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Trigonometric functions (Circular functions) are generally stated as functions of an angle of a triangle. They form the connection between the sides of the triangle and the angle they make with each other. There are six trigonometric functions, sine, cosine, tangent, cosecant, secant, and cotangent for a particular angle. For finding the values of Trigonometric Functions for an angle greater than 90 degrees, we have to redefine these trigonometric functions using a unit circle. For many angles, we can find the values of trigonometric functions for sine and cosine on the unit circle chart and memorize them. Still, we have to deduce using the standard trigonometric functions for angles not given in the unit circle chart.
Signs of Trigonometric Functions
Before jumping into finding the values of Trigonometric Functions without using a trigonometry calculator, we need to understand the signs of trigonometric functions. They depend on the quadrant in which the angle of trigonometric functions falls. To easily remember whether a trigonometric function is positive or negative in a particular quadrant, a mnemonic sentence “A Smart Trig Class” can be used. A represents the first quadrant where all trigonometric functions are positive. The second word, “Smart,” falls in the second quadrant, which suggests only sine functions are positive. In the third quadrant, ‘Trig’ refers to only tangential functions to be positive. Lastly, in the fourth quadrant, “Class” refers to all cos functions being positive.
The values of Trigonometric Functions from 0 to 90 degrees
You must memorize the below values of trigonometric functions between 0 to 90 degrees and must calculate trigonometric functions beyond 90 degrees using the unit circle method.
Trigonometric Ratios | 0 ° | 30 ° | 45 ° | 60° | 90° |
Sin µ | 0 | 1/2 | 1/√2 | √3/2 | 1 |
Cos µ | 1 | √3/2 | 1/√2 | 1/2 | 0 |
Tan µ | 0 | 1/√3 | 1 | √3 | ∞ |
Cosec µ | ∞ | 2 | √2 | 2/√3 | 1 |
Sec µ | 1 | 2/√3 | √2 | 2 | ∞ |
Cot µ | ∞ | √3 | 1 | 1/√3 | 0 |
Correlation between Trigonometric Functions and Unit Circle
The trigonometric functions defined using the triangle only go up to 90 degrees. To find the values of trigonometric functions beyond 90 degrees, we shall redefine the trigonometric functions in terms of a circle unit.
The X and Y-axis divide the unit circle centered at the origin into four quadrants. These quadrants are named I, II, III, and IV and mimic the direction of a positive angle sweep. For an angle t, the intersection of its side and the unit circle by its coordinates x and y. Now x and y coordinates become the output of trigonometric functions.
f(t)= cos t
f(t)= sin t
This means x= cos t, y= sin t.
For example: Consider a unit circle with coordinates (-√2/2, √2/2)
As per our previous discussion, we can calculate the values of trigonometric functions only up to 90 degrees. Using the unit circle method, we can find the values beyond 90 degrees without using a trigonometry calculator.
The values of Trigonometric Functions on the Unit Circle Diagram
For finding the values of trigonometric functions, we refer to the unit circle diagram. The coordinates of specific points and measures of every angle in degrees and radians are represented in the diagram. The unit circle diagram enables us to observe each angle using the trigonometric functions.
We will be able to locate the coordinate of any point on the unit circle diagram. For any angle t, we can find the x and y coordinate using the trigonometric formula, x= cos t, y= sin t. The unit circle diagram shows the periodicity of the trigonometric functions. Periodicity reveals how trigonometric functions output a repeated set of values at regular intervals. Look at the x values of the coordinates in the diagram for values of t from 0 to 2π.
As we can see, a pattern emerges which moves between +1 and -1. The same pattern will repeat for higher values of t. Remember the x values corresponding to the cosine of t. It indicates cosine function periodicity.
Importance of memorizing trigonometric functions
In most of the exams, a scientific calculator will not be allowed and hence using a trigonometry calculator is out of the question. Hence a thorough understanding of the unit circle and memorizing the different values of the trigonometric functions can take you a long way in solving any type of problem-related to trigonometric functions quickly.
Conclusion
We analyzed the relationship between the trigonometric functions and the unit circle and how a deep understanding of the unit circle diagram can help find the values of trigonometric functions. Before jumping into solving trigonometric functions using the unit circle method, we have understood the essential trigonometric functions and values between 0 and 90 degrees. Along with finding the values of trigonometric functions, the sign of the trigonometric functions is also essential; hence learning the phrase “A Smart Trig Class” can be an excellent trick to assign the negative or positive sign to the calculated value of trigonometric functions. Initially, it looks complicated, but proper practice and memorizing the unit circle diagram will set you on the right path of learning how to calculate the exact values of trigonometric functions.
