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How do you find the exact value of $\tan \left( \theta \right)=-4$ ?

Last updated date: 05th Mar 2024
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IVSAT 2024
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Hint: We solve the given problem with the help of a graphing calculator. This is done by pressing the “$2ND$ ” key followed by the “$TAN$ ” key and then the value “ $-4$ “. After pressing the “ $ENTER$ ” key, we get the answer.

Complete step by step answer:
In the given problem, we have to find the value of $\theta $ at which $\tan \theta $ is $-4$ . We can do this on a graphical calculator like the $TI-84$ calculator. The $TI-84$ calculator provides a direct program to find the inverse trigonometric functions.
To evaluate the inverse trigonometric functions on $TI-84$ , we at first press the “ $2ND$ “ key to activate the alternative function of a key. We now press the “ $TAN$ ” key to enter the ${{\tan }^{-1}}$ function as the “ $2ND$ ” already activated the inverse operator before. Now, we can either enter a parenthesis or not enter it as both will yield the same answer is this case. We now press the “ $-$ ” key followed by pressing the “ $4$ “ key to enter $-4$ .
Having entered all the values, we now press the “ $ENTER$ ” key. After pressing the “$ENTER$ ” key, the calculator immediately presents the answer “ $-76.96$ ” which is in degrees as the default mode of the calculator for angle measurements in degrees. We can change the settings to radian by pressing the “ $MODE$ ” key.
Now, the range of ${{\tan }^{-1}}x$ is $\left( -{{90}^{\circ }},{{90}^{\circ }} \right)$ and our answer is $-{{76.96}^{\circ }}$ . Therefore, we can conclude that the exact value of $\tan \left( \theta \right)=-4$ is $\theta ={{76.96}^{\circ }}$ .

We should keep mind the range of various trigonometric functions like ${{\sin }^{-1}}x,{{\cos }^{-1}}x$ and should write the answers within that range only. The calculator always shows the answer which is closest to the $x-axis$ and therefore, we should transform these values to the values within range. This problem can also be solved by plotting the graph of $\tan x$ and drawing another line $y=-4$ . The point of intersection within the region $\left( -{{90}^{\circ }}
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