2. Preliminaries 0.1 THE REAL NUMBERS AND THE CARTESIAN PLANE 0.2
0.1
THE REAL NUMBERS AND THE CARTESIAN PLANE
0.2
LINES AND FUNCTIONS
0.3
GRAPHING CALCULATORS AND COMPUTER ALGEBRA SYSTEMS
0.4
TRIGONOMETRIC FUNCTIONS
0.5
TRANSFORMATIONS OF FUNCTIONS
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3. TRIGONOMETRIC FUNCTIONS
0.4
TRIGONOMETRIC FUNCTIONS
4.1
A function f is periodic of period T if f (x + T ) = f (x)
for all x such that x and x + T are in the domain of f.
The smallest such number T > 0 is called the fundamental period
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4. TRIGONOMETRIC FUNCTIONS
0.4
TRIGONOMETRIC FUNCTIONS
The Unit Circle, Sine, and Cosine
Measure θ in radians, given by the length of the arc indicated in the figure.
Define sin θ to be the y-coordinate
of the point on the circle and
cos θ to be the x-coordinate of the
point.
It follows that sin θ and cos θ are
defined for all values of θ, so that each has domain
−∞ < θ < ∞, while the range for each of these functions is the interval [−1, 1].
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5. TRIGONOMETRIC FUNCTIONS
0.4
TRIGONOMETRIC FUNCTIONS
4.1
Unless otherwise noted, we always measure angles in radians.
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6. TRIGONOMETRIC FUNCTIONS
0.4
TRIGONOMETRIC FUNCTIONS
4.1
The functions f (θ) = sin θ and g(θ) = cos θ are periodic, of period 2π.
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7. TRIGONOMETRIC FUNCTIONS
0.4
TRIGONOMETRIC FUNCTIONS
4.1
Solving Equations Involving Sines and Cosines
Find all solutions of the equations (a) 2 sin x − 1 = 0 and (b) cos2 x − 3 cos x + 2 = 0.
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8. TRIGONOMETRIC FUNCTIONS
0.4
TRIGONOMETRIC FUNCTIONS
4.1
Solving Equations Involving Sines and Cosines
(a) 2 sin x − 1 = 0
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9. TRIGONOMETRIC FUNCTIONS
0.4
TRIGONOMETRIC FUNCTIONS
4.1
Solving Equations Involving Sines and Cosines
(b) cos2 x − 3 cos x + 2 = 0
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10. TRIGONOMETRIC FUNCTIONS
0.4
TRIGONOMETRIC FUNCTIONS
4.2
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11. TRIGONOMETRIC FUNCTIONS
0.4
TRIGONOMETRIC FUNCTIONS
4.2
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12. TRIGONOMETRIC FUNCTIONS
0.4
TRIGONOMETRIC FUNCTIONS
4.2
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13. TRIGONOMETRIC FUNCTIONS
0.4
TRIGONOMETRIC FUNCTIONS
4.2
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14. TRIGONOMETRIC FUNCTIONS
0.4
TRIGONOMETRIC FUNCTIONS
4.2
Altering Amplitude and Period
Graph y = 2 sin x and y = sin 2x, and describe how each differs from the graph of y = sin x.
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15. TRIGONOMETRIC FUNCTIONS
0.4
TRIGONOMETRIC FUNCTIONS
4.2
Altering Amplitude and Period
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16. TRIGONOMETRIC FUNCTIONS
0.4
TRIGONOMETRIC FUNCTIONS
Amplitude and Frequency
A is the amplitude
f is the frequency:
p is the period: p = 1/f
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17. TRIGONOMETRIC FUNCTIONS
0.4
TRIGONOMETRIC FUNCTIONS
4.2
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18. TRIGONOMETRIC FUNCTIONS
0.4
TRIGONOMETRIC FUNCTIONS
4.6
Writing Combinations of Sines and Cosines as a Single Sine Term
Prove that 4 sin x + 3 cos x = 5 sin(x + β) for some constant β and estimate the value of β.
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19. TRIGONOMETRIC FUNCTIONS
0.4
TRIGONOMETRIC FUNCTIONS
4.6
Writing Combinations of Sines and Cosines as a Single Sine Term
This will occur if we can choose a value of β so that
cos β = 45 and sin β = 35. By the Pythagorean Theorem, this is possible only if sin2 β + cos2 β = 1.
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20. TRIGONOMETRIC FUNCTIONS
0.4
TRIGONOMETRIC FUNCTIONS
4.6
Writing Combinations of Sines and Cosines as a Single Sine Term
For the moment, the only way to estimate β is by trial and error.
Using your calculator or computer, you should find that one solution is β ≈ radians (about 36.9 degrees).
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21. TRIGONOMETRIC FUNCTIONS
0.4
TRIGONOMETRIC FUNCTIONS
4.8
Finding the Height of a Tower
A person 100 feet from the base of a radio tower measures an angle of 60o from the ground to the top of the tower. Find the height of the tower.
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22. TRIGONOMETRIC FUNCTIONS
0.4
TRIGONOMETRIC FUNCTIONS
4.8
Finding the Height of a Tower
Using the similar triangles:
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Slide 22