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Trigonometric Identities Part II

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Trigonometric Identities Part II - PDF

If you ever want to read it again as many times as you want, here is a downloadable PDF to explore more.

1. Introduction
2. Complement and Supplement of a Ratio
📥 Trigonometric Identities Part II - PDF

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3. Sum and Difference Formulas  4. Double Angle Formula 5. Half Angle Formula 7. Summary 8. Frequently Asked Questions 9. External References

21 December 2020

Reading Time: 9 Minutes

Introduction

This is a continuation of the first blog on Trigonometric Identities, we recommend you to read that first. To visit that please click here Trigonometric Identities Part 1

Most of us find it difficult to understand Trigonometry as it’s hard to remember so many related formulae and functions. But it can be easy if we understand what is Trigonometry and its functions how different Trigonometric identities can be proved or derived using the basic relationship of the angles and sides of the triangle.  

In this section, we continue the discussion on the rest of the Trigonometric identities. We will discuss the following:

  • Sum and Difference Formulas
  • Double Angle Formulas
  • Complement and Supplement of a Ratio

Complement and Supplement of a Ratio

Complementary Angles

Two angles are said to be complementary angles to each other if their sum is equal to 90°. In a right triangle, the sum of the two acute angles is equal to 90°. So, the two acute angles of a right triangle are always complementary to each other.
Let ABC be a right triangle, right-angled at B.  

complementary triangle

If  ∠ACB = θ, then  ∠BAC = 90° - θ and hence the angles  ∠BAC and  ∠ACB are complementary. The trigonometric identity associated with complementary ratios is as following:

 1 . For the angle θ

 2. For the angle (90° - θ)

 Comparing (1) and (2) we get     Trigonometric Ratios of   Complementary Angles

 sin θ =  AB/AC

 cos θ  =  BC/AC

 tan θ  =  AB/BC

 cosec θ =  AC/AB

 sec θ  =  AC/BC

 cot θ  =  BC/AB

 cos (90 - θ) =  AB/AC

 sin (90 - θ)  =  BC/AC

 tan (90 - θ)  =  BC/AB

cosec (90 - θ)  =  AC/AB

 sec (90 - θ)  =  AC/BC

 cot (90 - θ)  =  AB/BC

 sin θ  =  AB/AC  =  cos (90 - θ)

 cos θ  =  BC/AC  =  sin (90 - θ)

 tan θ  =  AB/BC  =  cot (90 - θ)

 cosec θ  =  AC/AB  =  sec (90 - θ)

 sec θ  =  AC/BC  =  cosec (90 - θ)

 cot θ  =  BC/AB  =  tan (90 - θ)

Supplementary Angles

Two angles are supplementary angles to each other if their sum is equal to 180°. Trigonometric-ratios of supplementary angles are given below

sin (180° - θ) = - sin θ

sin (180° + θ)  =  - sin θ

cos (180° - θ) = - cos θ

cos (180° + θ)  =  - cos θ

tan (180° - θ) = - tan θ

tan (180° + θ)  =  tan θ

sec (180° - θ) = - sec θ

csc (180° + θ)  =  - csc θ

csc (180° - θ) = -csc θ

sec (180° + θ)  =  - sec θ

cot (180° - θ) = - cot θ

cot (180° + θ)  =  cot θ


Sum and Difference Formulas

The fourth Trig identity that we will discuss about is sum and difference formulas also known as:

Sum and Difference Identities

Consider two angles, a and b, the trigonometric sum and difference formulas are as follows:
Sum:
Sin(a+b) = sin a cos b + cos a sin b
Cos(a+b) = cos a cos b – sin a sin b
Tan(a+b) = (tan a + tan b) /(1-tan a.tan b)

Difference: 
sin(a–b) = sin a cos b – cos a sin b
cos(a–b) = cos a cos b + sin a sin b
tan(a–b) = (tan a - tan b)/(1 + tan a.tan b)

 
There are 2 sum and difference formulas for each of the primary trigonometric functions Sine, Cosine, and Tangent.

