Trigonometry

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20

TRIGONOMETRIC IDENTITIES

Reciprocal identities

Tangent and cotangent identities

Pythagorean identities

Sum and difference formulas

Double-angle formulas

Half-angle formulas

Products as sums

Sums as products

AN IDENTITY IS AN EQUALITY that is true for any value of the variable. (An equation is an equality that is true only for certain values of the variable.)

In algebra, for example, we have this identity:

(x + 5)(x − 5) = x² − 25.

The significance of an identity is that, in calculation, we may replace either member with the other.  We use an identity to give an expression a more convenient form.  In calculus and all its applications, the trigonometric identities are of central importance.

On this page we will present the main identities.  The student will have no better way of practicing algebra than by proving them.  Links to the proofs are below.

Reciprocal identities

sin θ  =      1  
csc θ
        csc θ  =      1  
sin θ
 
cos θ  =      1  
sec θ
        sec θ  =      1  
cos θ
 
tan θ  =      1  
cot θ
        cot θ  =      1  
tan θ

Proof

Again, in calculation we may replace either member of the identity with the other.  And so if we see "sin θ", then we may, if we wish, replace

  it with "    1  
csc θ
";  and, symmetrically, if we see  "   1  
csc θ
", then we may

replace it with "sin θ".

Tangent and cotangent identities

tan θ  =   sin θ
cos θ
         cot θ  =   cos θ
sin θ

Proof

Pythagorean identities

a) sin²θ + cos²θ   =   1
 
b) 1 + tan²θ   =   sec²θ
 
c) 1 + cot²θ   =   csc ²θ

a')     sin²θ  =  1 − cos²θ.       cos²θ  =  1 − sin²θ.

These are called Pythagorean identities, because, as we will see in their proof, they are the trigonometric version of the Pythagorean theorem.

The two identities labeled a') -- "a-prime" -- are simply different versions of a).  The first shows how we can express sin θ in terms of cos θ; the second shows how we can express cos θ in terms of sin θ.

Note:  sin²θ -- "sine squared theta" -- means (sin θ)².

Problem.   A 3-4-5 triangle is right-angled.

right triangle

a)  Why?

To see the answer, pass your mouse over the colored area.
To cover the answer again, click "Refresh" ("Reload").

It obeys the Pythagorean theorem.

b)  Evaluate the following:

sin²θ = 16
25
  cos²θ =  9 
25
  sin²θ + cos²θ = 1.

Example 1.   Show:

trigonometric identities trigonometric identities  
 
Solution:  The problem means that we are to write the left-hand side, and then show, through substitutions and algebra, that we can transform it to look like the right hand side.  We begin:
 
trigonometric identities trigonometric identities Reciprocal identities
 
  trigonometric identities on adding the fractions
 
  trigonometric identities Pythagorean identities
 
  trigonometric identities  
 
  trigonometric identities Reciprocal identities

That is what we wanted to show.

Sum and difference formulas

sin (alpha + β)  =  sin alpha cos β + cos alpha sin β
sin (alphaβ)  =  sin alpha cos β − cos alpha sin β
cos (alpha + β)  =  cos alpha cos β − sin alpha sin β
cos (alphaβ)  =  cos alpha cos β + sin alpha sin β

Note:  In the sine formulas, + or − on the left is also + or − on the right.  But in the cosine formulas, + on the left becomes − on the right; and vice-versa.

Since these identities are proved directly from geometry, the student is not normally required to master the proof.  However, all the identities that follow are based on these sum and difference formulas.  The student should definitely know them.

Here is the proof of the sum formulas.

Example 2.   Evaluate sin 15°.

Solution. sin 15° trigonometric identities  
 
  trigonometric identities   Formulas
 
  trigonometric identities   Topics 4 and 5
 
  trigonometric identities  

Example 3.   Prove:

  trigonometric identities trigonometric identities  
 
Solution.  trigonometric identities trigonometric identities   Tangent identity
 
    trigonometric identities   Formulas
 
      We will now construct  tan alpha  by dividing the first term in the
numerator by cos alpha cos β.  But then we must divide every term by
cos alpha cos β:
    trigonometric identities    
 
    trigonometric identities    

That is what we wanted to prove.

Double-angle formulas

trigonometric identities

Proof

There are three versions of cos 2alpha.  The first is in terms of both cos alpha and sin alpha.  The second is in terms only of cos alpha.  The third is in terms only of sin alpha

Example 4.    Show:   sin 2alpha trigonometric identities    
 
Solution.  sin 2alpha  = 2 sin alpha cos alpha   Formulas
 
      We will now construct  tan alpha  by dividing by cos alpha.  But to preserve the equality, we must also multiply by cos alpha.
 
    trigonometric identities   Lesson 5 of Algebra
 
    trigonometric identities    
 
    trigonometric identities   Reciprocal identities
 
    trigonometric identities   Pythagorean identities

That is what we wanted to prove.

Example 5.   Show:   sin x trigonometric identities
Solution.    sin x trigonometric identities trigonometric identities
  -- according to the previous identity with alpha x
2
.

Half-angle formulas

The following half-angle formulas are inversions of the double-angle formulas, because alpha is half of 2alpha.

trigonometric identities

The plus or minus sign will depend on the quadrant.  Under the radical, the cosine has the + sign; the sine, the − sign.

Proof

  Example 6.   Evaluate cos  π
8
.
  Solution.   Since  π
8
 is half of   π
4
, then according to the half angle formula:
trigonometric identities trigonometric identities  
 
  trigonometric identities   Topic 4
 
  trigonometric identities   Lesson 23 of Algebra
 
  trigonometric identities    
 
  trigonometric identities   Lesson 27 of Algebra
  Example 7.   Derive an identity for tan  alpha
2
.
  Solution.   tan  alpha
2
 =  trigonometric identities Tangent identity
 
          =  trigonometric identities Half angle formulas
 
          =  trigonometric identities  
 
          =  trigonometric identities Lesson 19 of Algebra
 
          =  trigonometric identities Pythagorean identity a'
 
          =  trigonometric identities  
 
          =  trigonometric identities  

on dividing both numerator and denominator by cos alpha.

Products as sums

a)  sin alpha cos β   =   ½[sin (alpha + β) + sin (alphaβ)]
 
b)  cos alpha sin β   =   ½[sin (alpha + β) − sin (alphaβ)]
 
c)  cos alpha cos β   =   ½[cos (alpha + β) + cos (alphaβ)]
 
d)  sin alpha sin β   =   −½[cos (alpha + β) − cos (alphaβ)]

Proof

Sums as products

e)  sin A + sin B  =  2 sin ½ (A + B) cos ½ (AB)
 
f)  sin A − sin B  =  2 sin ½ (AB) cos ½ (A + B)
 
g)  cos A + cos B  =  2 cos ½ (A + B) cos ½ (AB)
 
h)  cos A − cos B  =  −2 sin ½ (A + B) sin ½ (AB)

In the proofs, the student will see that the identities e) through h) are inversions of a) through d) respectively, which are proved first.  The identity f) is used to prove one of the main theorems of calculus, namely the derivative of sin x.

The student should not attempt to memorize these identities.  Practicing their proofs -- and seeing that they come from the sum and difference formulas -- is enough.


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