25/05/2018, 15:53

Sum and Difference Identities

Mount McKinley, in Denali National Park, Alaska, rises 20,237 feet (6,168 m) above sea level. It is the highest peak in North America. (credit: Daniel A. Leifheit, Flickr) How can the height of a mountain be measured? What about the distance ...

Mount McKinley, in Denali National Park, Alaska, rises 20,237 feet (6,168 m) above sea level. It is the highest peak in North America. (credit: Daniel A. Leifheit, Flickr)

How can the height of a mountain be measured? What about the distance from Earth to the sun? Like many seemingly impossible problems, we rely on mathematical formulas to find the answers. The trigonometric identities, commonly used in mathematical proofs, have had real-world applications for centuries, including their use in calculating long distances.

The trigonometric identities we will examine in this section can be traced to a Persian astronomer who lived around 950 AD, but the ancient Greeks discovered these same formulas much earlier and stated them in terms of chords. These are special equations or postulates, true for all values input to the equations, and with innumerable applications.

In this section, we will learn techniques that will enable us to solve problems such as the ones presented above. The formulas that follow will simplify many trigonometric expressions and equations. Keep in mind that, throughout this section, the term formula is used synonymously with the word identity.

Finding the exact value of the sine, cosine, or tangent of an angle is often easier if we can rewrite the given angle in terms of two angles that have known trigonometric values. We can use the special angles, which we can review in the unit circle shown in [link].

The Unit Circle

We will begin with the sum and difference formulas for cosine, so that we can find the cosine of a given angle if we can break it up into the sum or difference of two of the special angles. See [link].

Sum formula for cosine cos( α+β )=cos α cos β−sin α sin β
Difference formula for cosine cos( α−β )=cos α cos β+sin α sin β

First, we will prove the difference formula for cosines. Let’s consider two points on the unit circle. See [link]. Point  P  is at an angle  α  from the positive x-axis with coordinates  ( cos α,sin α )  and point  Q  is at an angle of  β  from the positive x-axis with coordinates  ( cos β,sin β ).  Note the measure of angle  POQ  is  α−β. 

Label two more points:  A  at an angle of  ( α−β )  from the positive x-axis with coordinates  ( cos( α−β ),sin( α−β ) );  and point  B  with coordinates  ( 1,0 ).  Triangle  POQ  is a rotation of triangle  AOB  and thus the distance from  P  to  Q  is the same as the distance from  A  to  B.

We can find the distance from  P  to  Q  using the distance formula.

d PQ = (cos α−cos β) 2 + (sin α−sin β) 2        = cos 2 α−2 cos α cos β+ cos 2 β+ sin 2 α−2 sin α sin β+ sin 2 β

Then we apply the Pythagorean identity and simplify.

= ( cos 2 α+ sin 2 α)+( cos 2 β+ sin 2 β)−2 cos α cos β−2 sin α sin β = 1+1−2 cos α cos β−2 sin α sin β = 2−2 cos α cos β−2 sin α sin β

Similarly, using the distance formula we can find the distance from  A  to  B.

d AB = (cos(α−β)−1) 2 + (sin(α−β)−0) 2       = cos 2 (α−β)−2 cos(α−β)+1+ sin 2 (α−β)

Applying the Pythagorean identity and simplifying we get:

= ( cos 2 (α−β)+ sin 2 (α−β))−2 cos(α−β)+1 = 1−2 cos(α−β)+1 = 2−2 cos(α−β)

Because the two distances are the same, we set them equal to each other and simplify.

2−2 cos α cos β−2 sin αsin β = 2−2 cos(α−β)   2−2 cos α cos β−2 sin αsin β=2−2 cos(α−β)        

Finally we subtract  2  from both sides and divide both sides by  −2.

cos αcos β+sin αsin β=cos( α−β )  

Thus, we have the difference formula for cosine. We can use similar methods to derive the cosine of the sum of two angles.

A General Note
Sum and Difference Formulas for Cosine

These formulas can be used to calculate the cosine of sums and differences of angles.

cos( α+β )=cos αcos β−sin αsin β
cos( α−β )=cos αcos β+sin αsin β
How To

Given two angles, find the cosine of the difference between the angles.

  1. Write the difference formula for cosine.
  2. Substitute the values of the given angles into the formula.
  3. Simplify.
Finding the Exact Value Using the Formula for the Cosine of the Difference of Two Angles

Using the formula for the cosine of the difference of two angles, find the exact value of  cos( 5π 4 − π 6 ).

Use the formula for the cosine of the difference of two angles. We have

   cos(α−β)=cos αcos β+sin αsin β cos( 5π 4 − π 6 )=cos( 5π 4 )cos( π 6 )+sin( 5π 4 )sin( π 6 )                    =( − 2 2 )( 3 2 )−( 2 2 )( 1 2 )                    =− 6 4 − 2 4                    = − 6 − 2 4
Try It

Find the exact value of  cos( π 3 − π 4 ).

2 + 6 4

Finding the Exact Value Using the Formula for the Sum of Two Angles for Cosine

Find the exact value of  cos( 75 ∘ ).

