1. Trigonometric Functions
Chapter 5
Trigonometric Functions
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2. Trigonometric Functions of Any Angle: The Unit Circle
SECTION 5.3
Evaluate trigonometric functions of any angle.
Determine the signs of the trigonometric functions in each quadrant.
Find a reference angle.
Use reference angles to find trigonometric function values.
Define the trigonometric functions using the unit circle. 1 2 3 4 5

3. TRIGONOMETRIC FUNCTIONS OF ANGLES
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4. DEFINITION OF THE TRIGONOMETRIC FUNCTIONS OF ANY ANGLE θ
Let P(x,y) be any point on the terminal ray of an angle  in standard position (other than the origin) and let We define
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EXAMPLE 1
Finding Trigonometric Function Values
Suppose  is an angle whose terminal side contains the point P(–1,3). Find the exact values of the six trigonometric functions of .
Solution
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EXAMPLE 1
Finding Trigonometric Function Values
Solution continued
Now, with x = –1, y = 3, and r = , we have
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7. TRIGONOMETRIC FUNCTION VALUES OF QUADRANTAL ANGLES
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COTERMINAL ANGLES
Because the value of each trigonometric function of an angle in standard position is completely determined by the position of the terminal side, the following statements are true.
Coterminal angles are assigned identical values by the six trigonometric functions.
The signs of the values of the trigonometric functions are determined by the quadrant containing the terminal side.
For any integer n, θ, and θ + n360° are coterminal angles and θ and θ + 2πn are coterminal angles.
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9. TRIGONOMETRIC FUNCTION VALUES OF COTERMINAL ANGLES
 in degrees
 in radians
These equations hold for any integer n.
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10. SIGNS OF THE TRIGONOMETRIC FUNCTIONS
Suppose the angle  is not quadrantal and its terminal side contains the point (x,y). The only value other than x and y used in defining the trigonometric functions, , is always positive. Therefore, the signs of x and y determine the signs of the trigonometric functions.
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Determining the Quadrant in Which an Angle Lies
EXAMPLE 4
If tan θ > 0 and cos θ < 0, in which quadrant does θ lie?
Solution
Because tan θ > 0, θ lies either in quadrant I or in quadrant III. However, cos θ > 0 for θ in quadrant I; so θ must lie in quadrant III.
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EXAMPLE 5
Evaluating Trigonometric Functions
Solution
Since tan θ > 0 and cos θ < 0, θ lies in Quadrant III; both x and y must be negative.
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EXAMPLE 5
Evaluating Trigonometric Functions
Solution continued ,
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14. DEFINITION OF A REFERENCE ANGLE
Let  be an angle in standard position that is not a quadrantal angle.
The reference angle for  is the positive acute angle  (“theta prime”) formed by the terminal side of  and the x-axis.
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15. DEFINITION OF A REFERENCE ANGLE
Quadrant I
Quadrant II
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16. DEFINITION OF A REFERENCE ANGLE
Quadrant III
Quadrant IV
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EXAMPLE 6
Identifying Reference Angles
Find the reference angle  for each angle .
a.  = 250º b.  = c.  = 5.75
Solution
Because 250º lies in quadrant III,
 =  - 180º. So  = 250º - 180º = 70º.
Because lies in quadrant II,  = π - .
So  = π
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EXAMPLE 6
Identifying Reference Angles
Solution continued
Since no degree symbol appears in θ = 5.75,  has radian measure. Now ≈ 4.71 and 2π ≈ So  lies in quadrant IV and
 = 2π - . So
 = 2π – 5.75 ≈ 6.28 – 5.75 = 0.53.
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19. USING REFERENCE ANGLES TO FIND TRIGONOMETRIC FUNCTION VALUES
Step 1 Assuming that  > 360º or θ < 0°, find a coterminal angle for  with degree measure between 0º and 360º. Otherwise, go to Step 2.
Step 2 Find the reference angle  for the angle resulting from Step 1. Write the trigonometric function of  .
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20. USING REFERENCE ANGLES TO FIND TRIGONOMETRIC FUNCTION VALUES
Step 3 Choose the correct sign for the trigonometric function value of θ based on the quadrant in which it lies. Write the given trigonometric function of θ in terms of the same trigonometric function of θ with the appropriate sign.
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Using the Reference Angle to Find Values of Trigonometric Functions
EXAMPLE 8
Find the exact value of each expression.
Solution
Step 1 0º < 330º < 360º; find its reference angle. a.
Step º is in Q IV; its reference angle  is .
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Using the Reference Angle to Find Values of Trigonometric Functions
EXAMPLE 8
Solution continued
Step 3 In Q IV, tan θ is negative, so .
b. Step 1
is between 0 and 2π coterminal with .
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23. is in Q IV; its reference angle  is
Using the Reference Angle to Find Values of Trigonometric Functions
EXAMPLE 8
Solution continued
Step 2
is in Q IV; its reference angle  is .
Step 3 In Q IV, sec θ > 0; so .
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24. TRIGONOMETRIC FUNCTIONS AND THE UNIT CIRCLE
A circle with radius 1 centered at the origin of a rectangular coordinate system is a unit circle.
In a unit circle, s = rθ = 1·θ = θ, so the radian measure and the arc length of an arc intercepted by a central angle in a unit circle are numerically identical.
The correspondence between real numbers and endpoints of arcs on the unit circle is used to define the trigonometric functions of real numbers, or the circular functions.
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25. UNIT CIRCLE DEFINITIONS OF THE TRIGONOMETRIC FUNCTIONS OF REAL NUMBERS
Let t be any real number and let P = (x,y) be the point on the unit circle associated with t. Then
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