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

1. In Quadrant I, all Sinθ, Cosθ & Tan θ are all positive.
2. In Quadrant II only Sinθ is positive
3. In Quadrant III only Tanθ is positive
4. In Quadrant IV only Cosθ is positive

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 360^o, 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

• 1 + tan² θ = sec² θ
• 1 + cot² θ = cosec² θ