Solving More Difficult Trig Equations General Solutions
Solving More Difficult Trig Equations General Solutions
When a special value (k) of the trigonometric ratios is given, inverse trigonometric functions can be used to
determine a solution to an equation involving sine, cosine, or tangent. To write a generalized solution, sele smallest positive solution between 0 to 2K here denoted by s. Equation
cosx = /c sinx= /f
General Solution
Principal Solution
x = cos~\k) = s
x = ± s + 2jin X-S + iTW, or
X - s\n\k) = s
X-{K-S)
tanx = /f
x = tan\k)^s
+
Inn
x-s + nn
Solve each of the following equations using exact values. Solve if necessary, and give the smallest positive solution between 0 tothe In . Write general solution to each equation, using the chart above.
a. 2cosx = ->/3
d. -2sinx = 1
b. tanx = 1
e.
2cosx=V2
c.
-tanx=>/3
f.
-2sinx=-2
The following examples show general solutions to equations involving sine and tangent. When taking the invers
trigonometric function, the inverse function can be found on the calculator, especially when special values are n available. A) Equation:( f x
B) Equation:( f ( x -1))
+ jin
(7t 2Kn ;tn) /\
-1 X
X = -
Use the general solutions calculated for equations A and B in the previous problem to determine specific solutions, rounded to the nearest hundredth. n =1 n-2 n =3 A) 1.35...,
B)
n = l
n = 2
n = 3
1.59...,
2.64...,
In the preceding examples, at what point in the process of solving the equation is the general form (with th first written down?
For each equation, provide the general solution, as well as the first three positive solutions.
A) 18-i-11cos(f x) = 16
B) 12sin(^x) + 5 = 11
For problems 14-17 find all values of x on the interval 0 <x< 10 that solve each equation.
14)
10-12cos(f(x-4)) = 14
15)
2 + 3cos(^x) = 4
16)
20sin2x+3 = 19
17)
4tan(f(x-2)) = 20
Practice Problems Solve each algebraically on the given interval. 1)
0 = l-l-2cosx if 0<x
When a special value (k) of the trigonometric ratios is given, inverse trigonometric functions can be used to
determine a solution to an equation involving sine, cosine, or tangent. To write a generalized solution, sele smallest positive solution between 0 to 2K here denoted by s. Equation
cosx = /c sinx= /f
General Solution
Principal Solution
x = cos~\k) = s
x = ± s + 2jin X-S + iTW, or
X - s\n\k) = s
X-{K-S)
tanx = /f
x = tan\k)^s
+
Inn
x-s + nn
Solve each of the following equations using exact values. Solve if necessary, and give the smallest positive solution between 0 tothe In . Write general solution to each equation, using the chart above.
a. 2cosx = ->/3
d. -2sinx = 1
b. tanx = 1
e.
2cosx=V2
c.
-tanx=>/3
f.
-2sinx=-2
The following examples show general solutions to equations involving sine and tangent. When taking the invers
trigonometric function, the inverse function can be found on the calculator, especially when special values are n available. A) Equation:( f x
B) Equation:( f ( x -1))
+ jin
(7t 2Kn ;tn) /\
-1 X
X = -
Use the general solutions calculated for equations A and B in the previous problem to determine specific solutions, rounded to the nearest hundredth. n =1 n-2 n =3 A) 1.35...,
B)
n = l
n = 2
n = 3
1.59...,
2.64...,
In the preceding examples, at what point in the process of solving the equation is the general form (with th first written down?
For each equation, provide the general solution, as well as the first three positive solutions.
A) 18-i-11cos(f x) = 16
B) 12sin(^x) + 5 = 11
For problems 14-17 find all values of x on the interval 0 <x< 10 that solve each equation.
14)
10-12cos(f(x-4)) = 14
15)
2 + 3cos(^x) = 4
16)
20sin2x+3 = 19
17)
4tan(f(x-2)) = 20
Practice Problems Solve each algebraically on the given interval. 1)
0 = l-l-2cosx if 0<x
