How to Use Sum and Difference Identities to Find Exact Trig Values (With Step-by-Step Examples)

Example

a) Find the exact value of sin(165°)

b) Find the exact value of cos(π/12)

Solution a)

Since 165° is not an angle that we can easily work with using the special triangles, we must apply one of the sum and difference identities.

sin(a + b) = sin(a)cos(b) + cos(a)sin(b)

sin(a - b) = sin(a)cos(b) - cos(a)sin(b)

cos(a + b) = cos(a)cos(b) - sin(a)sin(b)

cos(a - b) = cos(a)cos(b) + sin(a)sin(b)

tan(a + b) = (tan(a) + tan(b)) / (1 - tan(a)tan(b))

tan(a - b) = (tan(a) - tan(b)) / (1 + tan(a)tan(b))

Step 1: write the angle as a sum or difference of two "friendly" angles; that is, angles that we can easily work with using the special triangles of the unit circle.

Strategy: take the given angle and subtract from it some of the "friendly" angles. For instance:

165° - 45° = 120°

So, 165° = 120° + 45°

120° and 45° are "friendly" angles, so they work just fine.

Hence, we can write:

sin(165°) = sin(120° + 45°) Step 2: apply the appropriate sum and difference identity.

In this case, we want:

sin(a + b) = sin(a)cos(b) + cos(a)sin(b)

Substituting in a = 120° and b = 45°, we get

sin(120° + 45°) = sin(120°)cos(45°) + cos(120°)sin(45°)

Step 3: Evaluate each trigonometric ratio and simplify; this task is now easy since we have rewritten the original angle as a sum of two "friendly" angles.

sin(120° + 45°) = sin(120°)cos(45°) + cos(120°)sin(45°)

sin(120° + 45°) = (√3 / 2)(√2 / 2) + (-1 / 2)(√2 / 2)

sin(120° + 45°) = (√6 / 4) + (-√2 / 4)

sin(120° + 45°) = (√6 - √2) / 4

Solution b)

Step 1: write the angle (π/12) as a sum or difference of two "friendly" angles. The idea here is that the angles you pick will simplify to reveal they are "friendly" angles that lie in the domain [0, 2π).

Observe:

(3π/12) - (2π/12) simplifies to (π/12)

However, we have that:

• (3π/12) = π/4

• (2π/12) = π/6

And we know π/4 and π/6 are "friendly" angles that lie in the domain [0, 2π). We can easily evaluate these using special triangles.

Hence, we can write:

cos(π/12) = cos((3π/12) - (2π/12)) = cos((π/4) - (π/6))

Step 2: apply the appropriate sum and difference identity.

In this case, we want:

cos(a - b) = cos(a)cos(b) + sin(a)sin(b)

Substituting in a = π/4 and b = π/6, we get

cos((π/4) - (π/6)) = cos(π/4)cos(π/6) + sin(π/4)sin(π/6)

Step 3: Evaluate each trigonometric ratio and simplify;  this task is now easy since we have rewritten the original angle as a difference of two "friendly" angles.

cos((π/4) - (π/6)) = cos(π/4)cos(π/6) + sin(π/4)sin(π/6)

cos((π/4) - (π/6)) = (√2 / 2)(√3 / 2) + (√2 / 2)(1 / 2)

cos((π/4) - (π/6)) = (√6 / 4) + (√2 / 4)

cos((π/4) - (π/6)) = (√6 + √2) / 4

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