trigonometric functions: the unit circle math 109 - precalculus s. rook
TRANSCRIPT
Trigonometric Functions: The Unit Circle
MATH 109 - PrecalculusS. Rook
Overview
• Section 4.2 in the textbook:– Circular trigonometric functions– Properties of sine & cosine– Even & odd functions
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Circular Trigonometric Functions
Unit Circle
• Unit Circle: a special circle with a radius of 1, center of (0, 0), and equation of x2 + y2 = 1
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Real Number Line & Unit Circle
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• Starting at (1, 0) consider “wrapping” the positive real number line around the unit circle– Each number on the real number
line corresponds to ONE point on the unit circle
– Each point on the unit circle corresponds to MANY points on the real number line• Recall that the circumference of a circle is 2πr• Thus, each revolution around the unit circle constitutes
2π ≈ 6.28 units
Circular Functions
• Now consider a point (x, y) on the circumference of the unit circle where t is the length of the arc from (1, 0) to (x, y)
• Then – The central angle is equivalent
to the length of the arc it cuts on the Unit Circle
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tt
r
s1
Circular Functions (Continued)
• All points encountered on the unit circle can then be written as a function of t where t is the distance traveled from (1, 0)
t can be positive (counterclockwise) or negative (clockwise)
• There are six trigonometric functions – defined with respect to the Unit Circle they are:
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xt cosyt sin 0,tan xx
yt
0,1
sec xx
t0,1
csc yy
t 0,cot yy
xt
Circular Functions (Continued)• We call these the circular functions
– The radian measure of θ is the same as the arc length from (1, 0) to a point P on the terminal side of θ on the circumference of the unit circle
• Any point (x, y) on the unit circle can be written as (cos t, sin t)
• It is to your advantage to memorize the values in Quadrant I and the quadrantal angles (next slide)
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Common Angles and Function Values on the Unit Circle
Degrees Radians cos θ sin θ
0° 0 1 0
30°
45°
60°
90° 0 1
180° -1 0
270° 0 -1
360° 1 0
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6
4
3
2
2
3
2
2
3
2
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2
2
1
2
2
2
1
2
1
2
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Circular Functions (Example)
Ex 1: Use the Unit Circle to find the six trigonometric functions of:
a)
b)
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4
7
3
4
Circular Functions (Example)
Ex 2: Use the Unit circle to find all values of t, 0 < t < 2π where
a)
b)
c) 11
2
1cos t
2
2sin t
1tan t
Circular Functions (Example)
Ex 3: If t is the positive distance from (1, 0) to point P along the circumference of the unit circle, sketch t on the circumference of the unit circle and then find the value of:
a) P = (0.8560, -0.5169); find i) cos t, ii) csc t, and iii) cot t
b) P = (0.0432, 0.9782); find i) sin t, ii) sec t, and iii) tan t
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Properties of Sine & Cosine
Domain & Range of Sine & Cosine
• Recall that domain is the allowable set of input values for a function:– Given (x, y) = (cos t, sin t) on the unit circle, there are no
places on the unit circle where cos t or sin t are undefined• Domain of f(t) = sin t and f(t) = cos t is (-oo, +oo)
• Recall that range is the acceptable output values for a function:– All points (x, y) = (cos t, sin t) on the unit circle must
satisfy: -1 ≤ (x, y) ≤ 1 • Range of f(t) = sin t and f(t) = cos t is [-1, 1]
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Period of Sine & Cosine
• We have already seen that ONE revolution around the unit circle occurs when t assumes values in the interval [0, 2π)
• Given a function f(t), the period is the smallest value c, c > 0 such that f(t + c) = f(t) for all t in the domain of f– i.e. when the function values start to repeat
• Given the point t = (x, y) on the unit circle, what value added to t results in the same point (x, y)? – Thus: sin(t + 2πn) = sin(t) and cos(t + 2πn) = cos(t), n is an
integer– What is the period of f(t) = sin t and f(t) = cos t
2π
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Period of Sine & Cosine (Example)
Ex 4: Evaluate the trigonometric function using its period [rewrite in the form cos(t + 2πn) or sin(t + 2πn)] and the unit circle:
a)
b)
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3
8cos
2
15sin
Even and Odd Functions
Even and Odd Functions
• Recall the definition of even and odd functions:– If f(-t) = f(t), f(t) is an even function– If f(-t) = -f(t), f(t) is an odd function
• Examine the Unit Circle at the right:
cos(-θ) = cos θ meaning?sin(-θ) = -sin θ meaning?
Even Odd
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tt coscos
tt secsec
tt sinsin
tt csccsc
tt tantan tt cotcot
Even & Odd Functions (Example)
Ex 5: Use to evaluate:
a) cos t
b) sec(-t)
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5
1cos t
Summary
• After studying these slides, you should be able to:– Define the six trigonometric functions in terms of the unit
circle– Given a value of t or a point (x, y) evaluate the six
trigonometric functions– State the domain, range, and period of the sine and cosine– Understand even & odd functions
• Additional Practice– See the list of suggested problems for 4.2
• Next lesson– Right Triangle Trigonometry (Section 4.3)
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