Simulation · Maths · Class 11
Doubling the angle is not doubling the value
From the lesson Double and triple angle formulas in Trigonometric Functions. Change the values and watch what happens.
Doubling the angle is not doubling the valueMaths · Class 11
The idea behind it
NCERT §3.4
- Putting y = x in the sum formulas gives the double angle results: sin 2x = 2 sin x cos x and cos 2x = cos² x − sin² x.
- Using sin² x + cos² x = 1, cos 2x also equals 2 cos² x − 1 and 1 − 2 sin² x. Rearranged, cos² x = (1 + cos 2x)/2 and sin² x = (1 − cos 2x)/2.
- In terms of tan x: cos 2x = (1 − tan² x)/(1 + tan² x) and sin 2x = 2 tan x/(1 + tan² x), both for x ≠ nπ + π/2; tan 2x = 2 tan x/(1 − tan² x) for 2x ≠ nπ + π/2, n ∈ Z.
- Triple angles: sin 3x = 3 sin x − 4 sin³ x and cos 3x = 4 cos³ x − 3 cos x, each proved by writing 3x = 2x + x.
- tan 3x = (3 tan x − tan³ x)/(1 − 3 tan² x), for 3x ≠ nπ + π/2, n ∈ Z.
- Worked example: if sin x = 4/5 with x acute, cos x = 3/5, so sin 2x = 2 × 4/5 × 3/5 = 24/25, cos 2x = 9/25 − 16/25 = −7/25, tan 2x = −24/7, and sin 3x = 12/5 − 4 × 64/125 = 44/125. The negative cos 2x shows that 2x is obtuse.
- Half angles: writing x/2 for x gives cos x = 2 cos² (x/2) − 1 = 1 − 2 sin² (x/2), so sin² (x/2) = (1 − cos x)/2 and cos² (x/2) = (1 + cos x)/2, with the sign fixed by the quadrant of x/2. For example sin² 22.5° = (1 − 1/√2)/2 = (2 − √2)/4, so sin 22.5° = √(2 − √2)/2.
- Check at x = π/6: sin 3x = sin π/2 = 1, and 3 × 1/2 − 4 × 1/8 = 3/2 − 1/2 = 1.