Download LESSON 35 (5.3) SOLVING TRIGONOMETRIC

LESSON 35 (5.3) SOLVING TRIGONOMETRIC EQUATIONS
You should learn to:
1. Use standard algebraic techniques to solve trigonometric equations.
2. Solve trigonometric equations with a calculator.
3. Use trigonometric equations to solve real-life problems.
Terms to know: trigonometric equation
A trigonometric equation is an equation involving one or more trigonometric functions. Solving trigonometric
equations involves the use of basic algebraic techniques, and “backward trig”.
When solving trigonometric equations, your main goal is to isolate a single trigonometric function. Then use
“backwards” (inverse) trig to solve for the angles. You may have to use factoring, in which case you must make one
side of the equation equal to zero before you factor.
Example 2: Solve the following trigonometric equations without using a calculator. Find all solutions in the
interval  0, 2  .
a. 2cos x  1  0
b. sin x  2   sin x
sin x  2   sin x
2 cos x  1  0
2 cos x  1
2sin x  2
2
3
2 cos x 1

2
2
1
cos x 
2
 5
x ,
3 3
2 sin x
2

2
2
1
sin x 
2
1
 3
2
x
c. 2 tan 2 x  6  0
2 2
3
2
2 tan x 6

2
2
2
tan x  3
 3
tan 2 x  3
x
 2 4 5
3
,
3
,
3
,
3
1
1
1
 3
,
4 4
1
2 2
1
3
 3
sin 2 x  2sin x
sin 2 x  2sin x  0
sin x(sin x  2)  0
sin x  0 or sin x  2  0
sin x  2
x  0R , 
tan x   3
2
d. sin 2 x  2sin x
2 tan 2 x  6  0
2 tan 2 x  6
2
1
(1,0)
(1, 0)
e. 2cos2 x  cos x  1
2 cos 2 x  cos x  1
2 cos x  cos x  1  0
(2 cos x  1)(cos x  1)  0
2 cos x  1  0 or cos x  1  0
1
cos x  
or cos x  1
2
2 4
x  0R ,
,
3 3
2
2
3
 3
1
(1, 0)
2
For trigonometric equations which are difficult or impossible to solve using algebraic methods, you can still use a
calculator to solve them. Make sure to build a window which will show only the solutions that you want if solving by
graphing.
Example 3: Use a calculator to find the solutions to 4cos x  x in the interval   ,   . Express your answers to 3
decimal place accuracy.
y  4 cos x
yx
x  1.2524,  2.1333
window :
  ,  
X  5,5
Example 4: A batted baseball leaves the bat at an angle of  with respect to the horizontal, and an initial velocity of
v0  100 feet per second. The ball is caught by an outfielder 300 feet from home plate. Find
 (in degrees) if the equation for the distance traveled by the ball is d 
1 2
v0 sin(2 ) .
32
1 2
v0 sin(2 )
32
1
300  (100) 2 sin(2 )
32
d
1
(100) 2 sin(2 )
32
y  300
y
  36.87 , 53.13
window :
0,90
X  200,600
ASSIGNMENT 35 (5.3): Pages 376-379 (8, 9, 10, 12, 14, 17, 20, 23, 36, 38, 39, 48, 51, 54, 59 (just graph using
each side of the equation as a separate function, and then use intersect), 82, 98, 99, 101, 103, 111, 112, 116) Page
348 (178) + Page 295 (76, 92)