Trigonometric functions can be simply defined as the functions of an angle i.e. the relationship between the angles and sides of a triangle are given by these trig functions. In trigonometry, there are 6 main functions which are sine, cosine, tangent, secant, cosecant and cotangent angles.
What are Trigonometric Functions
The angles of sine, cosine, and tangent are the primary classification of trigonometric functions. Consider the following diagram as a reference for an explanation of these three primary functions. This diagram can be referred to as the sin-cos-tan triangle.
Trigonometric Functions Formula
Formulas for right angled triangle are as follows:
Sine, cosine, and tangent
Sine Function
Sine function of an angle is the ratio between the opposite side length to that of the hypotenuse. From the above diagram, the value of sin will be:
Sin a = \(\frac{Opposite}{Hypotenuse}\) = \(\frac{CB}{CA}\)
Cos Function
Cos of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse. From the above diagram, the cos function will be derived as follows.
Cos a = \(\frac{Adjacent}{Hypotenuse}\) = \(\frac{AB}{CA}\)
Tan Function
Tangent function is the ratio of the length of the opposite side to that of the adjacent side. It should be noted that the tan can also be represented in terms of sine and cos as their ratio. From the diagram taken above, the tan function will be the following.
Tan a = \(\frac{Opposite}{Adjacent}\) = \(\frac{CB}{BA}\)
Also, in terms of sine and cos, tan can be represented as:
Tan a = \(\frac{sin\ a}{cos\ a}\)
Secant, cosecant, and cotangent
Secant, cosecant (csc) and cotangent are the three additional trigonometric functions which are derived from the primary functions of sine, cos, and tan. The reciprocal of sine, cos, and tan are cosecant (csc), secant (sec), and cotangent (cot) respectively. The formula of each of these functions are given as:
Sec a = 1/(cos a) = Â \(\frac{Hypotenuse}{Adjacent}\) = \(\frac{CA}{AB}\)
Cosec a = 1/(sin a) = \(\frac{Hypotenuse}{Opposite}\) = \(\frac{CA}{CB}\)
cot a = 1/(tan a) = \(\frac{Adjacent}{Opposite}\) = \(\frac{BA}{CB}\)
Trigonometric Functions Table
The trigonometric table for six functions, Sin, Cos, Tan, Cosec, Sec, Cot, are as :
Trigonometric Ratios/ angle= a degrees |
0 Â° | 30 Â° | 45 Â° | 60 Â° |
90 Â° |
Sin a | 0 | 1/2 | 1/âˆš2 | âˆš3/2 | 1 |
Cos a | 1 | âˆš3/2 | 1/âˆš2 | 1/2 | 0 |
Tan a | 0 | 1/âˆš3 | 1 | âˆš3 | Not Defined |
Cosec a | Not Defined | 2 | âˆš2 | 2/âˆš3 | 1 |
Sec a | 1 | 2/âˆš3 | âˆš2 | 2 | Not Defined |
Cot a | Not defined | âˆš3 | 1 | 1/âˆš3 | 0 |
Inverse Trigonometric Functions
Inverse functions are used to obtain an angle from any of the angle’s trigonometric ratios. Basically inverses of the sine, cosine, tangent, cotangent, secant, and cosecant functions. These functions are represented as arc sine, arc cosine, arc tangent, arc cotangent, arc secant, and arc cosecant.
Trigonometry Examples with Answers
Example 1: Find the values of Sin 45^0, Cos 30^0 and Tan 60^0.[all angles are in degrees)
Solution: Using trigonometric table, we have
Sin 45^0 = 1/âˆš2
Cos 30^0 = 1/2
Tan 60^0 = âˆš3
Example 2: Evaluate Sin 105^0 degrees.
Solution: Sin 105^0 can be written as sin (60^0 + 45^0) which is similar to sin (A + B).
We know that, the formula for sin (A + B) = sin A x cos B + cos A x sin B
Therefore, sin 105^0 = sin (60^0 + 45^0) = sin 60^0 x cos 45^0 + cos 60^0 x sin 45^0
= âˆš3/2 x 1/âˆš2 + 1/2 x 1/âˆš2
= âˆš3/2âˆš2 + 1/2âˆš2
= (âˆš2 + âˆš6)/4
Example 3: A boy sees a bird sitting on a tree at an angle of elevation of 20^0. If boy is standing 10 miles away from the tree, at what height bird is sitting?
Solution: Consider ABC a right triangle, A is a bird’s location, B = tree is touching the ground and C = boy’s location.
So BC 10 miles, angle C = 20^0 and let AB = x miles
We know, tan C = opp/adj
tan(20^0) = x/10
or x = 10 x tan(20^0)
or x = 10 x 0.36 = 3.6
Bird is sitting at the height of 3.6 miles from the ground.
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