Trigonometric identities are very useful for right angles triangles, where you can calculate the value of its sides and angles in just minutes. Moreover, these identities are also useful in practical life situations, for example, calculation of the height of a building etc. Let’s find more about such identities in this section.
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Trigonometric Identities
In algebraic form, an identity in x is satisfied by some particular value of x. For example (x+1)2=x2+2x+1 is an identity in x. It is satisfied for all values of x. The same applies to trigonometric identities also.
The equations can be seen as facts written in a mathematical form, that is true for “right angle triangle”. Any trigonometric identity dealing with any variable of a right angle triangle will be satisfied by any value within an acceptable range of that variable.
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Trigonometric Relations
Reciprocal Relationship
As the name suggests, these relations involve two trigonometric ratios which are connected by inverse relations between them. For example,
- sin θ = 1/ cosec θ or sin θ x cosec θ = 1
- cos θ = 1/ sec θ or cos θ x sec θ = 1
- tan θ = 1/cot θ or tan θ x cot θ = 1
Quotient Relations
Again, as the name suggests, quotient relations involve three trigonometric ratios; where one is the quotient obtained after division operation between the other two. For example,
- tan θ = sin θ /cos θ
- cot θ = cos θ / sin θ
Pythagoras Relations
The following are the relations or identities derived from Pythagoras theorem, that’s why they are called Pythagorean relations.
Trigonometric Identity 1
- (sin2 θ) + (cos2 θ) = 1
Derivation
Before derivation let’s understand what is Pythagoras Theorem? For any right-angled triangle, let ABC be our right angle triangle. If angle C is at 90 degrees then AB will be the hypotenuse. Assume AC be base and BC be perpendicular. According to Pythagorean theorem for any right angle triangle:
Pythagoras Theorem
(Hypotenuse) 2= (Base) 2 + (Perpendicular) 2
(AB) 2= (AC) 2+ (BC)2
Step 1: Consider a right angle triangle ABC, let angle C be of 90.
Step 2: According to Pythagoras Theorem,
(AB) 2= (AC) 2+ (BC) 2 let be equation (1)
Dividing equation 1 by square of AB on both the sides,
(AB) 2/ (AB) 2= (AC) 2/ (AB)2+ (BC) 2/ (AB)2
=> 1= (AC) 2/ (AB)2+ (BC) 2/ (AB)2
Step 3: Now, sin θ =Perpendicular/ Hypotenuse= (AC) /(AB)
cos θ= Base/ Hypotenuse= (BC) / (AB)
Substituting the values,
1= sin2θ + cos2θ
Trigonometric Identity 2
- 1+ tan2 θ = sec2 θ
Derivation
Step 1: Divide the equation 1 by (AC) 2 we get,
(AB) 2= (AC) 2+ (BC) 2
(AB) 2/ (AC) 2= (AC) 2/(AC)2+ (BC) 2/ (AC)2
Step 2: Now,
Secθ = Hypotenuse/Base
tanθ= Perpendicular/Base
Substituting the values we get,
1+ tan2 θ = sec2 θ
Trigonometric Identity 3
- 1+ cot2 θ = cosec2 θ
The above relation can also be derived in the same way by dividing the Pythagorean equation,
(AB) 2= (AC) 2+ (BC) 2
this time divide the equation by (BC) 2 and substitute the values.
Identities do not only rule the world of algebra but also have a colony in the province of Geometry as we have seen above. Trigonometric identities are very useful for Right Angle triangles where you can calculate the value of its sides and angles in just minutes. Moreover, these identities are also useful for practical life situations, for example, calculation of heights of buildings that we have just calculated.
Solved Example for You
Q: For a right angle triangle ABC right angle at C for which angle BAC = θ and sinθ = 4/5 Find the value of cosθ.
Solution: Using the identity, sin2θ + cos2θ = 1
Putting value of sin in the identity,
(4/5)^2 + cos2θ =1
cosθ =√1-(4/5)^2 = 3/5
Question- What are Trigonometric identities?
Answer- Trigonometric identities are quite beneficial for right angles triangles. In here, you can calculate the value of its sides and angles within minutes. Furthermore, these identities are also beneficial in practical life circumstances, for instance, calculating the height of a building and so on.
Question- What are quotient relations?
Answer- As evident from the name, quotient relations comprise three trigonometric ratios. In here, one is the quotient we get after division operation between the other two. For instance, tan θ = sin θ /cos θ and cot θ = cos θ / sin θ.
Question- Who is the father of trigonometry?
Answer- Hipparchus is referred to as the father of trigonometry and was a Greek astronomer. Trigonometry is a branch of mathematics that studies the angles of sides of triangles. Moreover, Hipparchus also created the first accurate star map.
Question- What are the laws of trigonometry?
Answer- The trigonometric laws of identity comprise these significant reciprocal equalities. The tangent is the sine we divide by the cosine. The cotangent is equal to one over the tangent, or the cosine we divide by the sine. The secant is equal to one over the cosine and the cosecant is equal to one over the sine.
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