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Math 8 2.5: Unit Circle Approach; Properties of the Trigonometric Functions
Consider a circle given by the equation
2 2 x + y = 1 , called a unit circle.
On a unit circle , the measure of a central angle θ , and the length of its arct, can be represented by the same real number. Thus θ = t. So sin θ = sin t.
Definition of the Trig Functions of Real Numbers – These should look familiar!
sin t = b
csc t , b 0 b
cos t = a
sec t , a 0 a
= ≠ P a b ( , ) =( cos , sin t t )
tan , 0
b t a a
= ≠ cot , 0
a t b b
**Fill in the Unit Circle Handout
EX: Find the exact value of each function.
Steps: Make a sketch, determine which quadrant the terminal side of the angle lies, and then
use the reference angle to determine the value.
a)
sin 4
⎛ π⎞ ⎜ ⎟ ⎝ ⎠
b)
sec 3
⎛ π⎞ ⎜− ⎝ ⎠
Domain and Range of the Trigonometric Functions
Function Domain Range
sin θ All real numbers,^ ( −∞ ∞,^ ) − 1 ≤ sin θ≤ 1
cos θ All real numbers,^ ( −∞ ∞,^ ) − 1 ≤ cos θ≤ 1
tan θ ( −∞ ∞, )except odd integer multiples of 90
or
π D
−∞ < tan θ< ∞
csc θ ( −∞ ∞,^ )except odd integer multiples of^ π^ or^180
D (^) csc θ≤ − 1 or csc θ≥ 1
sec θ ( −∞ ∞, )except odd integer multiples of 90
or
π D
sec θ≤ − 1 or sec θ≥ 1
cot θ ( −∞ ∞,^ )except odd integer multiples of^ π^ or^180
D −∞ < cot θ< ∞
Periodic Properties
A functionf is called periodic if there is a positive numberp such that, whenever θ is in the
domain off, so is θ + p , and f ( θ+ p ) = f ( θ). The smallest such numberp, is called the
(fundamental) period off.
Sine, cosine and cosecant and secant all have a period of 2 π.
Tangent and cotangent have a period of π.
EX: Use the fact that the trig functions are periodic to find the exact value of:
a) co s 420 b)
D 9
sin 4
Use Even-Odd Properties to Find the Exact Values of the Trigonometric Functions
Recall: Tests for even-odd polynomial functions?
Even-Odd Properties
sin sin tan tan
csc csc cot cot
cos cos
sec sec
θ θ θ θ
θ θ θ θ
θ θ
θ θ
