Identities, Sum and Difference Forumulas

Trigonometric Identities
• sin(angle) = y/r
• cos(angle)= x/r
• tan(angle) = y/x = sin(angle)/cos(angle)
• sin²(angle) + cos²(angle) = 1
• Rewriting the above identity gives sin²(angle) = 1 - cos²(angle)
• or cos²(angle) = 1 - sin²(angle)

  We use identities to alter the form of an expression or an equation to make it easier to work with in solving problems or modeling some phenomenon. Even a very simple problem can get quite complex very quickly. The figure on the right shows an airborne vehicle at a given point in the sky. The distance to vehicle, slant range, may be determined by a radar. The radar is point up at some elevation angle but it has also been turned counterclockwise to some azimuth angle. If we want to get the ground range from the radar to a point directly under the vehicle we could write

  • cos(elevation angle) = ground range/slant range
  • ground range = slant range * cos(elevation angle).

  If we now want to get the x and y components of the ground range we write

  • cos(azimuth) = x component/ground range
  • x component = ground range * cos(azimuth)
  • x component = slant range*cos(elevation)*cos(azimuth).

  As you can see this simple situation has started to get complex, we already have a cos times another cos. If would be simple to generate much more complex situations. So we want identities and we want to be able to manipulate and simplify trigonometric expressions and functions. Fortunately, many of the rules that you learned in Algebra will also work in trigonometry. Let's first look at expressions and some basic manipulations that we can perform.

  1. Factoring: sin(B)² - sin(B) * Cos(B) = sin(B) * [sin(B) - 1]
  2. Substitution: tan(B) * cos(B) = sin(B)/cos(B) * cos(B) = sin(B) (we used the identity tan(B)=sin(B)/cos(B)
  3. Common demoninator: 1/sin(B) + 1/cos(B) = cos(B)/[sin(B)*cos(B)] + sin(B)/[sin(B)*cos(B)] = [cos(B) + sin(B)]/[sin(B)*cos(B)]
  4. Squaring and substitution: sin(B) + cos(B) can be squared to give sin²(B) + 2 sin(B) cos(B) + cos²(B)
  5. Next we can group terms: sin²(B) + cos²(B) + 2 sin(B) cos(B)
  6. Finally we can use an identity to write: 1 + 2 sin(B) cos(B)

  Sum and Difference Formulas

  

  Expanding the squared terms gives:

  • Cos(B)² - 2Cos(A)Cos(B) + Cos(A)² =
  • On the right hand side: Cos(B-A)² - 2Cos(B-A) + 1 + Sin(B-A)²
  • On the left hand side: Cos(B)² - 2Cos(B)Cos(A) + 1 + Sin(B)² - 2Sin(B)Sin(A) + Sin(A)²
  • Simplifying the right-hand side: 2 - 2Cos(B - A)
  • Simplifying the left-hand side: 2 - 2Cos(A)Cos(B) - 2Sin(A)Sin(B)
  • Equating the two results and simplifying gives: -2Cos(B-A) = -2Cos(A)Cos(B) - 2Sin(A)Sin(B)
  • Dividing through by -2: Cos(B-A) = Cos(A)Cos(B) + Sin(A)Sin(B), which is our final formula

  The sum and difference formulas for sine, cosine and tangent are shown below:

  1. Cos(A - B) = Cos(A)Cos(B) + Sin(A)Sin(B)
  2. Cos(A + B) = Cos(A)Cos(B) - Sin(A)Sin(B)
  3. Sin(A - B) = Sin(A)Sin(B) - Cos(A)Cos(B)
  4. Sin(A + B) = Sin(A)Sin(B) + Cos(A)Cos(B)
  5. Tan(A - B) = [Tan(A) - Tan(B)]/[1 + Tan(A)Tan(B)]
  6. Tan(A + B) = [Tan(A) + Tan(B)]/[1 - Tan(A)Tan(B)]

  We can use the formulas in several ways. One use is to derrive other formulas that we can use to solve common problems. For example in a right triangle one if one of the angles is A and one is 90^o the third angle must be 180^o - 90^o - A = 90^o - A If we wanted a formula for Sin(90^o - A) we could use the sum and differenece formulas as follows:

  1. Sin(90^o - A) = Sin(90^o )Cos(A) - Cos(90^o)Sin(A)
  2. Since Cos(90^o) = 0 and Sin(90^o) = 1 this reduces to
  3. Sin(90^o - A) = Cos(A)