As   75 ∘ = 45 ∘ + 30 ∘ , we can evaluate  cos( 75 ∘ )  as  cos( 45 ∘ + 30 ∘ ).  Thus,

cos( 45 ∘ + 30 ∘ )=cos( 45 ∘ )cos( 30 ∘ )−sin( 45 ∘ )sin( 30 ∘ )                        = 2 2 ( 3 2 )− 2 2 ( 1 2 )                        = 6 4 − 2 4                        = 6 − 2 4
Try It

Find the exact value of  cos( 105 ∘ ).

2 − 6 4

The sum and difference formulas for sine can be derived in the same manner as those for cosine, and they resemble the cosine formulas.

A General Note
Sum and Difference Formulas for Sine

These formulas can be used to calculate the sines of sums and differences of angles.

sin( α+β )=sin α cos β+cos α sin β
sin( α−β )=sin α cos β−cos α sin β
How To

Given two angles, find the sine of the difference between the angles.

  1. Write the difference formula for sine.
  2. Substitute the given angles into the formula.
  3. Simplify.
Using to Evaluate the Difference of Angles

Use the sum and difference identities to evaluate the difference of the angles and show that part a equals part b.

  1. sin( 45 ∘ − 30 ∘ )
  2. sin( 135 ∘ − 120 ∘ )
  1. Let’s begin by writing the formula and substitute the given angles.
           sin(α−β)=sin α cos β−cos α sin β sin( 45 ∘ − 30 ∘ )=sin( 45 ∘ )cos( 30 ∘ )−cos( 45 ∘ )sin( 30 ∘ )

    Next, we need to find the values of the trigonometric expressions.

    sin( 45 ∘ )= 2 2 , cos( 30 ∘ )= 3 2 , cos( 45 ∘ )= 2 2 , sin( 30 ∘ )= 1 2

    Now we can substitute these values into the equation and simplify.

    sin( 45 ∘ − 30 ∘ )= 2 2 ( 3 2 )− 2 2 ( 1 2 )                        = 6 − 2 4
  2. Again, we write the formula and substitute the given angles.
              sin(α−β)=sin α cos β−cos α sin β sin( 135 ∘ − 120 ∘ )=sin( 135 ∘ )cos( 120 ∘ )−cos( 135 ∘ )sin( 120 ∘ )

    Next, we find the values of the trigonometric expressions.

    sin( 135 ∘ )= 2 2 ,cos( 120 ∘ )=− 1 2 ,cos( 135 ∘ )= 2 2 ,sin( 120 ∘ )= 3 2

    Now we can substitute these values into the equation and simplify.

    sin( 135 ∘ − 120 ∘ )= 2 2 ( − 1 2 )−( − 2 2 )( 3 2 )                            = − 2 + 6 4                            = 6 − 2 4 sin( 135 ∘ − 120 ∘ )= 2 2 ( − 1 2 )−( − 2 2 )( 3 2 )                            = − 2 + 6 4                            = 6 − 2 4
Finding the Exact Value of an Expression Involving an Inverse Trigonometric Function

Find the exact value of  sin( cos −1   1 2 + sin −1   3 5 ).

The pattern displayed in this problem is  sin( α+β ).  Let  α= cos −1 1 2   and  β= sin −1 3 5 .  Then we can write

cos α= 1 2 ,0≤α≤π sin β= 3 5 ,− π 2 ≤β≤ π 2

We will use the Pythagorean identities to find  sin α  and  cos β.

sin α= 1− cos 2 α        = 1− 1 4        = 3 4        = 3 2 cos β= 1− sin 2 β        = 1− 9 25        = 16 25        = 4 5

Using the sum formula for sine,

sin( cos −1   1 2 + sin −1   3 5 )=sin( α+β )                                        =sin α cos β+cos α sin β                                        = 3 2 ⋅ 4 5 + 1 2 ⋅ 3 5                                        = 4 3 +3 10

Finding exact values for the tangent of the sum or difference of two angles is a little more complicated, but again, it is a matter of recognizing the pattern.

Finding the sum of two angles formula for tangent involves taking quotient of the sum formulas for sine and cosine and simplifying. Recall,  tan x= sin x cos x ,cos x≠0.

Let’s derive the sum formula for tangent.

tan( α+β )= sin( α+β ) cos(α+β)                  = sin α cos β+cos α sin β cos α cos β−sin α sin β                  = sin α cos β+cos α sin β cos α cos β cos α cos β−sin α sin β cos α cos β   Divide the numerator and denominator by cos α cos β                  = sin α  cos β cos α  cos β + cos α  sin β cos α  cos β cos α  cos β cos α   cos β − sin α sin β cos α cos β                  = sin α cos α + sin β cos β 1− sin α sin β cos α cos β                  = tan α+tan β 1−tan α tan β

We can derive the difference formula for tangent in a similar way.

A General Note
Sum and Difference Formulas for Tangent

The sum and difference formulas for tangent are:

tan( α+β )= tan α+tan β 1−tan α tan β
tan( α−β )= tan α−tan β 1+tan α tan β
How To

Given two angles, find the tangent of the sum of the angles.

