Grade by Grade
B. Trigonometric Functions · Trigonometric Identities and Equations

Compound Angle Formulas for Cosine

B3.2 B3.1About 70 minutes
Learning goals
  • I can explain why cos(a + b) is not equal to cos a + cos b, using a counterexample.
  • I can follow the development of cos(a − b) = cos a cos b + sin a sin b from the unit circle, and get cos(a + b) from it.
  • I can verify the cosine compound angle formulas numerically with a calculator.
  • I can use the formulas to find exact values such as cos(π/12), and to simplify expressions.

Words to know

compound angle
An angle formed by adding or subtracting two angles, such as a + b or a − b.
compound angle formula
An identity that expresses a ratio of a + b or a − b in terms of ratios of a and b, such as cos(a − b) = cos a cos b + sin a sin b.
exact value
A value written with integers, fractions and radicals, not a rounded decimal; for example cos(π/12) = (√6 + √2)/4.
distance formula
The distance between (x₁, y₁) and (x₂, y₂) is √((x₂ − x₁)² + (y₂ − y₁)²).

Can you split a cosine?

It is tempting to think that cos(a + b) = cos a + cos b, the way 3(a + b) = 3a + 3b. Test it with a = π/3 and b = π/6:

  • cos(π/3 + π/6) = cos(π/2) = 0
  • cos(π/3) + cos(π/6) = 1/2 + √3/2 ≈ 1.37

One counterexample is enough: cosine does not distribute over addition. (Cosine is a function, not a number you multiply by.) So what is cos(a + b)? In this lesson you will find out, and the answer will let you compute exact values for angles like π/12 (15°) and 7π/12 (105°) that are not on the special-angle list.

Following the development of cos(a − b)

The curriculum asks you to follow this argument, not to reproduce it from memory. Read it slowly with the diagram.

Step 1. Put two points on the unit circle: P at angle a and Q at angle b. Their coordinates are P(cos a, sin a) and Q(cos b, sin b). The angle between OP and OQ is a − b.

Step 2: cosine law in △POQ, with OP = OQ = 1:

PQ² = 1² + 1² − 2(1)(1) cos(a − b) = 2 − 2 cos(a − b)

Step 3: distance formula between P and Q:

  • PQ² = (cos a − cos b)² + (sin a − sin b)²
  • = cos²a − 2 cos a cos b + cos²b + sin²a − 2 sin a sin b + sin²b
  • = (cos²a + sin²a) + (cos²b + sin²b) − 2 cos a cos b − 2 sin a sin b
  • = 2 − 2(cos a cos b + sin a sin b), using sin²θ + cos²θ = 1 twice.

Step 4. Both expressions equal PQ², so 2 − 2 cos(a − b) = 2 − 2(cos a cos b + sin a sin b). Subtract 2 from both sides and divide by −2:

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

P(cos a, sin a)Q(cos b, sin b)ba − bO11Cosine law in △POQ:PQ² = 2 − 2 cos(a − b)Distance formula:PQ² = (cos a − cos b)²+ (sin a − sin b)²Set them equal:cos(a − b) =cos a cos b + sin a sin b
P and Q sit on the unit circle at angles a and b. Computing PQ² two ways (cosine law and distance formula) and setting the results equal gives the formula for cos(a − b).

From a − b to a + b

Write a + b as a − (−b) and use the formula you just found, with −b in place of b:

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

Cosine is even, so cos(−b) = cos b. Sine is odd, so sin(−b) = −sin b. Therefore:

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

Notice the pattern for cosine: cos cos, sin sin, and the sign in the middle is the opposite of the sign in the angle.

abcos(a + b)cos a cos b − sin a sin bcos a + cos b
10.50.07070.07071.4179
1.50.75−0.6282−0.62820.8024
Numerical check in radian mode (rounded to 4 decimal places). The formula matches every time; the tempting "split" does not.
Worked example

Example 1: An exact value by subtraction

Find the exact value of cos(π/12).

  1. Write π/12 as a difference of special angles: π/3 − π/4 = 4π/12 − 3π/12 = π/12.
  2. cos(π/3 − π/4) = cos(π/3) cos(π/4) + sin(π/3) sin(π/4).
  3. = (1/2)(√2/2) + (√3/2)(√2/2) = √2/4 + √6/4.
  4. Check with a calculator: (√6 + √2)/4 ≈ (2.4495 + 1.4142)/4 ≈ 0.9659, and cos(π/12) ≈ 0.9659 ✓.

cos(π/12) = (√6 + √2)/4

Worked example

Example 2: An exact value by addition

Find the exact value of cos(7π/12).

  1. 7π/12 = π/3 + π/4 (4π/12 + 3π/12).
  2. cos(π/3 + π/4) = cos(π/3) cos(π/4) − sin(π/3) sin(π/4).
  3. = (1/2)(√2/2) − (√3/2)(√2/2) = (√2 − √6)/4.
  4. Sense check: 7π/12 is in quadrant II, where cosine is negative. (√2 − √6)/4 ≈ −0.2588 is negative ✓.

cos(7π/12) = (√2 − √6)/4

π/4π/37π/12 = π/4 + π/3cos(7π/12)= cos(π/3 + π/4)= (√2 − √6)/4≈ −0.2588Negative: the point isleft of the y-axis.
Build 7π/12 from two special angles: π/4 then π/3 more. The point lands in quadrant II, so its cosine must be negative.
Worked example

Example 3: Recognize the pattern

Find the exact value of cos(5π/12) cos(π/12) + sin(5π/12) sin(π/12).

