B. Trigonometric Functions · Trigonometric Identities and Equations

Compound Angle Formulas for Sine

B3.2 B3.1About 70 minutes

Learning goals

I can follow how the sine compound angle formulas come from the cosine formulas and the cofunction identities.
I can use sin(a ± b) = sin a cos b ± cos a sin b to find exact values such as sin(π/12) and sin(7π/12).
I can find sin(a ± b) and cos(a ± b) from given ratios, using the quadrant of each angle.
I can recognize the pattern of a compound angle formula inside an expression and simplify it.

Words to know

compound angle formula for sine
sin(a + b) = sin a cos b + cos a sin b and sin(a − b) = sin a cos b − cos a sin b.

terminal arm
The ray from the origin that ends an angle in standard position; a point (x, y) on it at distance r gives sin = y/r and cos = x/r.

quadrant
One of the four regions of the plane. In radians: I is 0 to π/2, II is π/2 to π, III is π to 3π/2, IV is 3π/2 to 2π.

sinusoidal function
A function whose graph is a transformed sine curve, such as y = 2 sin(x + π/3).

From cosine to sine

You already know the cosine formulas. The cofunction identity from lesson 1 turns sine into cosine: sin θ = cos(π/2 − θ). Follow the steps:

sin(a + b) = cos(π/2 − (a + b))
= cos((π/2 − a) − b), regrouping the angle
= cos(π/2 − a) cos b + sin(π/2 − a) sin b, by the formula for cos(a − b)
= sin a cos b + cos a sin b, by the cofunction identities

So sin(a + b) = sin a cos b + cos a sin b. Replacing b with −b (sine is odd, cosine is even) gives sin(a − b) = sin a cos b − cos a sin b.

Three formulas in three lessons, and each one came from the one before. That is how much of mathematics is built: one solid result, then careful reasoning.

Sine mixes sine and cosine and keeps the sign of the angle. Cosine pairs cos with cos and sin with sin, and flips the sign.

A quick self-check

If you forget which formula has which sign, set b = 0. The formula must give back sin a (or cos a). For example, sin(a + 0) = sin a cos 0 + cos a sin 0 = sin a ✓. A second check: set a = b = π/4 in sin(a + b) and you must get sin(π/2) = 1: (√2/2)(√2/2) + (√2/2)(√2/2) = 1/2 + 1/2 = 1 ✓.

Worked example

Example 1: Exact value of sin(π/12)

Find the exact value of sin(π/12).

π/12 = π/3 − π/4.
sin(π/3 − π/4) = sin(π/3) cos(π/4) − cos(π/3) sin(π/4).
= (√3/2)(√2/2) − (1/2)(√2/2) = √6/4 − √2/4.
Calculator check: (√6 − √2)/4 ≈ 0.2588 and sin(π/12) ≈ 0.2588 ✓.

sin(π/12) = (√6 − √2)/4

Worked example

Example 2: Exact value of sin(5π/12), and a cofunction check

Find the exact value of sin(5π/12). Then compare with cos(π/12) from the last lesson.

5π/12 = π/4 + π/6.
sin(π/4 + π/6) = sin(π/4) cos(π/6) + cos(π/4) sin(π/6) = (√2/2)(√3/2) + (√2/2)(1/2).
= (√6 + √2)/4.
Last lesson, cos(π/12) = (√6 + √2)/4. These match, as the cofunction identity predicts: 5π/12 and π/12 are complements.

sin(5π/12) = (√6 + √2)/4

Worked example

Example 3: Spot the pattern

Evaluate sin(π/9) cos(2π/9) + cos(π/9) sin(2π/9).

This is sin a cos b + cos a sin b with a = π/9 and b = 2π/9: the pattern of sin(a + b).
a + b = 3π/9 = π/3.

sin(π/3) = √3/2

Worked example

Example 4: Using ratios in two quadrants

Given sin x = 3/5 with π/2 < x < π, and cos y = 5/13 with 3π/2 < y < 2π, find sin(x + y), sin(x − y) and cos(x + y). In which quadrant does x + y end?

Sketch each angle with a reference triangle (see the diagram). x is in quadrant II with point (−4, 3) and r = 5, so cos x = −4/5.
y is in quadrant IV with point (5, −12) and r = 13, so sin y = −12/13.
sin(x + y) = sin x cos y + cos x sin y = (3/5)(5/13) + (−4/5)(−12/13) = 15/65 + 48/65 = 63/65.
sin(x − y) = sin x cos y − cos x sin y = 15/65 − 48/65 = −33/65.
cos(x + y) = cos x cos y − sin x sin y = (−4/5)(5/13) − (3/5)(−12/13) = −20/65 + 36/65 = 16/65.
sin(x + y) > 0 and cos(x + y) > 0, so the terminal arm of x + y is in quadrant I (after any full turns).

sin(x + y) = 63/65, sin(x − y) = −33/65, cos(x + y) = 16/65; x + y ends in quadrant I.

