Question

What trig functions are positive in which quadrants?

Answer 1

In this question, we need to explain the trig functions are positive in the four quadrants. Trigonometric functions are also known as a Circular Functions can be simply defined as the functions of an angle of a triangle. It means that the relationship between the angles and sides of a triangle are given by these trig functions.

Complete step by step solution:
The distance from a point to the origin is always positive, but the signs of the$x$and $y$ coordinates may be positive or negative. Thus, in the first quadrant, where$x$ and $y$ coordinates are all positive, all six trigonometric functions have positive values. In the second quadrant, only sine and cosecant (the reciprocal of sine) are positive. In the third quadrant, only tangent and cotangent are positive. Finally, in the fourth quadrant, only cosine and secant are positive.
The signs of the functions in the four quadrants are represented as a graph is given as follows.

Additional information:
The values of quadrantal angles, when an angle lies along an axis, the values of the trigonometric functions are either $0,1, - 1,$ or undefined. When the value of a trigonometric function is undefined, it means that the ratio for that given function involves division by zero. The values of a function are undefined are technically not in the domain of that function. Therefore, the domain of sine and cosine is all real numbers. The domain of tangent and secant is all real numbers except $\dfrac{\pi }{2} + k\pi$ where $k$ is an integer. The domain of cosecant and cotangent are all real numbers except $k\pi$, where $k$ is an integer.

Note:
We note that, in the first quadrant, where $x$ and $y$ coordinates are all positive, all six trigonometric functions have positive values. In the second quadrant, only sine and cosecant (the reciprocal of sine) are positive. In the third quadrant, only tangent and cotangent are positive. Trigonometric functions are elementary functions, the argument of which is an angle. Trigonometric functions describe the relation between the sides and angles of a right triangle. For each of these functions, there is an inverse trigonometric function. The trigonometric functions can be defined using the unit circle.