Preparing for the SAT II
Trigonometry
Trigonometry
Trigonometry begins in the right triangle, but it doesn’t have to be restricted to triangles. The trigonometric functions carry the ideas of triangle trigonometry into a broader world of real-valued functions and wave forms.
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©Carolyn C. Wheater, 2000
Trigonometry Topics
Radian Measure
The Unit Circle Trigonometric Functions
Larger Angles
©Carolyn C. Wheater, 2000
Graphs of the Trig Functions Trigonometric Identities Solving Trig Equations
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Radian Measure
To talk about trigonometric functions, it is
©Carolyn C. Wheater, 2000
helpful to move to a different system of angle measure, called radian measure. A radian is the measure of a central angle whose intercepted arc is equal in length to the radius of the circle.
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Radian Measure
There are 2 radians in a full rotation --
©Carolyn C. Wheater, 2000
once around the circle There are 360° in a full rotation To convert from degrees to radians or radians to degrees, use the proportion degrees radians 360 2
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Sample Problems
Find the degree Find the radian
measure equivalent 3 of radians.
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measure equivalent of 210°
degrees radians 360 2 210 r 360 2 360r 420 420 7 r 360 6
©Carolyn C. Wheater, 2000
degrees radians 360 2 d 3 4 360 2 2d 270 d 135
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The Unit Circle
Imagine a circle on the
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coordinate plane, with its center at the origin, and a radius of 1. Choose a point on the circle somewhere in quadrant I.
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The Unit Circle
Connect the origin to the
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point, and from that point drop a perpendicular to the x-axis. This creates a right triangle with hypotenuse of 1.
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The Unit Circle
The length of its legs are
©Carolyn C. Wheater, 2000
the x- and y-coordinates of the chosen point. Applying the definitions of the trigonometric ratios to this triangle gives y x sin( ) y cos x 1 1
is the angle of rotation 1 x
y
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The Unit Circle
The coordinates of the chosen point are the
cosine and sine of the angle .
This
provides a way to define functions sin() and cos() for all real numbers .
©Carolyn C. Wheater, 2000
y sin( ) y 1
The
x cos x 1
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other trigonometric functions can be defined from these.
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Trigonometric Functions
sin( ) y
1 csc y
1 sec x
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is the angle of rotation 1 x
cos x
©Carolyn C. Wheater, 2000
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y
y tan x
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x cot y
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Around the Circle
As that point
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moves around the unit circle into quadrants II, III, and IV, the new definitions of the trigonometric functions still hold.
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Reference Angles
The angles whose terminal sides fall in
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quadrants II, III, and IV will have values of sine, cosine and other trig functions which are identical (except for sign) to the values of angles in quadrant I. The acute angle which produces the same values is called the reference angle.
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Reference Angles
The reference angle is the angle between the
©Carolyn C. Wheater, 2000
terminal side and the nearest arm of the xaxis. The reference angle is the angle, with vertex at the origin, in the right triangle created by dropping a perpendicular from the point on the unit circle to the x-axis.
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Quadrant II
Original angle
For an angle, , in
quadrant II, the reference angle is In quadrant II,
©Carolyn C. Wheater, 2000
Reference angle
is positive cos() is negative tan() is negative
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sin()
Quadrant III
Original angle
For an angle, , in
quadrant III, the reference angle is - In quadrant III,
©Carolyn C. Wheater, 2000
Reference angle
sin()
is negative cos() is negative
tan()
is positive
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Quadrant IV
For an angle, , in
Reference angle
quadrant IV, the reference angle is 2 In quadrant IV,
is negative cos() is positive
sin()
©Carolyn C. Wheater, 2000
Original angle
tan()
is negative
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All Star Trig Class
Use the phrase “All Star Trig Class” to
remember the signs of the trig functions in different quadrants.
Star
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All
Sine is positive All functions are positive
Trig
Tan is positive
Class
Cos is positive
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Graphs of the Trig Functions
Sine The most fundamental sine wave, y=sin(x), has the graph shown. It fluctuates from 0 to a high of 1, down to –1, and back to 0, in a space of 2.
©Carolyn C. Wheater, 2000
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Graphs of the Trig Functions
The graph of y a sin b x h k is
cb g h
determined by four numbers, a, b, h, and k.
amplitude, a, tells the height of each peak and the depth of each trough. The frequency, b, tells the number of full wave patterns that are completed in a space of 2. 2 The period of the function is b The two remaining numbers, h and k, tell the translation of the wave from the origin.
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The
©Carolyn C. Wheater, 2000
Sample Problem
5 4 3 2 1 2 1 1 2 ©Carolyn C. Wheater, 2000 3 4 5 1 2
Which of the following
equations best describes the graph shown?
