|Class 11 Maths Trigonometric Functions||Trigonometric Functions (Any Angle)|
Trigonometric Functions (Any Angle)
In trigonometric ratios, we have studied trigonometric ratios for acute angles as the ratio of sides of a right angled triangle. We will now extend the definition of trigonometric ratios to any angle in terms of radian measure and study them as trigonometric functions.
Let’s take a xy plane & draw a circle with radius (PO) 1 cm & center as center of xy plane. Since one complete revolution subtends an angle of 2π radian at the centre of circle, ∠AOB = π/2, ∠AOC = π and ∠AOD = 3π/2 . All angles which are integral multiples of π/2 are called quadrantal angles. Let us name these quadrants as Quadrant I, II, III & IV.
In Triangle POM (Quadrant I), Sinθ = b/1 = b , cosθ =a/1 = a & tanθ = b/a.
Now rotate the line PO anticlockwise & observe values of Sinθ, Cosθ & Tan θ.
You will observe that
Signs of Cosec θ, Sec θ & Cot θ can easily be determined using signs of Sinθ, Cosθ & Tan θ respectively.
Memory Tip to remember Signs: Add sugar to coffee
Trigonometric Function for θ > 360 degree
If we rotate (clockwise or anticlockwise) line OP by 360o, it will come back to same position. Thus if θ increases (or decreases) by any integral multiple of 2π, the values of sine and cosine functions do not change. Thus,
sin(2nπ + θ) = sinθ , n ∈ Z ,
cos(2nπ + θ) = cosθ, n ∈ Z
tan(2nπ + θ) = tanθ, n ∈ Z
Note that, in the above scenario, Sinθ = b/1 = b , cosθ =a/1 = a & tanθ = b/a.
Also, in right Triangle POM , a2 + b2 =1
Using these 2 equations we can say that sin2 θ+ cos2 θ= 1
Also we can prove that