Solving Trig Equations Objective: Solve many different Trig equations.

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Solving Trig Equations
Objective: Solve many different Trig equations.
Introduction• We are going to use standard Algebra techniques to
solve a trig equation. Your primary goal in solving a trig equation is to isolate the trig function. For example, given , we would divide both sides by 2 and have .2/1sin x
1sin2 x
Introduction• We are going to use standard Algebra techniques to
solve a trig equation. Your primary goal in solving a trig equation is to isolate the trig function. For example, given , we would divide both sides by 2 and have
• We can use the unit circle and find that the in two places, x = /6 and x = 5/6. Since we aren’t
working with inverse functions, we have no domain restriction.
.2/1sin x1sin2 x
2/1sin x
Introduction• These are not the only two places where this is true.
Look at the graph below. There are an infinite number of places that the We express all of these answers by writing
.2/1sin x
n 26
n 265
Introduction• We can also look at these answers as coterminal
angles. We can move around the circle an infinite number of times and land on /6 and 5/6.
Example 1• Solve xx sin2sin
Example 1• Solve• Add the sinx to both sides 02sinsin xx
xx sin2sin
Example 1• Solve• Add the sinx to both sides
• Subtract from both sides
02sinsin xx
xx sin2sin
2 2sin2 x
Example 1• Solve• Add the sinx to both sides
• Subtract from both sides
• Divide both sides by 2
02sinsin xx
xx sin2sin
2 2sin2 x
2/2sin x
Example 1• Using the reference angle of 450 or /4, the in the third and fourth quadrant. The answers are
n00 360315 n00 360225
2/2sin x
n 245
n 247
Example 2• Solve 01tan3 2 x
Example 2• Solve• Add 1 to both sides
• Divide both sides by 3
• Take the square root of both sides
01tan3 2 x
1tan3 2 x
3/1tan2 x
33
31
31tan x
Example 2• Using the unit circle and a reference angle of /6, the
answers are in all four quadrants.
n
6n
65
33tan x
n18030 n180150
Example 3• Solve xxx cot2coscot 2
Example 3• Solve• Subtract 2cotx from both sides
• Take out the common factor
• Set each term equal to zero and solve
xxx cot2coscot 2
0cot2coscot 2 xxx
0)2(coscot 2 xx
02cos2 x0cot x
Example 3• Again, use the unit circle to solve.
• The cotx = 0 where cosx = 0. n
2 n18090
Example 3• Again, use the unit circle to solve.
• The cotx = 0 where cosx = 0.
There are no solutions to this equation since the range of cosx is [1,1]
02cos2 x
n
2
2cos2 x
2cos x
n18090
Example 4• Solve by factoring
01sinsin2 2 xx
Example 4• Solve by factoring
• Look at this as the quadratic function
01sinsin2 2 xx
012 2 xx
Example 4• Solve by factoring for [0,2)• Look at this as the quadratic function
0)1)(12( xx
01sinsin2 2 xx
2/112012
xxx
012 2 xx
101
xx
Example 4• Now use these answers to solve for the sinx for [0,2)
2/1sin x
2/112012
xxx
1sin x
101
xx
Example 4• Now use these answers to solve for the sinx for [0,2)
1sin x2/1sin x
2/112012
xxx
101
xx
67
x611
x2
x
o210 o330 o90
Class Work
• Page 558• 2, 8, 10, 12, 22, 26
Homework
• Page 558• 1, 515 odd• 21, 23, 25