(A)
y = 3sin(2x) - 1 (B) y = 2sin(4x) (C) y = 2sin(2x) - 1 (D) y = 4sin(2x) - 1 (E) y = 3sin(4x)
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Sample Problem
5 4 3 2 1 2 1 1 2 ©Carolyn C. Wheater, 2000 3 4 5 1 2
Find the baseline between
the high and low points.
Graph is translated -1 vertically.
Find height of each peak. Amplitude is 3
Count number of waves in
y = 3sin(2x) - 1
2
Frequency
is 2
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Graphs of the Trig Functions
Cosine The graph of y=cos(x) resembles the graph of y=sin(x) but is shifted, or translated, units to 2 the left. It fluctuates from 1 to 0, down to –1, back to 0 and up to 1, in a space of 2.
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©Carolyn C. Wheater, 2000
Graphs of the Trig Functions
The values of a, b, h, and k change the
shape and location of the wave as for the sine. y a cos b x h k
cb g h
©Carolyn C. Wheater, 2000
Amplitude Frequency Period Translation
a b 2/b h, k
Height of each peak Number of full wave patterns Space required to complete wave Horizontal and vertical shift
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Sample Problem
Which of the following
8 6 4 2 2 1 1 2
equations best describes the graph?
©Carolyn C. Wheater, 2000
y = 3cos(5x) + 4 (B) y = 3cos(4x) + 5 (C) y = 4cos(3x) + 5 (D) y = 5cos(3x) +4 (E) y = 5sin(4x) +3
(A)
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Sample Problem
Find the baseline Vertical translation + 4
Find the height of
8 6 4 2 2 1 1 2
peak
©Carolyn C. Wheater, 2000
Amplitude = 5
Number of waves in
2
Frequency
y = 5cos(3x) + 4
=3
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Graphs of the Trig Functions
Tangent The tangent function has a discontinuous graph, repeating in a period of .
©Carolyn C. Wheater, 2000
Cotangent Like the tangent, cotangent is discontinuous.
• Discontinuities of the cotangent are 2 units left of those for tangent.
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Graphs of the Trig Functions
Secant and Cosecant The secant and cosecant functions are the reciprocals of the cosine and sine functions respectively. Imagine each graph is balancing on the peaks and troughs of its reciprocal function.
©Carolyn C. Wheater, 2000
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Trigonometric Identities
An identity is an equation which is true for
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all values of the variable. There are many trig identities that are useful in changing the appearance of an expression. The most important ones should be committed to memory.
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Trigonometric Identities
Reciprocal Identities
1 sin x csc x
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Quotient Identities
sin x tan x cos x cos x cot x sin x
1 cos x sec x 1 tan x cot x
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Trigonometric Identities
Cofunction Identities The function of an angle = the cofunction of its complement.
sin x cos(90 x)
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sec x csc(90 x) tan x cot(90 x)
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Trigonometric Identities
Pythagorean Identities
The
fundamental
Pythagorean identity
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sin x cos x 1
2 2
Divide the first by
sin2x
Divide the first by cos2x
1 cot 2 x csc2 x tan2 x 1 sec2 x
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Solving Trig Equations
Solve trigonometric equations by following
these steps:
If
©Carolyn C. Wheater, 2000
there is more than one trig function, use identities to simplify Let a variable represent the remaining function Solve the equation for this new variable Reinsert the trig function Determine the argument which will produce the desired value
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Solving Trig Equations
To solving trig equations:
Use Let
©Carolyn C. Wheater, 2000
identities to simplify variable = trig function for new variable the trig function the argument
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Solve
Reinsert
Determine
Sample Problem
Solve 3 3 sin x 2 cos2 x 0 Use the Pythagorean 3 3 sin x 2 cos2 x 0 identity
• (cos2x = 1 - sin2x)
Distribute
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3 3 sin x 2 1 sin 2 x 0 3 3 sin x 2 2 sin 2 x 0 1 3 sin x 2 sin 2 x 0 2 sin 2 x 3 sin x 1 0
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c
h
like terms Order terms
Combine
Sample Problem
Solve 3 3 sin x 2 cos2 x 0 2 Let t = sin x 2 sin x 3 sin x 1 0
Factor
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and solve.
2t 2 3t 1 0 (2t 1)(t 1) 0 2t 1 0 2t 1 1 t 2
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t 1 0 t 1
Sample Problem
Solve 3 3 sin x 2 cos2 x 0 Replace t = sin x.
t t
©Carolyn C. Wheater, 2000
= sin(x) = ½ when = sin(x) = 1 when
6 x 2
x
or
5 6
5 , , So the solutions are x 6 6 2
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