TRIGONOMETRY

Trigonometry is the study of relationships that deal with angles, lengths and heights of triangles and relations between different parts of circles and other geometrical figures. Applications of trigonometry are also found in engineering, astronomy, Physics and architectural design.
Trigonometric identities are very useful and learning the below formulae help in solving the problems better. There is an enormous number of fields where these identities of trigonometry and formula of trigonometry are used.
Now to get started let us start with noting the difference between Trigonometric identities and Trigonometric Ratios.

• Trigonometric Identities are some formulas that involve the trigonometric functions. These trigonometry identities are true for all values of the variables.
• Trigonometric Ratio is known for the relationship between the measurement of the angles and the length of the side of the right triangle.

Now let us start with the basic formulas of trigonometry and see the basic relationships on which the whole concept is based on.
In a right-angled triangle, we have Hypotenuse, Base and Perpendicular. The longest side is known as the hypotenuse, the other side which is opposite to the angle is Perpendicular and the third side is Base. The six trigonometric functions are sine, cosine, secant, cosecant, tangent and cotangent. So now all the trigonometric ratios are based on the lengths of these lengths of the side of the triangle and the angle of the triangle.

The Trigonometric properties are given below:

Square law of Trigonometry
The basic trigonometric identities based on the Pythagoras Theorem are listed here. You can use the basic definition and Pythagoras theorem to prove these.

Negative Angles
Trigonometric ratios for negative angles can be derived using the circular concept of negative angles and can be derived using cartesian notation and conventions.

Periodicity and Periodic Identities
The basic concept of trigonometry is based on the repetition of the values of sine, cos and tan after 360° due to their periodic nature.
If n is an integer and  in radians (if  in degrees the replace  with 360)

Reduction formulas
If the angles are given in any of the four quadrants then the angle can be reduced to the equivalent first quadrant by changing signs and trigonometric ratios:

• First Quadrant
  
  
   
• Second Quadrant
  
• Third Quadrant
  
  
   
• Fourth Quadrant
  
  

Sum to product rules
Some identities related to sum and difference of two angles can be listed as follows:

Product to sum rules
And from the above identities we can derive the multiplication rules and the summation rules of two trigonometric ratios:



Inverse trigonometric functions
Now let us talk about inverse trigonometric relations. They are sometimes denoted with a -1 in the superscript of the trigonometric ratios and sometimes also denoted using arc as a prefix, for example, sin^-1, cos^-1, arctan etc.

Angle value table

SOME EXAMPLE AND SOLUTION
Example 1: If tan 45° + cosec 30° = x, then find the value of x.
1)   √3
2)   (1 - 2√2)/√2
3)   (v3 - 4)/2√3
4)    3 

Solution: 
We know that,
tan 45° = 1
cosec 30° = 2
According to the given information,
x = tan 45° + cosec 30° = 1 + 2 = 3

Example 2: What is the value of [sin(90° - A) + cos(180° - 2A)] / [cos(90° - 2A) + sin(180° - A)]?
1) sin(A/2).cosA
2) cot(A/2)
3) tan(A/2)
4) sinA.cos(A/2)

Solution: 
[sin(90° - A) + cos(180° - 2A)] / [cos(90° - 2A) + sin(180° - A)]
(cosA - cos2A) / (sin2A + sinA)
[2sin{(A + 2A)/2}.sin{(2A - A)/2}] / [2sin{(A + 2A)/2}.cos{(2A - A)/2}]
(sin 3A/2) . (sinA/2) /(sin 3A/2) . (cos A/2)
tan(A/2)

Example 3: If cot(A/2) = x, then the value of x is?
1) √[(1 + cosA)/(1 - cosA)]
2) cosecA - cotA
3) √[(1 - cosA)/2]
4) √[(1 + cosA)/2]

Solution: 
As per the given data,
cot (A/2) = x
cos (A/2)/sin (A/2) = x
we know that:

√(1+cosA)/2/√(1-cosA)/2) = x
x = √(1+cosA)/√(1-cosA)

Example 4:  The value of sin 10° sin 30° sin50° sin 70° will be -
a) 4/25
b) 1/16
c) 1/8
d) 3/16

Solution: 
sin 10° sin 30° sin 50° sin 70° = (1/2) sin10° sin50° sin70°
Multiplying and dividing by 2cos10° we get,
 sin 10° sin 30° sin50° sin 70° = 2 × cos10° sin10° sin50° sin70° × (1/(4cos10°))
= sin20° sin50° sin70° × (1/(4cos10°))      ----( sin2A = 2sinAcosA)
Multiply and divide by 2,
sin 10° sin 30° sin50° sin 70° = 2sin20° sin50° sin70° × (1/8cos10°)
= 2sin20° sin50° sin(90 - 20)° × (1/8cos10°)
= 2sin20° sin50° cos20° × (1/8cos10°)
= Sin40° sin50° × (1/8cos10°)      ----( sin2A = 2sinAcosA)
Multiply and divide by 2,
= 2 × Sin40° sin(90 - 40)° × (1/16cos10°)
= 2 × Sin40° cos40° × (1/16cos(90 - 80)°)
= sin80°/(16sin80°) = (1/16)
 sin 10° sin 30° sin50° sin 70° = 1/16