Reference triangles for Example 4. The signs of the coordinates give the signs of the ratios: cos x is negative in quadrant II, sin y is negative in quadrant IV.

Worked example

Example 5: Two waves become one

Show that sin x + √3 cos x = 2 sin(x + π/3). What is the maximum value of sin x + √3 cos x?

Expand the right side: 2 sin(x + π/3) = 2[sin x cos(π/3) + cos x sin(π/3)].
= 2[(1/2) sin x + (√3/2) cos x] = sin x + √3 cos x ✓.
So y = sin x + √3 cos x is a single sinusoidal function with amplitude 2, shifted π/3 to the left.
Its maximum is 2, reached when x + π/3 = π/2, that is x = π/6.

The identity holds, and the maximum value is 2 (at x = π/6). You will use this idea again in the unit Combining Functions and Modelling, where sums of functions are graphed.

Just for fun

Riddle: Why does sin(a + b) never go anywhere alone?

Because it always brings a cosine along: sin a cos b + cos a sin b. Cosine, on the other hand, sticks with its own kind: cos a cos b − sin a sin b.

Hands-on activity

Two waves, one wave

You need: graph paper, a pencil, a ruler and a calculator in radian mode.

Make a table with x = 0, 0.5, 1, 1.5, 2, 2.5, 3, 3.5, 4, 4.5, 5, 5.5 and 6.
For each x, calculate A = sin x + √3 cos x and B = 2 sin(x + π/3). Round to two decimal places.
Plot both sets of points on the same grid (x from 0 to 6.3, y from −2 to 2) and join each set with a smooth curve.

What to look for: columns A and B are identical, so the two curves are one curve. The maximum is 2 near x ≈ 0.52 (that is π/6) and the minimum is −2 near x ≈ 3.67 (that is 7π/6). Sample values: x = 0 gives 1.73 (√3); x = 2 gives about 0.19.

Your turn

Expand sin(x − π/4) and simplify.

Find the exact value of sin(7π/12).

Simplify sin 3x cos x − cos 3x sin x.

Use a compound angle formula to simplify sin(π/2 + x).

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

Evaluate sin(5π/18) cos(π/9) − cos(5π/18) sin(π/9).

More practice: warm-up

State the formula for sin(a + b).

State the formula for sin(a − b).

Expand sin(x + π) and simplify.

Use the formula to evaluate sin(π/4 + π/4).

Which special angles add to 5π/12?

More practice: on your own

Find the exact value of sin(−7π/12).

cos a = 12/13 with 3π/2 < a < 2π, and sin b = −4/5 with π < b < 3π/2. Find sin(a + b) and sin(a − b).

Prove that sin(x + y) + sin(x − y) = 2 sin x cos y.

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

Show that √3 sin x − cos x = 2 sin(x − π/6). What is its maximum value?

Challenge

Prove that sin(a + b) / (cos a cos b) = tan a + tan b.

Simplify sin(x + π/4) + sin(x − π/4).

x and y are acute, sin x = 3/5 and sin y = 5/13. Find sin(x + y). Is x + y acute or obtuse? Explain why sin(x + y) alone cannot tell you.

Check your understanding

Quiz

Which formula is correct for every a and b?

What is the exact value of sin(π/12)?

Simplify sin 3x cos x − cos 3x sin x.

Which expression is equivalent to sin(x − π/2)?

Which expression equals sin x + √3 cos x?

a and b are acute with sin a = 3/5 and cos b = 4/5. What is sin(a + b)?

For parents and teachers

Big idea (B3.2): sin(a + b) = sin a cos b + cos a sin b and sin(a − b) = sin a cos b − cos a sin b. They follow from the cosine formulas and the cofunction identity, and they give exact values for angles such as π/12 (15°) and 5π/12 (75°).

How to help: Ask your teen to say the two patterns out loud ("sine: sin cos, cos sin, same sign"; "cosine: cos cos, sin sin, opposite sign") and to use the b = 0 check whenever they are unsure. For problems that give one ratio, insist on a quick sketch of the angle in its quadrant before any arithmetic.

Common difficulties:

Mixing the sine and cosine patterns.
Losing a negative sign when an angle is in quadrant II, III or IV.
Deciding the quadrant of a + b from its sine alone; the cosine is needed too.
Forgetting that sin(5π/12) and cos(π/12) must agree (a good check).