17)

Question

Select the answer which is equivalent to the given expression using your calculator.  tan(arccos ((8)/(9)))  (sqrt17)/(8)  (9)/(8)  (sqrt17)/(9)  (9)/(sqrt17)

Bytelearn · Accepted Answer

Understand Relationship: To find the value of tan(arccos(8/9)), we need to understand the relationship between the tangent and cosine functions in a right triangle. The cosine of an angle is the adjacent side over the hypotenuse, and the tangent of an angle is the opposite side over the adjacent side.Consider Right Triangle: Let's consider a right triangle where the angle we're interested in is θ, such that cos(θ) = 8/9. By the definition of cosine, the adjacent side to angle θ is 8, and the hypotenuse is 9.Use Pythagorean Theorem: To find the opposite side, we can use the Pythagorean theorem: hypotenuse^2 = adjacent^2 + opposite^2. Plugging in the values we have, we get 9^2 = 8^2 + opposite^2.Calculate Opposite Side: Calculating the opposite side, we have 81 = 64 + opposite^2, which simplifies to opposite^2 = 81 - 64, and further to opposite^2 = 17.Find All Triangle Sides: Taking the square root of both sides, we find that the opposite side is √17. Now we have all three sides of the right triangle: adjacent = 8, opposite = √17, and hypotenuse = 9.Calculate Tangent Value: The value of tan(θ) is the opposite side over the adjacent side. So, tan(θ) = (√17)/8. This is the value of tan(arccos(8/9)).

Select the answer which is equivalent to the given expression using your calculator. $\newline$ $\tan \left(\arccos \frac{8}{9}\right)$ $\newline$ $\frac{\sqrt{17}}{8}$ $\newline$ $\frac{9}{8}$ $\newline$ $\frac{\sqrt{17}}{9}$ $\newline$ $\frac{9}{\sqrt{17}}$

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