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9.2 Sum and difference identities

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In this section, you will:
  • Use sum and difference formulas for cosine.
  • Use sum and difference formulas for sine.
  • Use sum and difference formulas for tangent.
  • Use sum and difference formulas for cofunctions.
  • Use sum and difference formulas to verify identities.
Photo of Mt. McKinley.
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 .

Using the sum and difference formulas for cosine

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] .

Diagram of the unit circle with points labeled on its edge. P point is at an angle a from the positive x axis with coordinates (cosa, sina). Point Q is at an angle of B from the positive x axis with coordinates (cosb, sinb). Angle POQ is a - B degrees. Point A is at an angle of (a-B) from the x axis with coordinates (cos(a-B), sin(a-B)). Point B is just at point (1,0). Angle AOB is also a - B degrees. Radii PO, AO, QO, and BO are all 1 unit long and are the legs of triangles POQ and AOB. Triangle POQ is a rotation of triangle AOB, so the distance from P to Q is the same as the distance from A to B.
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   P O Q   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   P O Q   is a rotation of triangle   A O B   and thus the distance from   P   to   Q   is the same as the distance from   A   to   B .

Diagram of the unit circle with points labeled on its edge. P point is at an angle a from the positive x axis with coordinates (cosa, sina). Point Q is at an angle of B from the positive x axis with coordinates (cosb, sinb). Angle POQ is a - B degrees. Point A is at an angle of (a-B) from the x axis with coordinates (cos(a-B), sin(a-B)). Point B is just at point (1,0). Angle AOB is also a - B degrees. Radii PO, AO, QO, and BO are all 1 unit long and are the legs of triangles POQ and AOB. Triangle POQ is a rotation of triangle AOB, so 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 P Q = ( 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 .

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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