Question

Which of the following is greatest?
A.\[\tan 1\]
B.\[\tan 4\]
C.\[\tan 7\]
D.\[\tan 10\]

Answer 1

Hint: The basic three trigonometric functions are sine, cosine, and tangent. A tangent can also be written as a ratio of sine and cosine functions. One of the important things that we need to know is that the value of the tangent function increases in all four quadrants.
Formula:
To solve this problem we need to know a trigonometric ratio of the function tangent:
\[\tan (\pi + \theta ) = \tan \theta\]
\[\tan (2\pi + \theta ) = \tan \theta \]
\[\tan (3\pi + \theta ) = \tan \theta \]
And the value of \[\pi \]\[ \simeq 3.14\]

Complete Step by step answer:
Here we aim to find the greatest value among the given tangent function. Since we can’t find the values of the tangent function directly, we will make use of the \[\pi \] value to modify the given functions for our convenience.
Let’s take the first option \[\tan 1\] since one is the smallest number, we will leave this function as it is.
Now let us consider the second given function \[\tan 4\] , let us add and subtract \[\pi \] to the degree of the function as \[4 > \pi \] .
\[\tan 4\]\[ = \tan (\pi + 4 - \pi )\]\[ = \tan (\pi + (4 - \pi ))\]
We know that \[\tan (\pi + \theta ) = \tan \theta \] . Here \[\theta = 4 - \pi \] thus we get
\[\tan 4\]\[ = \tan (4 - \pi )\]
Now let us substitute the value of \[\pi \] in the above function.
\[\tan 4\]\[ = \tan (4 - 3.14) = \tan (0.86)\]
Now let us modify the third given function \[\tan 7\] , let us add and subtract \[2\pi \] to the degree of the function as \[7 > 2\pi \] .
\[\tan 7\] \[ = \tan (2\pi + 7 - 2\pi ) = \tan (2\pi + (7 - 2\pi ))\]
We know that \[\tan (2\pi + \theta ) = \tan \theta \] . Here \[\theta = 7 - 2\pi \] thus we get
\[\tan 7 = \tan (7 - 2\pi )\]
Now let us substitute the value of \[\pi \] in the above function.
\[\tan 7 = \tan (7 - (2 \times 3.14)) = \tan (0.72)\]
Now let us consider the fourth given function \[\tan 10\] , let us add and subtract \[3\pi \] to the degree of the function since \[10 > 3\pi \] .
\[\tan 10 = \tan (3\pi + 10 - 3\pi ) = \tan (3\pi + (10 - 3\pi ))\]
We know that \[\tan (3\pi + \theta ) = \tan \theta \] . Here \[\theta = 10 - 3\pi \] thus we get
\[\tan 10 = \tan (10 - 3\pi )\]
Now let us substitute the value of \[\pi \] in the above function.
\[\tan 10 = \tan (10 - (3 \times 3.14)) = \tan (0.58)\]
Since the values \[1,0.86,0.72,0.58\] are all lie in the first quadrant as, they are positive. And also, we know that the tangent function increases in all four quadrants.
\[\tan 1 > \tan (0.86) > \tan (0.72) > \tan (0.58)\]
Now we can see that the function \[\tan 1\] is the greatest.
Let us see the options, option (a) \[\tan 1\] is the correct option as we got the same answer in the above calculation.
Option (b) \[\tan 4\] is an incorrect answer since we got that \[\tan 1\] is the greatest of all.
Option (c) \[\tan 7\] is an incorrect answer since we got that \[\tan 1\] is the greatest of all.
Option (d) \[\tan 10\] is an incorrect answer since we got that \[\tan 1\] is the greatest of all.

Hence option (a) \[\tan 1\] is the correct option.

Note:
Here we can’t find the value of tangent functions for small degrees so we modified it by using the trigonometric ratios since we have some standard identities. Then we also have to check whether that function is an increasing function or a decreasing function in the quadrant where those degrees lie.