Question

Using the periodic properties of trigonometric functions, how do you find the exact value of the expression $\tan 17\pi$ ?

Answer 1

Hint : Tangent is a trigonometric function; all the trigonometric functions are periodic. In these functions angle is the input value and numerical value. If after a fixed interval of input values, the output values repeat their values then the function is said to be periodic. The trigonometric functions are periodic means that after a fixed interval of angles, the numerical values repeat.

Complete step-by-step answer :
We have to find the exact value of $\tan 17\pi$
We can write it as $\tan (16\pi + \pi )$
Tangent function is periodic after $2\pi$ , that means the output value comes out to be the same after every $2\pi$ radians. So, $\tan [2(8\pi ) + \pi ] = \tan \pi$
Now, $\tan \pi = \tan (\pi - 0) = \tan 0 = 0$
Hence the exact value of the expression $\tan 17\pi$ is $0$ .
So, the correct answer is “0”.

Note : Trigonometric ratios tell us the relation between the two sides of a right-angled triangle and one of its angles other than the right angle. All the trigonometric functions have different signs in different quadrants but their magnitude repeats. All the trigonometric functions are positive in the first quadrant; sine and cosecant are positive in the second quadrant while all other functions are negative; tangent and cotangent are positive in the third quadrant while all other functions are negative; and cosine and secant are positive in the fourth quadrant while all other functions are negative. To solve such questions, we must know the values of trigonometric functions at some basic angles like $0,\dfrac{\pi }{6},\dfrac{\pi }{4},\,\dfrac{\pi }{3},\dfrac{\pi }{2},etc.$