TRIGONOMETRIC FUNCTIONS

Introduction

Trigono + Metry = Triangle + Measure

Triangle is any figure that has 3 angles. Angle is the measure of rotation of a line about a fixed point.

Trigonometry is used in many areas such as the science of seismology, designing electric circuits, describing the state of an atom, predicting the heights of tides in the ocean, analyzing a musical tone and in many other areas.

Parts of Angles:   Initial side, Terminal Side & Vertex

 Two types of Angles:  Positive & Negative

Measure of angle is amount of rotation performed to get the terminal side from the initial side.

Type of measure of angle:  Degree Measure & Radian Measure

Degree Measure

• 1° = 1/360th of rotation. ( 1 complete rotation = 360 degree)
• 1° = 60′   (1 degree =60 minutes)
• 1′ = 60″  ( 1 minute = 60 seconds)

Lets draw 360°,180°, 270°, 410°, – 30°, 50°

Radian Measure:

• 1 radian is the Angle subtended at centre by arc of length 1 unit in a circle of radius 1 unit.
• One revolution = 2π radian

Relationship between Degree & Radian

2π radian =360o = One revolution

Or 1 radian = 180°/ π = 57° 16′ approx.

Also,  1° = (π / 180) radian = 0.01746 radian approx.

Degree to Radian Formula

• Radian measure = (π/ 180) × Degree measure
• Degree measure = (180/π) x Radian measure

Numerical:  Convert 60° to radian form & Convert π into degree.

Solution: Lets first convert 60° to radian

Radian measure = (π/ 180) × Degree measure = (π/ 180) * 60°   = π/3 radian

Now let’s convert π into degree.

Degree measure = (180/π) x Radian measure = 180/π * π  = 180o

Trigonometric Ratios (Acute Angles)

Mantra : Pandit Badri Prasad.. Hare hare Bol. 

Using this logic ,In right ΔABC

sine of ∠ A = P/H = BC/AC

cosine of ∠ A = B/H  = AB/AC

tangent of ∠ A = P/B = BC/AB

cosecant of ∠ A = 1/ sine of A = AC/BC

secant of ∠ A = 1/cosine of A  = AC/AB

cotangent of ∠ A = 1/tangent of A = AB/BC

The value of sin A or cos A never exceeds 1, whereas the value of sec A or cosec A is always greater than or equal to 1.

Trigonometric ratios of an acute angle in a right triangle express the relationship between the angle and the length of its sides.

The values of the trigonometric ratios of an angle do not vary with the lengths of the sides of the  triangle, if the angle remains the same.

Greek letter θ (theta) is also used to denote an angle

Let’s see the values of trigonometric ratios for angles 0°, 30°, 45°, 60° and 90°.

Trigonometric Function for θ > 360 degree

From the table above you can observe that as ∠ A increases from 0° to 90°, sin A increases from 0 to 1 and cos A decreases from 1 to 0.


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

Memory Tip to remember Signs: Add sugar to coffee

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

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