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

- In Quadrant I, all Sinθ, Cosθ & Tan θ are all positive.
- In Quadrant II only Sinθ is positive
- In Quadrant III only Tanθ is positive
- 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**

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Note that, in the above scenario, Sinθ = b/1 = b , cosθ =a/1 = a & tanθ = b/a.

Also, in right Triangle POM , a^{2} + b^{2} =1

Using these 2 equations we can say that **sin ^{2} **

Also we can prove that

**1 + tan**^{2 }**θ = sec**^{2}**θ****1 + cot**^{2}**θ = cosec**^{2}**θ**

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