  1. Write the sum formula for tangent.
  2. Substitute the given angles into the formula.
  3. Simplify.
Finding the Exact Value of an Expression Involving Tangent

Find the exact value of  tan( π 6 + π 4 ).

Let’s first write the sum formula for tangent and substitute the given angles into the formula.

     tan( α+β )= tan α+tan β 1−tan α tan β tan( π 6 + π 4 )= tan( π 6 )+tan( π 4 ) 1−( tan( π 6 ) )( tan( π 4 ) )

Next, we determine the individual tangents within the formula:

tan( π 6 )= 1 3 ,tan( π 4 )=1

So we have

tan( π 6 + π 4 )= 1 3 +1 1−( 1 3 )(1)                   = 1+ 3 3 3 −1 3                   = 1+ 3 3 ( 3 3 −1 )                   = 3 +1 3 −1
Try It

Find the exact value of  tan( 2π 3 + π 4 ).

1− 3 1+ 3

Finding Multiple Sums and Differences of Angles

Given  sin α= 3 5 ,0<α< π 2 ,cos β=− 5 13 ,π<β< 3π 2 , find

  1. sin( α+β )
  2. cos( α+β )
  3. tan( α+β )
  4. tan( α−β )

We can use the sum and difference formulas to identify the sum or difference of angles when the ratio of sine, cosine, or tangent is provided for each of the individual angles. To do so, we construct what is called a reference triangle to help find each component of the sum and difference formulas.

  1. To find  sin( α+β ), we begin with  sin α= 3 5   and  0<α< π 2 .  The side opposite  α  has length 3, the hypotenuse has length 5, and  α  is in the first quadrant. See [link]. Using the Pythagorean Theorem, we can find the length of side  a:
    a 2 + 3 2 = 5 2 ​         a 2 =16           a=4

    Since  cos β=− 5 13   and  π<β< 3π 2 , the side adjacent to  β  is  −5, the hypotenuse is 13, and  β  is in the third quadrant. See [link]. Again, using the Pythagorean Theorem, we have

    ( −5 ) 2 + a 2 = 13 2          25+ a 2 =169                        a 2 =144                          a=±12

    Since  β  is in the third quadrant,  a=–12.

    The next step is finding the cosine of  α  and the sine of  β.  The cosine of  α  is the adjacent side over the hypotenuse. We can find it from the triangle in [link]:  cos α= 4 5 .  We can also find the sine of  β  from the triangle in [link], as opposite side over the hypotenuse:  sin β=− 12 13 .  Now we are ready to evaluate  sin( α+β ).

    sin(α+β)=sin αcos β+cos αsin β                 =( 3 5 )( − 5 13 )+( 4 5 )( − 12 13 )                 =− 15 65 − 48 65                 =− 63 65
  2. We can find  cos( α+β )  in a similar manner. We substitute the values according to the formula.
    cos(α+β)=cos α cos β−sin α sin β                  =( 4 5 )( − 5 13 )−( 3 5 )( − 12 13 )                  =− 20 65 + 36 65                  = 16 65
  3. For  tan( α+β ), if   sin α= 3 5   and  cos α= 4 5 , then
    tan α= 3 5 4 5 = 3 4

    If  sin β=− 12 13   and  cos β=− 5 13 , then

    tan β= −12 13 −5 13 = 12 5

    Then,

    tan(α+β)= tan α+tan β 1−tan α tan β                 = 3 4 + 12 5 1− 3 4 ( 12 5 )                 =    63 20 − 16 20 ​                =− 63 16
  4. To find  tan( α−β ), we have the values we need. We can substitute them in and evaluate.
    tan( α−β )= tan α−tan β 1+tan α tan β                 = 3 4 − 12 5 1+ 3 4 ( 12 5 )                 = − 33 20 56 20                 =− 33 56
Analysis

A common mistake when addressing problems such as this one is that we may be tempted to think that  α  and  β  are angles in the same triangle, which of course, they are not. Also note that

tan( α+β )= sin( α+β ) cos( α+β )

Now that we can find the sine, cosine, and tangent functions for the sums and differences of angles, we can use them to do the same for their cofunctions. You may recall from Right Triangle Trigonometry that, if the sum of two positive angles is   π 2 , those two angles are complements, and the sum of the two acute angles in a right triangle is   π 2 , so they are also complements. In [link], notice that if one of the acute angles is labeled as  θ, then the other acute angle must be labeled  ( π 2 −θ ).

Notice also that  sin θ=cos( π 2 −θ ):  opposite over hypotenuse. Thus, when two angles are complimentary, we can say that the sine of  θ  equals the cofunction of the complement of  θ.  Similarly, tangent and cotangent are cofunctions, and secant and cosecant are cofunctions.

From these relationships, the cofunction identities are formed.

A General Note
Cofunction Identities

The cofunction identities are summarized in [link].

sin θ=cos( π 2 −θ ) cos θ=sin( π 2 −θ )
tan θ=cot(
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