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Sin + Cos + Tan Mixer

Trigonometry · Class XI

Blend sin, cos and tan with three weight sliders. Useful for spotting Fourier-like sums.

1
type a value
-33
0
type a value
-33
0
type a value
-22
Live values
  • f(π/4)0.7071
  • f(0)0
xy
  • f(x) = a sin x + b cos x + c tan x

Formulas in this lab

  • f(π/4)
    f(π/4)=a+b2+cf(\pi/4) = \tfrac{a+b}{\sqrt 2} + c
  • f(0)
    f(0)=bf(0) = b
Tip: Pure sin (a=1, others 0) vs pure cos (b=1) , note the π/2 phase shift.

Frequently asked questions

▶What does $f(x) = a\sin x + b\cos x + c\tan x$ look like?

It is a weighted blend of the three basic trig curves. The $a\sin + b\cos$ part can be rewritten as $R\sin(x+\alpha)$ with $R = \sqrt{a^2+b^2}$ — a key JEE identity. Adding $c\tan x$ injects asymptotes wherever $\cos x = 0$.

▶How do I use the sin-cos-tan mixer lab?

Slide $a$, $b$, $c$ between $-3$ and $3$. Set $c = 0$ to focus on the smooth $a\sin x + b\cos x$ sum. Try $a = 1$, $b = 1$ and check that $f(\pi/4) = \sqrt{2}$ — exactly $\sqrt{a^2+b^2}$, confirming the $R$-formula.

▶How does this preview Fourier series in engineering?

Any periodic signal can be built from sines and cosines of multiple frequencies — the heart of Fourier analysis. This lab is the simplest possible Fourier mixer (one frequency, three weights). Audio codecs, image compression and signal filters in WhatsApp voice notes all use the same idea.

▶Why do students lose marks on $a\sin x + b\cos x = R\sin(x+\alpha)$?

They forget $R = \sqrt{a^2+b^2}$ and $\tan\alpha = b/a$. The maximum value of $a\sin x + b\cos x$ is therefore $\sqrt{a^2+b^2}$, not $a + b$. JEE Main asks this every other year. The lab lets you eyeball the peak and check $R$.