Precalculus
Trigonometry Review
Final Review #5
Name:_________________________________________
Reciprocal Identities
1
secӨ=𝑐𝑜𝑠Ө
1
cscӨ = 𝑠𝑖𝑛Ө
Ratio Identities
Pythagorean Identities
𝑠𝑖𝑛Ө
sin2Ө + cos2Ө = 1
𝑐𝑜𝑠Ө
tan2Ө +1 = sec2Ө
tanӨ =𝑐𝑜𝑠Ө
cotӨ= 𝑠𝑖𝑛Ө
1
1+ cot2Ө = csc2Ө
cotӨ = 𝑡𝑎𝑛Ө
Summary of Cofunctions
Sine and Cosine
sinӨ=cos(90-Ө)
cosӨ=sin(90-Ө)
Tangent and Cotangent
tanӨ=cot(90-Ө)
cotӨ=tan(90-Ө)
Secant and Cosecant
secӨ=csc(90-Ө)
cscӨ=sec(90-Ө)
COFUNCTIONS
COTERMIANL ANGLES AND REFERENCE ANGLES
TRIGONOMETRIC EQUATIONS
1) Substitute equal values when necessary (usually for double angle, more than one term in the same
equation…) – your goal is to get the original equation written in terms of one trigonometric function if
possible
2) Once all of the substitutions are made, solve as if the trig function is a variable… I.e. sin=y, cos =x… this
allows for easier manipulation
3) When the trig functions are solved for values, solve for the theta Ө. Most equations have two theta values
that will satisfy---
** so if the original is quadratic you may potentially have 4 solutions
Identities
Verifying an identity1) Start with the most complex side of the equation
2) Substitute in equal values if necessary
3) DO NOT move terms to the other side of the equal sign
4) Simplify using all valid algebraic manipulations
1 + sin(x)
cos(x)
+
cos(x) 1 + sin(x)
=
2 sec(x)
Graphs of Trig Functions
y = a sin b(x+c)+k
a is amplitude
y = sin x
y= a cos b(x+c)+k
b is frequency
c is the phase shift
Different Amplitude--y= 2 sin x
y= cos x
y = 3 cos x
y= tan x
y = 2 tan x
k is the vertical shift
Changes in frequency
y= sin 2x
y= cos 3x
Vertical shifts
y = sin 2x + 3
1
y = cos 2 x – 2
Mixed:
Graph:
𝜋
y = 2 sin 2(x- 2 ) -2
Phase shifts
𝜋
y = sin 2( x+ 2 )
𝜋
y= cos 3( x – 3 )
Law of sines or Law of cosines