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Trigonometry is an integral part of mathematics. The study of trigonometry is not just limited to finding the functions or using the trigonometry calculator, but it also involves knowing the signs of trigonometric functions as an integral concept. The signs of trigonometric functions will depend on the terminal side’s angle points. If you know in which quadrant the terminal side of the angle lies, you can determine its sign.
There are eight regions that can have the terminal side of an angle — in the four quadrants or along the axes in the quadrantal angles (positive or negative direction). Each situation has a different meaning for the signs of trigonometric functions.
What are trigonometric functions?
Let us take a quick moment to refresh our memory of the trigonometric functions. Trigonometric functions, also called circular functions, are the various angles of a triangle. These trigonometric functions establish relationships between the angles and sides of a triangle. The essential trigonometric functions are sine, secant, tangent, cosine, cosecant, and cotangent.
The angles sine, cosine, and tangent are the primary functions, while the others are derived from these primary functions.
The six functions of trigonometry are generally defined using the right-angled triangle.
The basic trigonometry formulas are as follows:
Sine function: The sine function is the ratio of the length opposite to angle A to that of the hypotenuse.
– Sin A= Opposite/Hypotenuse
Cos function: The cos function is the ratio of the length of the side adjacent to the angle A to that of the hypotenuse.
– Cos A= Adjacent/Hypotenuse
Tan function: The tan function is the ratio of the length of the opposite side to that of the adjacent side to angle A.
– Tan A= Opposite/Adjacent
Sine and cos can also be used to determine Tan.
– Tan A= Sin A/Cos A
The other three functions, Secant, Cosecant, and Cotangent, are derived from the inverse of cos, sine, and tan.
– Sec A= 1/ Cos A= Hypotenuse/ Adjacent
– Cosec A= 1/ Sin A= Hypotenuse/ Opposite
– Cot A= 1/ Tan A= Adjacent/ Opposite
A trigonometry calculator can determine the values of the various functions. You can enter two values and easily answer the third one. However, it is necessary to memorize trigonometry formulas.
The signs of Trigonometric Functions
Trigonometric functions of sine, cosine, tangent and cotangent are based on the signs of x and y coordinates in the respective four quadrants.
The distance from the origin to a point is always positive. However, the signs of the x and y coordinates may be positive or negative.
- In the first quadrant, all functions are positive, as x and y coordinates are all positive.
- In the second quadrant, sine along with cosecant are positive. All the other four are negative.
- In the third quadrant, tangent and cotangent are positive, with others being negative.
- In the fourth quadrant, cosine and secant are positive, and the other four functions are negative.
The reciprocal of a number has the same sign as that of the original number.
Here is a little trick to remember the functions according to the quadrants. Remember the ASTC rule, which can be remembered as ‘Add Sugar to Coffee.’ The first letter of the first word is ‘A,’ which denotes that all the trigonometric functions are positive in the first quadrant. The first letter of the second word, ‘S’, indicates that sine and its reciprocal cosec are positive in the second quadrant. The first letter of the third word, ‘T,’ indicates that tangent and reciprocal Cotan are positive in the third quadrant. Lastly, the first letter of the last word, ‘C,’ indicates that cosine and its reciprocal Secant are positive in the fourth quadrant.
Trigonometric Function Quadrant I Quadrant II Quadrant III Quadrant IV
sin α Positive Positive Negative Negative
cos α Positive Negative Negative Positive
tan α Positive Negative Positive Negative
cosec α Positive Positive Negative Negative
sec α Positive Negative Negative Positive
cot α Positive Negative Positive Negative
Table showing quadrants
This table sums up signs of different trigonometric functions in different quadrants.
Values of Quadrantal Angles
Trigonometric Function 0 π/2 π 3π/2
sin α 0 1 0 -1
cos α 1 0 -1 0
tan α 0 Undefined 0 Undefined
cosec α Undefined 0 Undefined 0
sec α 1 Undefined -1 Undefined
cot α Undefined 1 Undefined -1
The values of trigonometric functions are either 0, 1, -1 or undefined when an angle lies along an axis. When the value is undefined, the ratio for that particular function is divisible by zero. These points are technically not in the domain of that function.
It is seen from the table that the domain of sine and cos are all numbers.
Other kinds of trigonometry
The branch of trigonometry that studies the ratio between the sides of a right-angled triangle is called core trigonometry. Thanks to the progress of academics, there are three more kinds of trigonometry used today, which include plane, spherical and analytic.
Plane trigonometry calculates the angles for plain triangles. This angle has mainly three vertices on the surface, and the sides are all straight lines. The sum of the plane equals 180 instead of the conventional 90.
Spherical trigonometry is used to calculate the angles of triangles drawn on a sphere. The measure of all angles in this branch of trigonometry is more significant than 180 degrees. Latitude and longitude variables are used along with the trigonometric functions. Generally, to determine distances, mapmakers and navigation enthusiasts use spherical trigonometry regularly.
Analytic trigonometry is the study of calculations in relation to half and double angles. This branch of trigonometry, mainly used in engineering, seeks to determine values based on the x-y plane. One function of the sum of two angles is used to determine the output of a double angle. For example, the sine of the sum of two angles is used to obtain the sine of a double angle. Various trigonometry formulas are also used to determine the values of half angles by using square roots and divisions.
Conclusion
Trigonometry is undoubtedly a very interesting yet complex concept. Understanding trigonometric signs are vital, as it helps define trigonometric functions. The signs of trigonometric functions depend upon the terminal side of the angle, and with various techniques, you can quickly memorize them. It is necessary because you may not always have a trigonometry calculator handy.
