Download Chapter 5: Trigonometric Equations Section: 5.1: Solving

Chapter 5:
Section:
Trigonometric Equations
5.1: Solving Conditional Equations I
We have solved equations such as:
cos x  
3
for 0  x  
2
or
sin  
1
for  90    90
2
What if we expanded the intervals above? (0 < x < 2π or 0º < θ < 360º)
What if we place no restriction on the variable? (x any real number)
Solve:
cos x  
Solve:
sin θ = ½
3
for 0  x  2
2
for
and for x  R
0    360
and
A
Solve: tan    3
Solve:
sin x = 0.2
Solve:
tan x = -4.6
Solve:
sec x =1.5
0    180
for
for x when
for x when
for
0  x  360
0  x  360
0 < x < 2π
and
and
A
x A
x A
Approx to 4 decimal places.
Explain why the following have no solutions:
a) cos x = 3
b) csc x = -½
HW 5.1, pp295-296: 1 – 19 odd.
and
Section 5.2:
Solving Trigonometric Conditional Equations II
cos x  
3
for 0  x  
2
The equations we have solved so far have been simple equations in
immediately solvable form in that they can be solved by inspection
(recognizing a common angle from its functional value or by using
inverse functions). The equations in this section require more algebraic
manipulation to write in immediately solvable form. Appendix A has a
review on solving algebraic equations.
First Degree (Linear) for (0 < x < 2π)
2 cos x + 1 = 3
4 + 3 tan x = 2 tan x + 5
Second Degree (Quadratic)
sin2 x – sin x – 2 = 0 (0 < x < 2π)
cos2 t – 3 cos t = 2
(0 < x < 2π)
tan x cos x = tan x (0 < x < 2π)
3 tan2 A = 1
for exact values of A.
Find all solutions for
cos2 t – 3 cos t = 2
HW 5.2, pp302-226: 1 – 43 odd, 61.
0° < A < 360°
for t a real number
Section 5.3:
More Trigonometric Equations, Multi-Angle Equations
In this section we will solve trigonometric equations that require
both trig identities and algebraic manipulation.
3 cos x – 2 sin2 x = 0
(0 < x < 2π)
6 cos θ + 7 tan θ = sec θ
0° < θ < 360°
sin t = cos t + 1
0 < t < 2π
for
sin 2x 
3
for 0  x  360
2
sin 2x  3sin x  0 for 0  x  360
cos 4x cos x + sin 4x sin x = 0
0° < x < 360°
cos2 2θ – 2 sin 2θ + 2 = 0
(0 < θ < 2π)
Solving Trigonometric Equations
1. If the equation involves different multiples of the angle, use
identities to obtain the same multiple.
2. If the equation involves more than one function, if possible, use
identities to reduce it to involve only one function. If not, try the
following:
a) Square both sides of the equation.
b) Multiply both sides by the common denominator to eliminate
fractions. Then, if possible, use identities to rewrite the
equation in terms of one function.
3. Identify the equation as linear or quadratic in form or as an
equation that is factorable. Then use established algebraic
procedures.
HW 5.3, pp311-312: 1 – 31 odd, 43 – 55 odd.
Section 5.4:
Parametric Equations
In algebra, graphs in the coordinate plane are usually represented
by an equation in the variables x, and y. These equations are called
Cartesian or rectangular. Ex: y = 2x2 + 3x +1.
In Chapter 1, we introduced the unit circle x2 + y2 = 1 and then
defined the x and y coordinates in terms of arc length t.
x = cos t and y = sin t We have introduced a third variable t
We now define the variable t as a parameter and the equations
x = f (t) = cos t and y = g (t) = sin t as parametric equations
A parameter can represent time, arc length or an angle. In Chapter 1 we
parametrically defined the x- and y-coordinates on the unit circle as
continuous functions of t.
x = cos t and y = sin t
Where
x2 + y2 = 1
Ex: Let’s let t be units of time (0 < t < 2π) for an object travelling along
the unit circle
a) Where is the object at t = 0 ?, t = π/2 ?
b) At what time will the object be at the point (0, -1)?
c) What is the direction of motion?
d) How long will it take object to return to its starting position?
t
x = cos t
y = sin t
0
π/2
π
3 π/2
2π
Use an arrow on the curve to indicate direction (orientation) of travel.
Ex: Describe the motion of x = sin 2t and y = cos 2t, where t
represents time and 0 < t < π, by finding the rectangular equation for the
curve. Then graph the motion indicating the starting and ending points,
the orientation and then find the time it takes to return to the starting
position.
cos2 2t + sin2 2t =1
y2 + x2 = 1
or x2 + y2 = 1
t
x = sin 2t
y = cos 2t
0
π/4
π/2
3 π/4
π
Steps for Sketching Parametric Equations
1. Eliminate the parameter (if possible) to find the rectangular
equation. If x and y are defined in terms of sine and/or cosine,
try to use the Pythagorean identity to eliminate. For other trig
functions, try to use an equivalent form of the Pythagorean
identity. If x and y are defined in other terms, try the method of
substitution.
2. Set up a table of t x, and y, with increasing (smallest to largest)
values of t.
3. Plot several points on the curve in order of the increasing values
of the parameter.
Ex: A particle moves in the plane with its x- and y-coordinates varying
with t according to x = 2cos t and y = 3sin t, and 0 < t < 2π.
1. Eliminate the parameter to find the rectangular equation.
2. Find the starting and ending points of motion on the indicated
interval of t.
3. Sketch the graph that describes the motion and use an arrow to
indicate the orientation (direction of increasing values of t).
t x = 2 cos t
y = 3 sin t
0
π/2
π
3π/2
2π
HW 5.4, pp320-321: 1 – 27, odd.