Chapter 10

The Trig Ratios of The Sum or Difference of Two Angles

It is possible to develop a formula for sin(a+b) in terms of sina, and sinb.  It is also possible to develop a formula for cos(a+b) in terms of cosa, and cosb.  The same can be done for tan(a+b) as well as cot(a+b).  In general, the purpose of this chapter is to calculate the trig ratios of a+b and a-b in terms of the trig ratios of just a and the trig ratios of just b.

A) The Trig Ratios of (a+b)

1) sin(a+b):

sin(a+b) = sina cosb + cosa sinb

Proof:

Pay attention to angles a (or xOy) and b (or yOz).  From A, line segment AC is drawn to Ox.  It is obvious that AC is the opposite side to angle (a + b).  Thus, sin(a+b) = AC/OA.  The rest is as follows:

Example 1: Calculate sin75°.

Solution:  sin(75°) = sin(45° + 30°) = sin45° cos30°  + cos45° sin30°

= (√2/2) (√3/2)      +  (√2/2) (1/2)

=  [√6 + √2] /4.

2) cos(a+b):

cos(a+b) = cosa cosb - sina sinb

Proof:

Pay attention to angles a (or xOy) and b (or yOz).  From P, normal PT is drawn to Oz.  It is clear that OT is the adjacent side to angle (a + b).  Thus, cos(a+b) = OT/OP.  The rest is as follows:

Example 2: Calculate cos75°.

Solution:  cos(75°) = cos(45° + 30°) = cos45° cos30°  - sin45° sin30°

= (√2/2) (√3/2)      -  (√2/2) (1/2)

=  [√6 - √2] /4.

3) tan(a+b):

Example 3: Calculate tan75°.

4) cot(a+b):

Example 4: Calculate cot75°.

B) The Trig Ratios of (a-b)

If in the formulas for (a+b), b is replaced with -b, the formulas for the difference (a-b) will simply result

1) sin(a-b):

Using sin(a+b) and replacing b with -b, we get:

sin[a-b] = sin[a+(-b)] = sina cos(-b) + cosa sin(-b)

Note that cos(-b) = cosb, but sin(-b) = -sinb ; thus the above becomes:

sin(a-b) = sina cosb - cosa sinb

2) cos(a-b):

Using cos(a+b) and replacing b with -b, we get:

cos[a-b] = cos[a+(-b)] = cosa cos(-b) - sina sin(-b)

Note that cos(-b) = cosb, but sin(-b) = -sinb ; thus the above becomes:

cos(a-b) = cosa cosb + sina sinb

3) tan(a-b):

4) cot(a-b):

Exercises:

1) Calculate the trig ratios of 15°.

Verify the following equalities:

2)  sin20° cos10° + cos20° sin10° = 1/2.

3)  sin80° cos20° - cos80° sin20° = 3/2.

4)  cos55° cos10° + sin55° sin10° = 2/2.

5)  cos85° cos35° - sin85° sin35° = -1/2.

6) (tan32° + tan28°) / (1 - tan32°tan28°) = 3.

Knowing that a and b are less than 90°, and that sina = 7/25, and tanb = 5/12, calculate the numerical value of each of the following:

7) sin(a + b),    8) cos(a - b),    9) tan(a + b),      10) cot(b - a),    11) sin(a - b)cos(a + b)

12)  Develop the formula for sin(a + b + c) in terms of the trig ratios of angles a, b, and c.

13)  Develop the formula for cos(a - b - c) in terms of the trig ratios of angles a, b, and c.

14) If tana = 3 and tanb = -2, show that   a - b = kπ + 3π/4.

Verify the following identities by using the trig ratios of the sum or difference of two angles:

15) cos(π/2 -a) = sina

16) cos(π/2 +a) = -sina

17) sin(π -a) + sin(π +a) = 0

18) cos(2π +a) + cos(2π -a) = 2cosa

19) tan(π +a) - tan(π -a) = 2tana

Verify the following identities:

20) sin3x cos2x + cos3x sin2x = sin5x

21) cos(a + 10)cos(a - 10) - sin(a +10)sin(a - 10) = cos2a

Verify the following identities:

Verify the following identities: