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3.1
3.2
3.3
3.4
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3.3
Circular Functions ▪ Finding Values of Circular Functions ▪
Determining a Number with a Given Circular Function Value ▪
Applying Circular Functions
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Circular Functions
A unit circle has its center at the origin and a radius of 1 unit.
The trigonometric functions of angle θ in radians are found by choosing a point ( x , y ) on the unit circle can be rewritten as functions of the arc length s .
When interpreted this way, they are called circular functions.
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Circular Functions
For any real number s represented by a directed arc on the unit circle,
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The Unit Circle
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The Unit Circle
 The unit circle is symmetric with respect to the x -axis, the y -axis, and the origin.
If a point ( a , b ) lies on the unit circle, so do
( a , – b ), ( – a , b ) and ( – a , – b ).
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The Unit Circle
 For a point on the unit circle, its reference arc is the shortest arc from the point itself to the nearest point on the x -axis.
For example, the quadrant I real number is associated with the point on the unit circle.
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The Unit Circle
Since sin s = y and cos s = x , we can replace x and y in the equation of the unit circle to obtain the Pythagorean identity
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Domains of Circular Functions
Sine and Cosine Functions:
Tangent and Secant Functions:
Cotangent and Cosecant Functions:
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Evaluating A Circular Function
Circular function values of real numbers are obtained in the same manner as trigonometric function values of angles measured in radians.
This applies both to methods of finding exact values (such as reference angle analysis) and to calculator approximations.
Calculators must be in radian mode when finding circular function values.
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Example 1
FINDING EXACT CIRCULAR FUNCTION
VALUES
Find the exact values of
Evaluating a circular function at the real number is equivalent to evaluating it at radians.
An angle of intersects the circle at the point (0, –1).
Since sin s = y , cos s = x , and
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Example 2(a)
FINDING EXACT CIRCULAR FUNCTION
VALUES
Use the figure to find the exact values of
The real number corresponds to the unit circle point
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Example 2(b)
FINDING EXACT CIRCULAR FUNCTION
VALUES
Use the figure and the definition of tangent to find the exact value of
Moving around the unit circle units in the negative direction yields the same ending point as moving around the circle units in the positive direction.
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Example 2(b)
FINDING EXACT CIRCULAR FUNCTION
VALUES corresponds to
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Example 2(c)
FINDING EXACT CIRCULAR FUNCTION
VALUES
Use reference angles and degree/radian conversion to find the exact value of
An angle of corresponds to an angle of 120 °.
In standard position, 120 ° lies in quadrant II with a reference angle of 60 °, so
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Cosine is negative in quadrant II.
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Example 3
APPROXIMATING CIRCULAR
FUNCTION VALUES
Find a calculator approximation for each circular function value.
(a) cos 1.85
≈ –.2756
(b) cos .5149
≈ .8703
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Example 3
APPROXIMATING CIRCULAR
FUNCTION VALUES (continued)
Find a calculator approximation for each circular function value.
(c) cot 1.3209
≈ .2552
(d) sec –2.9234
≈ –1.0243
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Caution
A common error in trigonometry is using a calculator in degree mode when radian mode should be used.
Remember, if you are finding a circular function value of a real number, the calculator must be in radian mode.
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Example 4(a)
FINDING A NUMBER GIVEN ITS
CIRCULAR FUNCTION VALUE
Approximate the value of s in the interval if cos s = .9685.
Use the inverse cosine function of a calculator.
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, so in the given interval, s ≈ .2517.
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Example 4(b)
FINDING A NUMBER GIVEN ITS
CIRCULAR FUNCTION VALUE
Find the exact value of s in the interval if tan s = 1.
Recall that , and in quadrant III, tan s is negative.
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Example 5
MODELING THE ANGLE OF ELEVATION
OF THE SUN
The angle of elevation of the sun in the sky at any latitude L is calculated with the formula where corresponds to sunrise and occurs if the sun is directly overhead. ω is the number of radians that Earth has rotated through since noon, when ω = 0. D is the declination of the sun, which varies because Earth is tilted on its axis.
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Example 5
MODELING THE ANGLE OF ELEVATION
OF THE SUN (continued)
Sacramento, CA has latitude L = 38.5
° or .6720 radian. Find the angle of elevation of the sun θ at 3
P.M. on February 29, 2008, where at that time,
D ≈ –.1425 and ω ≈ .7854.
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Example 5
MODELING THE ANGLE OF ELEVATION
OF THE SUN (continued)
The angle of elevation of the sun is about .4773 radian or 27.3
°.
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