Graphing Trig Functions

Table of Contents

Graphing Trig Functions

Sine

x	y
0	0
$\frac{\pi }{2}$	1
$\pi$	0
$\frac{3\pi }{2}$	-1
$2\pi$	0
![[Graphing Sine Graphs-20231210194237148.webp	265]]

Cosine

x	y
0	1
$\frac{\pi }{2}$	0
$\pi$	-1
$\frac{3\pi }{2}$	0
$2\pi$	1
![[Graphing Cosine Graphs-20231210194335167.webp	268]]

Tangent

x	y
$-\frac{\pi }{2}$	undef
$0$	0
$\frac{\pi }{2}$	undef
![[Graphing Tangent Graphs-20231210195021112.webp	177]]

Trigonometric Function Transformation Guide

Amplitude: Half the total range of the y axis the function holds Vertical Shift: The shift from y=0 to the middle of the function Phase Shift: The shift from x = 0 to the ‘first point’ of the function Period: The total range of a cycle of the function

Formula

y = $a\sin (k(cos-d))+c$ this applies to all trig functions. not just sin functions a: amplitude k: k is $\frac{2\pi }{period}$ d is the phase shift. if d is positive, go left. if d is negative, go right c is the vertical shift. if c is positive, go up. if c is negative, go down.

Mapping rule

Y Transform

depends on a and c stretch with a factor of a vertical shift at c (x, ay + c)

X Transform

period stretch of 1/k phase shift of -d (x/k-d,y)

Cos to SIn & Sin to Cos

$\cos x=\sin (x+\frac{period}{4})$ $\sin (x)=\cos (x-\frac{period}{4})$

Inverse Trigonometric Functions

Arcsin

Arccos

Arctan

Reciprocal Trigonometric Functions

Graph From Table Of Values

First, write the table for the original function, then find the reciprocals of all the values.

Sine → Cosecant function

x or y	0	$\frac{\pi }{6}$	$\frac{\pi }{3}$	$\frac{\pi }{2}$	$\frac{2\pi }{3}$	$\frac{5\pi }{6}$	$\pi$	$\frac{7\pi }{6}$	$\frac{4\pi }{3}$	$\frac{3\pi }{2}$	$\frac{5\pi }{3}$	$\frac{11\pi }{6}$	$2\pi$
y=sinx	0	1/2	$\frac{\sqrt{3}}{2}$	1	$\frac{\sqrt{3}}{2}$	1/2	0	-1/2	$\frac{-\sqrt{3}}{2}$	-1	$\frac{-\sqrt{3}}{2}$	-1/2	0
y=cscx	undef	2	$\frac{2}{\sqrt{3}}$	1	$\frac{2}{\sqrt{3}}$	2	undef	-2	$\frac{-2}{\sqrt{3}}$	-1	$\frac{-2}{\sqrt{3}}$	-2	undef

notice how the table is intervals with unit circle, but there is no xpi/4 ratios

Cosine → Secant function

x or y	0	$\frac{\pi }{6}$	$\frac{\pi }{3}$	$\frac{\pi }{2}$	$\frac{2\pi }{3}$	$\frac{5\pi }{6}$	$\pi$	$\frac{7\pi }{6}$	$\frac{4\pi }{3}$	$\frac{3\pi }{2}$	$\frac{5\pi }{3}$	$\frac{11\pi }{6}$	$2\pi$
y=cosx	1	$\frac{\sqrt{3}}{2}$	1/2	0	-1/2	$\frac{-\sqrt{3}}{2}$	-1	$\frac{-\sqrt{3}}{2}$	-1/2	0	1/2	$\frac{\sqrt{3}}{2}$	1
y=secx	1	$\frac{2}{\sqrt{3}}$	2	undef	-2	$\frac{-2}{\sqrt{3}}$	-1	$\frac{-2}{\sqrt{3}}$	-2	undef	2	$\frac{2}{\sqrt{3}}$	1

Tangent → Cotangent function

x or y	0	$\frac{\pi }{4}$	$\frac{\pi }{2}$	$\frac{3\pi }{2}$	$\pi$	$\frac{5\pi }{4}$	$\frac{3\pi }{2}$	$\frac{7\pi }{4}$	$2\pi$
y = tanx	0	1	undef	-1	0	1	undef	-1	0
y = cotx	undef	1	0	-1	undef	1	0	-1	undef

Memorizing Reciprocal Trig Functions

Cosecant

They opposites. Local minimas are 1. Local maximas are -1

Secant

Sad face. Local minmas are 1. Local maximas are -1

Cotangent

Like a flipped tangent graph that starts at 0 and ends at 2pi.