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Question: The incorrect statement is 1) \[\sin \theta =-\dfrac{1}{5}\] 2) \[\cos \theta =1\] 3) \[\sec \...

The incorrect statement is

  1. sin⁡θ=−15\sin \theta =-\dfrac{1}{5}
  2. cos⁡θ=1\cos \theta =1
  3. sec⁡θ=1/2\sec \theta =1/2
  4. tan⁡θ=20\tan \theta =20
Explanation

Solution

We are given the options and we have to find out which out of the four options is incorrect. We have trigonometric functions assigned some values, we have to tell whether they are correct or not. We will match the values assigned to the trigonometric function with the range of values the particular trigonometric functions can have. And accordingly, we will find the incorrect statement.

Complete step-by-step solution:
According to the given question, we are asked to find the incorrect statement from the following four options given. We will take it up one by one.

  1. sin⁡θ=−15\sin \theta =-\dfrac{1}{5}
    We have a sine function here and it is assigned the value −15≈−0.2-\dfrac{1}{5}\approx -0.2.
    And we know that sine function has the range from -1 to 1 and the assigned value is in that range. Therefore, the option is correct.
  2. cos⁡θ=1\cos \theta =1
    We now have a cosine function. The function is assigned the value 1. We know that cosine function has values ranging from -1 to 1, including -1 and 1. Since, the assigned value 1 is in the range of cosine function. Therefore, the option is correct.
  3. sec⁡θ=1/2\sec \theta =1/2
    We now have secant function. We can rewrite the function as,
    ⇒1cos⁡θ=12\Rightarrow \dfrac{1}{\cos \theta }=\dfrac{1}{2}
    That means,
    ⇒cos⁡θ=2\Rightarrow \cos \theta =2
    But as we know, cosine function ranges from -1 to 1 and 2 comes outside of this range. Therefore, the option is incorrect.
  4. tan⁡θ=20\tan \theta =20
    It is a correct answer as the tangent function ranges from −∞-\infty to ∞\infty .
    Therefore, the incorrect option is 3) sec⁡θ=1/2\sec \theta =1/2.

Note: The ranges of the required trigonometric functions should be not mistaken and should be correctly written. The comparison should be done clearly and distinctly. We must also remember that when we get functions secant, cosecant and cotangent, we have to convert them into cosine, sine and tangent by taking the reciprocal. Then, we can obtain the range easily.