B. Trigonometric Functions · Trigonometric Identities and Equations

Double Angle Formulas

B3.2 B3.3About 70 minutes

Learning goals

I can develop the double angle formulas for sine, cosine and tangent from the compound angle formulas.
I can choose among the three forms of cos 2x.
I can find sin 2x, cos 2x and tan 2x exactly from one given ratio and the quadrant of x.
I can recognize double angle patterns in expressions and evaluate them exactly.

Words to know

double angle formula
A compound angle formula with both angles equal: sin 2x = 2 sin x cos x, cos 2x = cos²x − sin²x, tan 2x = 2 tan x/(1 − tan²x).

Pythagorean identity
sin²x + cos²x = 1 for every x. It lets you swap sin²x for 1 − cos²x and the reverse.

half-angle idea
Reading a double angle formula backwards, for example cos²x = (1 + cos 2x)/2, to find ratios of half of a known angle.

Twice the angle, not twice the sine

Is sin 2x the same as 2 sin x? Try x = π/4: sin(π/2) = 1, but 2 sin(π/4) = √2 ≈ 1.41. Not the same. The graphs make the difference obvious: y = sin 2x squeezes two waves into the space of one, while y = 2 sin x stretches one wave taller.

To find the true value of sin 2x, use the compound angle formulas from the last three lessons with b = a.

y = sin 2x (solid) has period π and amplitude 1. y = 2 sin x (dashed) has period 2π and amplitude 2. They are clearly different functions.

Developing the formulas

Sine. sin 2x = sin(x + x) = sin x cos x + cos x sin x, so

sin 2x = 2 sin x cos x

Cosine. cos 2x = cos(x + x) = cos x cos x − sin x sin x, so cos 2x = cos²x − sin²x. Using sin²x + cos²x = 1 we get two more forms:

replace sin²x by 1 − cos²x: cos 2x = cos²x − (1 − cos²x) = 2cos²x − 1
replace cos²x by 1 − sin²x: cos 2x = (1 − sin²x) − sin²x = 1 − 2sin²x

Tangent. tan 2x = tan(x + x), so

tan 2x = 2 tan x / (1 − tan²x), for tan x ≠ ±1 and x ≠ π/2 + kπ.

These are identities: they are true for every x where both sides are defined. The "x" can be any angle. For example, sin 6x = 2 sin 3x cos 3x and cos x = 1 − 2 sin²(x/2).

Formula	Use it when ...
cos 2x = cos²x − sin²x	you know both sin x and cos x
cos 2x = 2cos²x − 1	you know only cos x, or you want everything in cosine
cos 2x = 1 − 2sin²x	you know only sin x, or you want everything in sine

Three forms of cos 2x. All three are equivalent; pick the one that makes the next step easiest.

A geometric view of sin 2x = 2 sin x cos x (for 0 < 2x < π). The isosceles triangle with equal sides 1 and top angle 2x splits into two right triangles with legs sin x and cos x. Computing its area two ways gives the formula.

Worked example

Example 1: All three double angle ratios

Given sin x = 3/5 with π/2 < x < π, find the exact values of sin 2x, cos 2x and tan 2x. In which quadrant is 2x?

x is in quadrant II, so cos x = −√(1 − 9/25) = −4/5.
sin 2x = 2 sin x cos x = 2(3/5)(−4/5) = −24/25.
cos 2x = 1 − 2 sin²x = 1 − 2(9/25) = 7/25.
tan 2x = sin 2x / cos 2x = (−24/25)/(7/25) = −24/7. (Check with the formula: tan x = −3/4, so 2(−3/4)/(1 − 9/16) = (−3/2)/(7/16) = −24/7 ✓.)
π < 2x < 2π. Sine negative and cosine positive means 2x is in quadrant IV.

sin 2x = −24/25, cos 2x = 7/25, tan 2x = −24/7; 2x is in quadrant IV.

Worked example

Example 2: Recognize the double angle

Find the exact value of each expression.

(a) 2 sin(π/8) cos(π/8); (b) cos²(π/12) − sin²(π/12); (c) 1 − 2 sin²(3π/8); (d) 2 tan(π/8)/(1 − tan²(π/8))

(a) The pattern 2 sin x cos x is sin 2x: sin(π/4) = √2/2.
(b) The pattern cos²x − sin²x is cos 2x: cos(π/6) = √3/2.
(c) The pattern 1 − 2 sin²x is cos 2x: cos(3π/4) = −√2/2.
(d) The pattern is tan 2x: tan(π/4) = 1.

(a) √2/2; (b) √3/2; (c) −√2/2; (d) 1

Worked example

Example 3: Starting from a tangent

Given tan x = 2 with π < x < 3π/2, find sin 2x, cos 2x and tan 2x.

