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3.3 Trig. Functions and Identities

Tags
Trigonometry
Trigonometric functions
Sine
Cosine
Tangent
Triangle
Radians
Degrees
Period
Amplitude
Principal axis
Inverse
Trigonometric identities
Angle
Reciprocal
Trigonometric functions are type of periodic functions where f(x+T)=f(x)f(x+T) = f(x) for certain TT.
There are three types of trigonometric functions you need to know:sin⁡x,cos⁡x,tan⁡x \sin x,\cos x,\tan x.
Function
Features
asin⁡(bx+c)+da\sin(bx+c)+d
Period: 2πb\frac {2π}b
Amplitude: ∣a∣|a|
Principal Axis: y=dy = d
1≤x≤11 ≤ x ≤ 1, thus find the min/max accordingly.
acos⁡(bx+c)+da\cos(bx+c)+d
Period: 2πb\frac {2π}{b}
Amplitude: ∣a∣|a|
Principal Axis: y=dy = d
1≤x≤11 ≤ x ≤ 1, thus find the min/max accordingly.
atan⁡(bx+c)+da\tan(bx+c)+d
Period: πb\frac πb
Vertical asymptotes:x=±2n−12π x = ±\frac{ 2n- 1}{2π} translated accordingly.
xx does not have a min/max.
We now utilize a different measurement of angle, called radian, where πrad≡180˚π rad ≡ 180˚.
radian=degree⋅π180radian=degree\cdot\frac{\pi}{180}
These are extremely useful in geometry, and in drawing functions.
In addition, you also need to be aware of reciprocal/inverse trigonometric functions.
Function
sec⁡x=1cos⁡x\sec x=\frac1{\cos x}
Features
csc⁡x=1sin⁡x\csc x=\frac1{\sin x}
cot⁡x=1tan⁡x=cos⁡xsin⁡x\cot x=\frac1{\tan x}=\frac{\cos x}{\sin x}
arcsin⁡x=sin⁡−1x\arcsin x=\sin^{-1}x
Domain: {x∣−1≤x≤ 1x |-1 ≤ x ≤ 1}
Range: {y∣−π2≤y≤π2y | -\frac\pi2 ≤ y ≤\frac\pi 2}
arccos⁡x=cos⁡−1x\arccos x=\cos^{-1}x
Domain: {x∣−1≤x≤ 1x | -1 ≤ x ≤ 1}
Range: {y∣0≤y≤πy | 0 ≤ y ≤ \pi}
arctan⁡x=tan⁡−1x\arctan x=\tan^{-1}x
Domain: {x∣xϵRx | x \epsilon ℝ}
Range: {y∣−π2<y<π2y | -\frac\pi2 < y < \fracπ2}
Notice that the inverse trigonometric functions are no longer periodic.
Angle relationships:
1.
Negative angle formulae
a.
sin⁡(−θ)=−sin⁡θ\sin(-\theta)=-\sin\theta
b.
cos⁡(−θ)=cos⁡θ\cos(-\theta)=\cos\theta
2.
Supplementary angle formulae
a.
sin⁡(π−θ)=sin⁡θ\sin(\pi-\theta)=\sin\theta
b.
cos⁡(π−θ)=−cos⁡θ\cos(\pi-\theta)=-\cos\theta
3.
Complementary angle formulae
a.
sin⁡(π2−θ)=cos⁡θ\sin(\frac\pi2-\theta)=\cos\theta
b.
cos⁡(π2−θ)=sin⁡θ\cos(\frac\pi2-\theta)=\sin\theta
Important trigonometric identities:
1.
sin⁡2θ+cos⁡2θ=1\sin^2\theta+\cos^2\theta =1
2.
sin⁡(nπ2+θ)\sin(\frac{nπ}2+\theta)
a.
If nn is odd, this changes to according to the sign in the quadrant.
b.
If nn is even, this stays as according to the sign in the quadrant.
3.
cos⁡(nπ2+θ)\cos(\frac{nπ}2+\theta)
a.
If nn is odd, this changes to according to the sign in the quadrant.
b.
If nn is even, this stays as according to the sign in the quadrant.
4.
1+tan⁡2θ=sec⁡2θ1+\tan^2\theta=\sec^2\theta
5.
1+cot⁡2θ=csc⁡2θ1+\cot^2\theta=\csc^2\theta
6.
Double angle identities:
a.
sin⁡2θ=2sin⁡θcos⁡θ\sin2\theta=2\sin\theta \cos\theta
b.
cos⁡2θ=cos⁡2θ−sin⁡2θ=1−2sin⁡2θ=2cos⁡2θ−1\cos2\theta=\cos^2\theta-\sin^2\theta=1-2\sin^2\theta=2\cos^2\theta-1
c.
tan⁡2θ=2tan⁡θ1−tan⁡2θ\tan2\theta=\frac{2\tan\theta}{1-\tan^2\theta}
7.
Compound angle identities:
a.
cos⁡(A±B)=cos⁡Acos⁡B∓sin⁡Asin⁡B\cos(A\pm B)=\cos A\cos B\mp \sin A\sin B
b.
sin⁡(A±B)=sin⁡Acos⁡B±cos⁡Asin⁡B\sin(A\pm B)=\sin A\cos B\pm \cos A\sin B
c.
tan⁡(A±B)=tan⁡A±tan⁡B1∓tan⁡Atan⁡B\tan(A\pm B)=\frac{\tan A\pm \tan B}{1\mp \tan A\tan B}
3.3 Trig functions and identities
done
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3.3 Trig functions and identities