FUNCTIONS 10
Trigonometric Functions
Trigonometric functions connect angles, circular motion and repeating change.
The main idea
The three basic trigonometric functions are
\[\boxed{\sin x,\qquad\cos x,\qquad\tan x}.\]They begin as ratios in triangles, but the unit circle extends them to all real inputs and reveals their periodic behaviour.
1. Right-triangle ratios
For an acute angle \(\theta\),
\[\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}},\quad \cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}},\quad \tan\theta=\frac{\text{opposite}}{\text{adjacent}}.\]adjacent / hypotenuse → cosine
opposite / adjacent → tangent
2. The unit circle
On the circle of radius \(1\), the point reached by angle \(\theta\) has coordinates
\[\boxed{(\cos\theta,\sin\theta)}.\]This gives sine and cosine meaning for every angle, not only angles in a right triangle.
3. Radians
Calculus uses radians because they measure angle directly from arc length.
\[180^\circ=\pi\text{ radians},\qquad360^\circ=2\pi.\]On a unit circle, an angle of \(\theta\) radians cuts off an arc of length \(\theta\).
full turn → \(2\pi\)
quarter turn → \(\pi/2\)
4. Sine and cosine graphs
Both functions repeat every \(2\pi\):
\[\sin(x+2\pi)=\sin x,\qquad\cos(x+2\pi)=\cos x.\]Sine begins at \(0\); cosine begins at \(1\). Their shapes are the same but shifted horizontally.
5. Exact key values
The graph highlights values that should become familiar:
\[\sin0=0,\quad\sin\frac{\pi}{2}=1,\quad\sin\pi=0,\]\[\cos0=1,\quad\cos\frac{\pi}{2}=0,\quad\cos\pi=-1.\]6. Tangent and its asymptotes
Because
\[\tan x=\frac{\sin x}{\cos x},\]tangent is undefined whenever \(\cos x=0\). Thus vertical asymptotes occur at
\[x=\frac{\pi}{2}+k\pi.\]7. The fundamental identity
From the unit-circle equation \(x^2+y^2=1\) and \((x,y)=(\cos\theta,\sin\theta)\),
\[\boxed{\sin^2\theta+\cos^2\theta=1}.\]\(x=\cos\theta,\ y=\sin\theta\)
↓
\(\cos^2\theta+\sin^2\theta=1\)
8. Amplitude and period
For
\[y=A\sin(Bx),\]the amplitude is \(|A|\), while the period is
\[\boxed{\frac{2\pi}{|B|}}.\]The graph compares \(\sin x\), \(2\sin x\), and \(\sin(2x)\).
9. Horizontal and vertical shifts
In
\[y=A\sin(Bx+C)+D,\]\(C\) changes the horizontal position and \(D\) changes the vertical position.
For example, \(\sin(x-\pi/2)\) is \(\sin x\) shifted right by \(\pi/2\).
10. Why trigonometric functions matter for calculus
Sine and cosine are linked directly through differentiation:
\[\boxed{\frac{d}{dx}\sin x=\cos x},\qquad \boxed{\frac{d}{dx}\cos x=-\sin x}.\]This makes them fundamental for modelling oscillation, waves and periodic change.
\(\cos x\xrightarrow{d/dx}-\sin x\)
Sine and cosine come from coordinates on the unit circle and repeat every \(2\pi\). Tangent is their ratio and has vertical asymptotes. Amplitude controls height, period controls repetition, and radians are the natural angle measure for calculus.