In quadrant III, both coordinates are negative. tan x = y/x = (−2)/(−1), so use the point (−1, −2) with r = √5.
sin x = −2/√5 and cos x = −1/√5.
sin 2x = 2(−2/√5)(−1/√5) = 4/5.
cos 2x = cos²x − sin²x = 1/5 − 4/5 = −3/5.
tan 2x = 2(2)/(1 − 4) = −4/3 (and (4/5)/(−3/5) = −4/3 ✓).

sin 2x = 4/5, cos 2x = −3/5, tan 2x = −4/3

Worked example

Example 4: Half an angle (extension)

Use cos 2x = 2cos²x − 1 to find the exact value of cos(π/8).

Let x = π/8, so 2x = π/4 and cos(π/4) = √2/2.
√2/2 = 2cos²(π/8) − 1, so cos²(π/8) = (1 + √2/2)/2 = (2 + √2)/4.
π/8 is in quadrant I, so take the positive root: cos(π/8) = √(2 + √2)/2.
Check: √(2 + 1.4142)/2 ≈ 0.9239, and cos(π/8) ≈ 0.9239 ✓.

cos(π/8) = √(2 + √2)/2

Just for fun

Doubling the angle does not double the sine: sin(π/2) = 1, but 2 sin(π/4) ≈ 1.41. If it did, the sine of a big enough angle would sail past 1, and the unit circle would have to change its name.

Hands-on activity

Two ways to find one area

You need: a sheet of paper, a ruler, a protractor, a pencil and a calculator.

Mark a point O near the top of the page. Draw two 10 cm segments from O with an angle of 70° between them (so 2x = 70° and x = 35°). Join their ends to make an isosceles triangle.
Draw the altitude from O to the base. Measure the base and the altitude.
Area 1 (two sides and the angle between them): ½ × 10 × 10 × sin 70°.
Area 2 (base and height): ½ × base × altitude.
Compare: the base should be 2 × 10 sin 35° and the altitude 10 cos 35°, which is the double angle formula in action.

What to look for: base ≈ 11.5 cm, altitude ≈ 8.2 cm, and both areas ≈ 47.0 cm². Small differences come from measuring.

Your turn

cos x = 4/5 with 3π/2 < x < 2π. Find sin 2x and cos 2x.

Find the exact value of 2 sin(π/12) cos(π/12).

Find the exact value of 2cos²(π/8) − 1.

Simplify sin 2x / sin x.

tan x = 3/4 with 0 < x < π/2. Find tan 2x.

Write cos 6x in terms of sin 3x and cos 3x (any one form).

More practice: warm-up

State the double angle formula for sine.

State all three forms of cos 2x.

State the double angle formula for tangent.

True or false: sin 2x = 2 sin x for all x.

Find the exact value of 1 − 2 sin²(π/12).

More practice: on your own

sin x = −5/13 with π < x < 3π/2. Find sin 2x, cos 2x and tan 2x.

Write sin 4x in terms of sin x and cos x.

Find the exact value of sin(π/8) cos(π/8).

Find the exact value of cos²(5π/12) − sin²(5π/12).

Simplify (1 − cos 2x)/sin 2x.

Challenge

Show that sin 3x = 3 sin x − 4 sin³x.

Use cos 2x = 1 − 2 sin²x to find the exact value of sin(π/8).

Show that cos 3x = 4cos³x − 3 cos x, and check it at x = π/3.

Check your understanding

Quiz

Which expression equals sin 2x for every x?

Which expression equals cos 2x for every x?

What is the exact value of 2 sin(π/8) cos(π/8)?

sin x = 3/5 with π/2 < x < π. What is sin 2x?

tan x = 3/4 and x is acute. What is tan 2x?

How do the graphs of y = sin 2x and y = 2 sin x compare?

For parents and teachers

Big idea (B3.2): the double angle formulas are the compound angle formulas with both angles the same: sin 2x = 2 sin x cos x, cos 2x = cos²x − sin²x = 2cos²x − 1 = 1 − 2sin²x, and tan 2x = 2 tan x/(1 − tan²x). They are used constantly in the rest of this unit for proofs and equations, and again in calculus.

How to help: Ask your teen to derive the three forms of cos 2x from cos(x + x) and sin²x + cos²x = 1, rather than memorizing them. Quiz them with a pattern: "What is 2 sin 5 cos 5?" (sin 10).

Common difficulties:

Thinking sin 2x = 2 sin x.
Forgetting the sign of cos x (or sin x) from the quadrant before doubling.
Using cos²x − sin²x when only one ratio is known; 1 − 2sin²x or 2cos²x − 1 is faster.
Losing track of what is doubled: sin 6x = 2 sin 3x cos 3x, not 2 sin 6x cos 6x.