How do I learn the Trigonometric Identities?
The derivation of the above trigonometric formulae can be a bit complicated because we may need a triangle, unit circle, and distance formulae and there is a little bit more to it so the best way to memorize is 

  1.  First remember - in all 3-equation right side starts with the same trig function as the left-hand side of the equation.
  2. The next thing to remember is the + and – sign used on the right side of the expansion. In the case of Sin, it is the same as the left-hand side of the expression but in the case of Cos and Tan, it’s the opposite. Look at the +/- sign highlighted in blue in the above 6 sum-difference equation.
  3. Once we place all the above we can place the two angles a & b be used in the equation as below then we just need to remember the other trig functions to be placed. 
  4. Now we can place trigonometric functions on the right - hand side. 
  5. For Sin equation we can remember it as a sandwich with outer sides both starts and ends with Sin and Cos is placed in the middle. 
  6. For Cos, we can remember that our Cos angles are together and Sin angles are together where a goes first each time and b goes second. 
  7. For Tan its more to memorize as we can think of it as very different from Cos and Sin. Here +/- is used twice in the right side of the equation where first it’s used the same as the left side of the equation and then 2nd time used as opposed to the left side. Only Tan function is used in the expansion of the equation.

This sum- difference trig identity is important as it can be used to simplify the expression and breaking it apart into the angles with the trig function instead of a compound angle for complicated calculations.

Example 1

 

 

Find the value of cos of 75° using a sum formula.

Solution: We know 75 is the sum of 30 and 45 so the Cosine sum formula can be used.as 
cos(a+b) = cos a cos b – sin a sin b 

cos (30 + 45) = cos 30 cos 45 – sin 30 sin 45

= √2/2 (√3/2) − √2/2 (1/2)   (Referring to the Trigonometric Ratio Table which can be found here: Trigonometry Table)

= √6/4 - √2/2

 

(√6–√2)/4 


Double Angle Formula

Fifth in our list of identities is Double Angle Identities or Double angle formulas. Double angle identities can be easily derived from the Sum and difference formulas. 
For example, using the sum identity for Sine we can get below Double angle identity 
sin2x = sin (x + x)
sin2x = sin x cos x + cos x sin x
sin2x = 2sinxcos x

OR

sin 2x = 2 sinx cosx

Similarly, for the cosine function, the double angle identity:
Cos2x = Cos (x + x)
Cos2x = Cos x Cos x - Sin x Sin x
Cos2x = Cos2x - Sin2

Referring from Pythagorean identity we can replace Sin2a or Cos2a and further break it down to below two double angle identities,
Cos2x = Cos2a – (1- Cos2a) referring from Pythagorean identity replace Sin2a
Cos2x = 2 Cos2x – 1……………. (a)
Or 
Cos 2x = 1 - Sin2x - Sin2x
Cos2a = 1 - 2Sin2a …………………(b)


Half Angle Formula

Half angle formulas can be derived from double angle identity. The half angle identity for the sine and cosine are derived from two of the cosine identities described earlier (a) and (b) with Double angle formula
Using Cos2x = 2 Cos2x – 1 …………. (a)

we can get below half angle identity
Cos 2(x/2) = 2 Cos2 (x/2) – 1
Cos(x) = 2 Cos2 (x/2) – 1 
2 Cos2 (x/2) = 1 + Cos(x) 
Cos2(x/2) = (1 + Cos x)/2

Cos(x/2) = ±√(1+Cosx)/2

Using Cos 2x = 1 - 2Sin2x …………………(b)
sin (x/2) = ±√(1- Cosx)/2
 

Example 2

 

 Use the double-angle formula for cosine to write Sine (6x) in terms Sin3x

Solution: Sin (6x) = Sin (3x +3x)

= Sin3x Cos3x + Cos3x Sin3x

= 2 (Sin3x Cos3x)

 

2sin3x cos3x


Summary

We have discussed most of the important Trigonometric identities and how to derive them or memorize them. But please remember it takes a lot of practice to master them.         

