Problem 1 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

Which of the following statements are correct/incorrect? i. \(\sin \theta=-\frac{1}{5} .\\{\) Ans. correct \(\\}\) ii. \(\cos \theta=1 .\\{\) Ans. correct \(\\}\) iii. \(\sec \theta=\frac{1}{2}\). \\{Ans. incorrect\\} iv. \(\tan \theta=20\). \\{Ans. correct\\}

Step-by-Step Solution

Answer

Statements i, ii, iv are correct while statement iii is incorrect as the value is within the range excluded by secant function.

Step 1: Analyzing Statement i

The sine function, \(\sin \theta\), can take on any value between -1 and 1 inclusive. Here, \(\sin \theta = -\frac{1}{5}\), which falls within this range, so the statement is correct.

Step 2: Analyzing Statement ii

The cosine function, \(\cos \theta\), can take on any value between -1 and 1 inclusive. Here, \(\cos \theta = 1\), which falls within this range, so the statement is correct.

Step 3: Analyzing Statement iii

The secant function, \(\sec \theta\), is the reciprocal of the cosine function. It takes on all the values not between -1 and 1 in its range. Here, \(\sec \theta = \frac{1}{2}\) which falls within the range excluded by secant (-1,1) hence the statement is incorrect.

Step 4: Analyzing Statement iv

The tangent function, \(\tan \theta\), can take on any real number. Here, \(\tan \theta = 20\) which falls within the possible values of the tangent function, so the statement is correct.

Key Concepts

Sine Function Range

The sine function, denoted as \(\sin\theta\), is a fundamental aspect of trigonometry with significant importance in the realm of IIT JEE mathematics. It maps the angle \(\theta\) to a real number, which represents the y-coordinate of a point on the unit circle.

The range of the sine function is the set of possible output values it can produce. For any angle \(\theta\), the sine function can yield values from -1 to 1, inclusive. This means the sine of an angle can never be less than -1 or greater than 1. The reason for this lies in the geometric interpretation of the sine function: it represents the height of a point on the unit circle, and since the radius of the unit circle is 1, the height can never exceed the radius.

For instance, in the provided problem, the value \(\sin\theta = -\frac{1}{5}\) is within the range of -1 to 1, confirming the statement as correct.

Cosine Function Range

The cosine function, denoted as \(\cos\theta\), is another core trigonometric function essential for solving IIT JEE problems. Similar to the sine function, it relates an angle \(\theta\) to the x-coordinate of a point on the unit circle.

The output values of the cosine function also lie within the range of -1 to 1, inclusive. Any value of \(\cos\theta\) outside of this range would contradict the basic properties of a unit circle. The cosine of an angle signifies the horizontal distance from the origin to the point on the unit circle, and like the sine function, this value is constrained by the radius of the circle.

In the textbook exercise, the value \(\cos\theta = 1\) satisfies the required condition for the cosine function's range, corroborating the given answer as correct.

Secant Function Values

The secant function, denoted as \(\sec\theta\), is the reciprocal of the cosine function. As the cosine function's values range between -1 and 1, the secant function's values take on all other real numbers except for the range between -1 and 1. This interval exclusion comes from the aspect that reciprocal values of numbers between -1 and 1 would lead to magnitudes greater than 1, and the secant function will tend toward infinity as the cosine approaches zero.

The secant function is undefined for \(\theta\) where \(\cos\theta\) equals 0, as this would require division by zero. Therefore, an assertion like \(\sec\theta = \frac{1}{2}\) is incorrect because it falls within the disallowed range for secant values. This concept is crucial when solving problems related to the secant function on trigonometric equations like the IIT JEE examination.

Tangent Function Values

The tangent function, denoted as \(\tan\theta\), is yet another pivotal trigonometric function in the toolkit of an IIT JEE aspirant. The function, defined as the ratio of the sine and cosine functions, can assume any real number as its value since it represents the slope of the line connecting the origin to a point on the unit circle. Unlike the sine and cosine functions, there is no upper or lower limit to the values that the tangent function can take, barring the instances where \(\cos\theta\) is zero—points at which the tangent function is undefined due to division by zero.

This limitless range is demonstrated in the textbook example where \(\tan\theta = 20\), a value well within the admissible scope of the tangent function, thus validating the third statement as correct.