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JEE Main 2026 Preparation: Question Papers, Solutions, Mock Tests & Strategy Unacademy » JEE Study Material » Mathematics » Sign of trigonometric functions

Sign of trigonometric functions

In this article, we will learn about trigonometric functions and their signs in various quadrants.

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Circular functions, sometimes known as trigonometric functions, can be simply described as functions of a triangle’s angle. This means that these trig functions determine the relationship between the angles and sides of a triangle. Sine, cosine, tangent, cotangent, secant, and cosecant are the basic trigonometric functions. A number of trigonometric formulas and identities define the relationship between the functions and aid in the determination of the triangle’s angles. 

Six trigonometric functions:

The sine, cosine, and tangent angles are the fundamental classifications of trigonometric functions. The cotangent, secant, and cosecant functions can all be deduced from the basic functions. In comparison to the fundamental trigonometric functions, the other three functions are frequently used. Take a look at the diagram below for an explanation of these three major functions. The sin-cos-tan triangle is the name for this diagram. 

Sine function:

The ratio of the opposite side length to the hypotenuse is the sine function of an angle. The value of sin, as shown in the diagram, is 

Sin a = Opposite/Hypotenuse

Cos function:

The cosine of an angle is the ratio of the neighboring side’s length to the hypotenuse’s length. The cos function can be calculated using the figure above. 

Cos a = Adjacent/Hypotenuse

Tan function:

The tangent function is the ratio of the opposing side’s length to the adjacent side’s length. It’s worth noting that the tan can also be expressed as the ratio of sine and cos. The tan function will be as follows, as seen in the diagram above. 

Tan a = Opposite/Adjacent

Tan can also be expressed in terms of sine and cos: 

Tan a = sin a/cos a

Secant, Cosecant, and Cotangent functions:

The three supplementary functions secant, cosecant (cosec), and cotangent are derived from the main functions sine, cos, and tan. Cosecant (cosec), secant (sec), and cotangent (cot) are the reciprocals of sine, cos, and tan, respectively. Each of these functions has the following formula: 

Sec a = 1/(cos a) = Hypotenuse/Adjacent 

Cosec a = 1/(sin a) = Hypotenuse/Opposite 

Cot a = 1/(tan a) = Adjacent/Opposite 

Sign of trigonometric functions in different quadrants:

In the diagram below, the sign of trigonometric functions in each quadrant is shown. 

All functions are positive in the first quadrant, only sin and cosec are positive in the second quadrant,

only tan and cot are positive in the third quadrant, and only cos and sec are positive in the fourth quadrant. As a result, the signs of various trigonometric functions in different quadrants can be tabulated as follows:

Example:

Find the value of cosec x if cot x = -5/12 and x lies in the fourth quadrant.

Solution:

Given, 

Cot x = -5/12 

We know that,

cosec²x – cot²x = 1

cosec ²x  = 1 + cot²x

                = 1 + (-5/12)²

                = 1 + (25/144)

                = (144 + 25)/144

               = 169/144 

cosec x = √(169/144) = ±13/12 

Given that x is in the fourth quadrant and cosec x in the fourth quadrant is negative. 

As a result, cosec x = -13/12 

In the same way, we may easily solve a variety of trigonometry problems. These can be used to calculate the values of trigonometry functions that are dependent on other functions as well as trigonometry angles. 

Conclusion:

The sign of a trigonometric function is determined by the coordinates of the points on the angle’s terminal side. You can figure out the signs of all trigonometric functions by knowing which quadrant the angle’s terminal side is in. The terminal side of an angle can be found in one of eight places: in any of the four quadrants, or along the axes in either a positive or negative direction (the quadrantal angles). For the signs of trigonometric functions, each scenario has a different meaning. 

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What are the 6 basic trigonometric functions?

Ans: In trigonometry, there are six functions of an angle that are ofte...Read full

How is trigonometry used in real life?

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Is cos positive in quadrant III?

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What quadrants is tan negative?

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What are the signs of the quadrants?

Ans. Both x and y have positive values in the first quadrant. X is negative while y is positive in the second...Read full

Ans:

In trigonometry, there are six functions of an angle that are often utilised. Sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), and cosecant (cosec) are their names. 

 

Ans:

 In our daily lives, trigonometry and its functions are used in a variety of ways. It’s used in geography to measure the distance between landmarks, astronomy to estimate the distance between nearby stars, and the satellite navigation system to calculate the distance between satellites. 

Ans:

The values of sin, cos, and tan in the first quadrant are all positive. The values for sin in the second quadrant are all positive. Only positive tan values are seen in the third quadrant. The values for cos in the fourth quadrant are all positive. 

 

Ans.

Because sine is positive and cosine is negative, tangent is negative for angles with their terminal arm in Quadrant II. Because sine and cosine are both negative, tangent is positive for angles with their terminal arm in Quadrant III.

Ans.

Both x and y have positive values in the first quadrant. X is negative while y is positive in the second quadrant. X and y are negative in the third quadrant, while x is positive and y is negative in the fourth quadrant. 

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