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MAT170
LIST OF CONCEPTS TEST 3
3.1 – Introduction:
 radian measure.
3.2
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The sine and cosine functions
Definition of sin t and cos t on the unit circle
Pythagorean identity
Periodic functions: understand the concept of a function that repeats its output on a regular basis (the
period).
Know the EXACT values of sine and cosine of /3, /4 and /6 and how to derive values for integer
multiples of these angles (e.g. 5/3, 11/6, 7/4 …)
Reference number
Trigonometric equations
3.3 Graphs of sine and cosine functions
 Graphs of sine and cosine: cosine function is even, the sine function is odd. Sine and cosine are
periodic of period 2.
 Sinusoidal functions (see box on page 208): know the meaning of A, B and C: |A|= amplitude, 2/|B| =
period, |C/B|=horizontal shift.
 Know how to sketch one period of the graph given the formula and how to find a possible formula
given the graph.
3.4 Other trigonometric functions
 Know the definitions of tangent, cotangent, secant and cosecant functions.
 Know the graph of the tangent function (tangent is periodic of period  since it is the slope of the ray
from the origin of an angle t).
3.5 Trigonometric identities
 Important identities: Pythagorean identities (page 225), Sum and difference formulas for the sine and
cosine (pages 223/224), Double angle formulas (page 228), Half angle formulas (page 230), Formulas
for products of sine and cosine (page 232).
 Know how to use sum/difference formulas and double angle formulas for sine and cosine to find exact
values of trig functions of special angles and to prove simple identities relating sine and cosine.
 Know how to solve trig equations using identities.
 Know how to verify (prove) an identity algebraically and graphically.
Sum/difference formulas, half- angle formulas and formulas for products of sine and cosine will be given
on the test. Pythagorean identities and double angle formulas will not be given. You need to memorize
them (or be able to derive them). Note that double angle formulas can be easily derived from
sum/difference formulas.
Strategies for proving an identity:
1. Work separately on the 2 sides of the identity. Begin with the more complicated expression and
modify it using algebra and known identities so that it looks like the other side.
2. If no other move suggests itself, convert the entire expression to one involving sine and cosine.
3. Combine fractions by writing them over a common denominator.
3.6
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(a  b)(a  b)  a 2  b 2 to set up applications of the Pythagorean
2
2
identities (e.g. (1  sin  )(1  sin  )  1  sin   cos  )
4.
Use the algebraic identity
5.
Always be mindful of the “target” expression and favor manipulation that brings you closer to
your goal.
Right angle trigonometry
Trigonometric functions of an angle in a right triangle.
Measuring angles using degrees
Conversion between radians and degrees
3.7 Inverse trigonometric functions
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Know the definitions of inverse sine, cosine and tangent functions. Think of sin
[
 
,
2 2
] whose sine is x. Think of cos
tan 1 x as the angle in [ 
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1
1
x as the angle in
x as the angle in [0,] whose cosine is x. Think of
 
, ] whose tangent is x.
2 2
KNOW YOUR BASIC VALUES FOR SINE, COSINE AND TANGENT IN EXACT FORM! E.g.
sin 1 (1 / 2)   / 6 radians (Examples 4,5,6).
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Know the domain and range of the inverse trigonometric functions.
Arcsine and arctangent are odd functions, arccosine is neither even nor odd.
Know how to solve a trigonometric equation graphically.
Know how to use the inverse sine, cosine and tangent functions to solve trigonometric equations.
Remember, these functions will only give one value back if you use your calculator, so you will need
to use your knowledge of the sine, cosine and tangent graphs to determine the other solutions, if they
are applicable.
3.8 Applications of trigonometric functions
 Know the Law of Cosines and the Law of Sines (these formulas will not be given in the test) and how
to apply them.
 Sine combination formulas, cosine combination formulas (optional)
 Herone’s formula (optional)