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Overview
• Section 4.4 in the textbook:– Trigonometric functions of any angle– Reference angles– Trigonometric functions of real numbers
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Trigonometric Functions of Any Angle
• Given an angle θ in standard position and a point (x, y) on the terminal side of θ, then the six trigonometric functions of ANY ANGLE θ are can be defined in terms of x, y, and the length of the line connecting the origin and (x, y) denoted as r
4
Trigonometric Functions of any Angle (Continued)
5
Function Abbreviation Definition
The sine of θ sin θ
The cosine of θ cos θ
The tangent of θ tan θ
The cotangent of θ cot θ
The secant of θ sec θ
The cosecant of θ csc θ
Where and x and y retain their signs from (x, y)
r
y
r
x
0, xx
y
0, yy
x
0, xx
r
0, yy
r
22 yxr
Trigonometric Functions of any Angle (Continued)
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Function Abbreviation Definition
The sine of θ sin θ
The cosine of θ cos θ
The tangent of θ tan θ
The cotangent of θ cot θ
The secant of θ sec θ
The cosecant of θ csc θ
Where and x and y retain their signs from (x, y)
r
y
r
x
0, xx
y
0, yy
x
0, xx
r
0, yy
r
22 yxr
Algebraic Signs of Trigonometric Functions
• The sign of the six trigonometric functions depends on which quadrant θ terminates in:
r is the distance from the origin to (x, y) so it is ALWAYS positive
– The signs of x and y depend on which quadrant (x, y) lies
– Remember the shorthand notation involving “the element of” symbol:
• i.e. means theta is a standard angle which terminates in Q IV
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QIV
Algebraic Signs of Trigonometric Functions (Continued)
Functions θ Є QI θ Є QII θ Є QIII θ Є QIV
and + + – –
and + – – +
and + – + –
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r
ysin
y
rcsc
r
xcos
x
rsec
x
ytan
y
xcot
Trigonometric Functions of any Angle (Example)
Ex 1: Find the value of all six trigonometric functions if:
a) (-1, 2) lies on the terminal side of θ
b) (-7, -1) lies on the terminal side of θ
9
Trigonometric Functions of any Angle (Example)
Ex 2: Given sec θ = -3⁄2 where cos θ < 0, find the exact value of tan θ and csc θ
10
Reference Angles
• An important definition is the reference angle– Allows us to calculate ANY angle θ using an
equivalent positive acute angle • We can now work in all four quadrants of the Cartesian
Plane instead of just Quadrant I!
• Reference angle: denoted θ’, the positive acute angle that lies between the terminal side of θ and the x-axis
θ MUST be in standard position
12
Reference Angles Summary
• Depending in which quadrant θ terminates, we can formulate a general rule for finding reference angles:– For any positive angle θ, 0° ≤ θ ≤ 360°:
• If θ Є QI:θ’ = θ
• If θ Є QII:θ‘ = 180° – θ
• If θ Є QIII:θ‘ = θ – 180°
• If θ Є QIV:θ’ = 360° – θ
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Reference Angles Summary (Continued)
– If θ > 360°:• Keep subtracting 360° from θ until 0° ≤ θ ≤ 360°• Go back to the first step on the previous slide
– If θ < 0°:• Keep adding 360° to θ until 0° ≤ θ ≤ 360°• Go back to the first step on the previous slide
– If θ is in radians:• Either replace 180° with π and 360° with 2π OR• Convert θ to degrees
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Reference Angles (Example)
Ex 3: i) draw θ in standard position ii) draw θ’, the reference angle of θ:
a) 312° b) π⁄8
c) 4π⁄5 d) -127°
e) 11π⁄3
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Reference Angle Theorem
• Reference Angle Theorem: the value of a trigonometric function of an angle θ is EQUIVALENT to the VALUE of the trigonometric function of its reference angle– The ONLY thing that may be different is the sign
• Determine the sign based on the trigonometric function and which quadrant θ terminates in
– The Reference Angle Theorem is the reason why we need to memorize the exact values of 30°, 45°, and 60° only in Quadrant I!
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Evaluating a Trigonometric Function Exactly
• To evaluate a trigonometric function of θ:– Ensure that 0 < θ < 2π when using radians or
0° < θ < 360° when using degrees– Find θ’ the reference angle of θ– Evaluate the function using the EXACT values of
the reference angle and the quadrant in which θ terminates
• Write the function in terms of sine or cosine if necessary
22
Evaluating a Trigonometric Function (Exactly)
Ex 4: Give the exact value:
a) sin 225° b) cos 750°
c) tan 120° d) sec -11π⁄4
23
Summary
• After studying these slides, you should be able to:– Calculate the trigonometric function of ANY angle θ– State the reference angle of an angle θ in standard
position– Evaluate a trigonometric function using reference angles
and exact values• Additional Practice
– See the list of suggested problems for 4.4• Next lesson
– Graphs of Sine & Cosine Functions (Section 4.5)
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