  1. The pattern cos a cos b + sin a sin b is cos(a − b), with a = 5π/12 and b = π/12.
  2. a − b = 4π/12 = π/3.

cos(π/3) = 1/2

Worked example

Example 4: Using given ratios

Given cos a = 3/5 with 0 < a < π/2, and sin b = 5/13 with π/2 < b < π, find the exact values of cos(a + b) and cos(a − b).

  1. Find the missing ratios. For a (quadrant I): sin a = √(1 − 9/25) = 4/5.
  2. For b (quadrant II, cosine negative): cos b = −√(1 − 25/169) = −12/13.
  3. cos(a + b) = cos a cos b − sin a sin b = (3/5)(−12/13) − (4/5)(5/13) = −36/65 − 20/65 = −56/65.
  4. cos(a − b) = cos a cos b + sin a sin b = −36/65 + 20/65 = −16/65.

cos(a + b) = −56/65 and cos(a − b) = −16/65

Worked example

Example 5: Proving equivalences from lesson 1 and 2

Use a compound angle formula to show that cos(π/2 − x) = sin x and cos(π + x) = −cos x for every x.

  1. cos(π/2 − x) = cos(π/2) cos x + sin(π/2) sin x = 0 · cos x + 1 · sin x = sin x.
  2. cos(π + x) = cos π cos x − sin π sin x = (−1) cos x − 0 · sin x = −cos x.
  3. These hold for every real x, which confirms what the triangle and the graphs suggested.

Both equivalences are true for all real x.

Just for fun

If cosine split over addition, then cos π = cos(π/3 + π/3 + π/3) would equal 1/2 + 1/2 + 1/2 = 3/2. A cosine bigger than 1 is about as likely as a snowball fight in Thunder Bay in July. (The real value is cos π = −1.)

Hands-on activity

Dice check of the formula

You need: two dice (or playing cards ace to 6), a calculator in radian mode, and paper for a table.

  1. Roll both dice. Let a = (first die) ÷ 2 and b = (second die) ÷ 4. These are angles in radians.
  2. Calculate three things and record them to 4 decimal places: cos(a + b), cos a cos b − sin a sin b, and cos a + cos b.
  3. Repeat for five rolls. Then do five more rolls checking cos(a − b) against cos a cos b + sin a sin b.

What to look for: the first two columns agree every time (any difference is only in the last digit, from rounding). The third column almost never agrees, and it can even be bigger than 1, which a cosine never is. Sample row: rolls 3 and 3 give a = 1.5, b = 0.75: cos(2.25) ≈ −0.6282, the formula gives −0.6282, and cos a + cos b ≈ 0.8024.

Your turn

1

Expand cos(x + π/3) and simplify.

2

Find the exact value of cos(5π/12) using 5π/12 = π/4 + π/6.

3

Simplify cos 2x cos x − sin 2x sin x.

4

Find the exact value of cos(11π/12) using 11π/12 = 2π/3 + π/4.

5

Evaluate cos(7π/18) cos(π/9) − sin(7π/18) sin(π/9).

6

Verify cos(a − b) = cos a cos b + sin a sin b for a = 0.8 and b = 0.3 (radians).

More practice: warm-up

1

State the formula for cos(a + b).

2

State the formula for cos(a − b).

3

True or false: cos(a + b) = cos a + cos b for all a and b.

4

Expand cos(x − π) and simplify.

5

Use the formula to evaluate cos(π/6 + π/3).

More practice: on your own

1

Find the exact value of cos(13π/12). (Hint: 13π/12 = 3π/4 + π/3.)

2

sin a = 4/5 with π/2 < a < π and cos b = −5/13 with π < b < 3π/2. Find cos(a + b) and cos(a − b).

3

Show that cos(x + y) + cos(x − y) = 2 cos x cos y.

4

Use a compound angle formula to show that cos(3π/2 − x) = −sin x.

5

Find the exact value of cos(−π/12).

Challenge

1

Find the exact value of cos(5π/12) cos(π/12). (Hint: use the result 2 cos x cos y = cos(x + y) + cos(x − y).)

2

Prove that cos(a + b) cos(a − b) = cos²a − sin²b.

3

a and b are acute angles with cos a = 1/7 and cos b = 13/14. Find the exact value of a − b.

Check your understanding

Quiz

1.

Which formula is correct for every a and b?

2.

What is the exact value of cos(π/12)?

3.

Simplify cos 2x cos x + sin 2x sin x.

4.

Which sum of special angles equals 7π/12?

5.

Using cos(a + b) with a = π, what is cos(π + x)?

6.

In the development of cos(a − b), the length PQ² is calculated in two ways. Which two?

For parents and teachers

Big idea (B3.2): cos(a + b) is not cos a + cos b. The correct formulas are cos(a + b) = cos a cos b − sin a sin b and cos(a − b) = cos a cos b + sin a sin b. The curriculum asks students to follow (not reproduce) the development, verify the formulas with numbers, and use them for exact values.

How to help: Have your teen explain the sign rule in their own words ("cos-cos, sin-sin, opposite sign"). Pick two angles, say a = 1 and b = 0.4, and let them check the formula on a calculator in radian mode; it builds trust that the formula is real.

Common difficulties:

  • Using the wrong sign in the middle. A quick check: put b = 0 and the formula must give cos a.
  • Choosing special angles that do not add to the target (π/12 = π/3 − π/4, 5π/12 = π/4 + π/6, 7π/12 = π/3 + π/4).
  • Forgetting the quadrant when finding a missing ratio from a given one (cos b is negative in quadrant II).
  • Radical arithmetic: (√3/2)(√2/2) = √6/4.