Here is a list of all of the main ones for reference:

Sin theta = 1/ Cosec theta   
Cos theta = 1/Sec theta       (Reciprocal)
Tan theta = 1/Cot theta        

sin (−x) = −sin (x)

cos (−x) = cos (x)

tan (−x) = −tan (x)

sec (−x) = sec (x)                   (odd/even identity)

cosec (−x) = −cosec (x)

cot (−x) = −cot (x)

Cos2x + Sin2x = 1 

1+ Tan2x = Sec2x                  (Pythagorean Identity)

Cot2x + 1 = Cosec2

Sin(a+b) = sin a cos b + cos a sin b
Cos(a+b) = cos a cos b – sin a sin b             
(Sum identity)
Tan(a+b) = (tan a + tan b) /(1-tan a.tan b)

sin 2x = 2 sinx cosx

Cos2x = 2 Cos2x – 1                (Double Angle Identity)

This is a continuation of the first blog on Trigonometric Identities, we recommend you to read that first. To visit that please click here Trigonometric Identities Part 1


Frequently Asked Questions

Complementary angle and Supplement angle identity?

 For the angle θ

 For the angle (90° - θ)

Trigonometric Ratios of  Complementary Angles

 sin θ =  AB/AC

 cos θ  =  BC/AC

 tan θ  =  AB/BC

 cosec θ =  AC/AB

 sec θ  =  AC/BC

 cot θ  =  BC/AB

 cos (90 - θ) =  AB/AC

 sin (90 - θ)  =  BC/AC

 tan (90 - θ)  =  BC/AB

cosec (90 - θ)  =  AC/AB

 sec (90 - θ)  =  AC/BC

 cot (90 - θ)  =  AB/BC

 sin θ  =  AB/AC  =  cos (90 - θ)

 cos θ  =  BC/AC  =  sin (90 - θ)

 tan θ  =  AB/BC  =  cot (90 - θ)

 cosec θ  =  AC/AB  =  sec (90 - θ)

 sec θ  =  AC/BC  =  cosec (90 - θ)

 cot θ  =  BC/AB  =  tan (90 - θ)

sin (180° - θ) = - sin θ

sin (180° + θ)  =  - sin θ

cos (180° - θ) = - cos θ

cos (180° + θ)  =  - cos θ

tan (180° - θ) = - tan θ

tan (180° + θ)  =  tan θ

sec (180° - θ) = - sec θ

csc (180° + θ)  =  - csc θ

csc (180° - θ) = -csc θ

sec (180° + θ)  =  - sec θ

cot (180° - θ) = - cot θ

cot (180° + θ)  =  cot θ

Sum and Difference Formulas?

Sin(a+b) = sin a cos b + cos a sin b
Cos(a+b) = cos a cos b – sin a sin b             

Tan(a+b) = (tan a + tan b) /(1-tan a.tan b)

sin(a–b) = sin a cos b – cos a sin b
cos(a–b) = cos a cos b + sin a sin b
tan(a–b) = (tan a - tan b)/(1 + tan a.tan b)

How are trigonometric ratios defined?

Sine A = Opposite side/Hypotenuse

Cos A = Adjacent side / Hypotenuse

Tan A = Opposite side / Adjacent side

Cot A = Adjacent side / Opposite side

Sec A = Hypotenuse / Adjacent side

Cosec A = Hypotenuse / Opposite side

Half Angle Formula?

Cos(x/2) = ±√(1+Cosx)/2

sin (x/2) = ±√(1- Cosx)/2

Cos2x formula

Cos2x = 2 Cos2x – 1 

Cot(a+b)?

cot(a+b) = [cot a. cot b - 1]/[cot b + cot a]


External References

Written by Rashi Murarka, Cuemath Teacher


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