Find the value of the ratios below in terms of and using
difference of two special angles. Simplify your answers.
Hint:
4.2
4.3
4.3
4.5
4.5

We can express these trigonometric values in “quarter‐form” (that is, with a denominator of 4) in terms of √2 and √3 by writing each as the difference (or sum) of two special angles. One very useful idea is to write an angle as a combination of 60° and 45° (or 60° and –45°) so that the sine and cosine of the pieces are known.

Below are the three problems and their solutions.

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Step 1. Use the cosine difference formula:
  cos(A – B) = cos A·cos B + sin A·sin B.

Let A = 60° and B = 45°. Then
  cos 15° = cos(60° – 45°)
      = cos60°·cos45° + sin60°·sin45°.

Step 2. Substitute the known exact values:
  cos60° = ½, cos45° = (√2)/2,
  sin60° = (√3)/2, sin45° = (√2)/2.

Thus,
  cos 15° = (½)(√2/2) + ((√3)/2)(√2/2)
      = (√2/4) + (√6/4)
      = (√6 + √2)/4.

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2) (Problem 4.3) Find cos 105°.
  Observation: 105° = 60° + 45°.

Step 1. Use the cosine sum formula:
  cos(A + B) = cos A·cos B – sin A·sin B.

Let A = 60° and B = 45°. Then
  cos 105° = cos(60° + 45°)
      = cos60°·cos45° – sin60°·sin45°.

Step 2. Substitute the known values:
  cos60° = ½, cos45° = (√2)/2,
  sin60° = (√3)/2, sin45° = (√2)/2.

Thus,
  cos 105° = (½)(√2/2) – ((√3)/2)(√2/2)
      = (√2/4) – (√6/4)
      = (√2 – √6)/4.

Note: Since √6 > √2, the answer is negative (as expected because cos 105° is negative).

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3) (Problem 4.5) Find sin 165°.
  Observation: 165° = 180° – 15°, and we know that sin(180° – θ) = sin θ. Therefore, sin 165° = sin 15°.

Now, write 15° = 60° – 45° and use the sine difference formula.

Step 1. Use the sine difference formula:
  sin(A – B) = sin A·cos B – cos A·sin B.

Let A = 60° and B = 45°. Then
  sin 15° = sin(60° – 45°)
      = sin60°·cos45° – cos60°·sin45°.

Step 2. Substitute the known values:
  sin60° = (√3)/2, cos45° = (√2)/2,
  cos60° = ½,    sin45° = (√2)/2.

Thus,
  sin 15° = ((√3)/2)(√2/2) – (½)(√2/2)
      = (√6/4) – (√2/4)
      = (√6 – √2)/4.

Since sin 165° = sin 15°, we have
  sin 165° = (√6 – √2)/4.

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Summary of Answers:

cos 15° = (√6 + √2)/4
  cos 105° = (√2 – √6)/4
  sin 165° = (√6 – √2)/4

These are the desired values expressed in terms of √2 and √3.