Type A Trigonometric Equations
For all sine equations, use the calculator angle + n.360˚ and 180˚ – calculator angle + n.360˚
For all cos equations, use the calculator angle + n.360˚ and – calculator angle + n.360˚
For all tan equations, use the calculator angle + n.180˚ only. (No second solution)
Note: Always start with the general solution method.
Once you have solved for x and simplified the result, the general solution is complete.
In this example you have been asked to solve for x 135 ;120 .
Determine the specific solution after you have determined the simplified general solution.
Worked Example 1: Solve for x if cos 3 x 45 0, 719 and x 135 ;120
General Solution:
3x 45 135,97 n 360 , n or 3 x 45 135,97 n 360 , n
3x 180,97 n 360 or 3x 90,97 n 360
x 60,32 n 120 x 30,32 n 120
Specific Solution: x 60,32 ; 59, 68 ; 89, 68 , 30,32 (not arranged in any order)
or x = 59, 68 ; 30,32 ; 60,32 or 89, 68 (arranged in ascending order)
Explanation
Starting with n 0, then n 1, n 2 and eventually n 1, n 2 etc. , continue to work out possible
values of x until your answers fall outside the given interval for this equation, namely x 135 ;120 .
A variation of the example above: Solve for x if cos x 45 tan 324, 28 , x 135 ;120
Use a calculator to determine the value of tan 324,28 , which gives cos 3 x 45 0, 719 .
The rest of the solution follows in the same way as in the example above.
Worked Example 2: Solve for x if sin 3 x 45 0, 468 and x 180 ;180
General Solution:
3x 45 27,90... n 360 , n or 3 x 45 180 27,90... n 360 , n
3x 17, 095... n 360 or 3 x 207,90... n 360
x 5, 70 n 120 x 69,30 n 120
Specific Solution: x 5, 70 ; 125,70 ; 114,3 ; 69,30 ; 50,70 ; 170,70 (not arranged in any order)
or x 170, 70 ; 114,3 ; 50, 70 ; 5, 70 ; 69,30 ;125,70 (arranged in ascending order)
Worked Example 3: Solve for x if tan 2x 35 0,38 and x 180 ;180
General Solution:
2 x 35 20,806... n 180 , n
2 x 55,806... n 180
x 27,90 n 90
Specific Solution: x 27,90 ; 62,10 ; 152,10 ; 117,90 (not arranged in any order)
or x 117,90 ; 27,90 ; 62,10 ; 152,10 (arranged in ascending order)
For all sine equations, use the calculator angle + n.360˚ and 180˚ – calculator angle + n.360˚
For all cos equations, use the calculator angle + n.360˚ and – calculator angle + n.360˚
For all tan equations, use the calculator angle + n.180˚ only. (No second solution)
Note: Always start with the general solution method.
Once you have solved for x and simplified the result, the general solution is complete.
In this example you have been asked to solve for x 135 ;120 .
Determine the specific solution after you have determined the simplified general solution.
Worked Example 1: Solve for x if cos 3 x 45 0, 719 and x 135 ;120
General Solution:
3x 45 135,97 n 360 , n or 3 x 45 135,97 n 360 , n
3x 180,97 n 360 or 3x 90,97 n 360
x 60,32 n 120 x 30,32 n 120
Specific Solution: x 60,32 ; 59, 68 ; 89, 68 , 30,32 (not arranged in any order)
or x = 59, 68 ; 30,32 ; 60,32 or 89, 68 (arranged in ascending order)
Explanation
Starting with n 0, then n 1, n 2 and eventually n 1, n 2 etc. , continue to work out possible
values of x until your answers fall outside the given interval for this equation, namely x 135 ;120 .
A variation of the example above: Solve for x if cos x 45 tan 324, 28 , x 135 ;120
Use a calculator to determine the value of tan 324,28 , which gives cos 3 x 45 0, 719 .
The rest of the solution follows in the same way as in the example above.
Worked Example 2: Solve for x if sin 3 x 45 0, 468 and x 180 ;180
General Solution:
3x 45 27,90... n 360 , n or 3 x 45 180 27,90... n 360 , n
3x 17, 095... n 360 or 3 x 207,90... n 360
x 5, 70 n 120 x 69,30 n 120
Specific Solution: x 5, 70 ; 125,70 ; 114,3 ; 69,30 ; 50,70 ; 170,70 (not arranged in any order)
or x 170, 70 ; 114,3 ; 50, 70 ; 5, 70 ; 69,30 ;125,70 (arranged in ascending order)
Worked Example 3: Solve for x if tan 2x 35 0,38 and x 180 ;180
General Solution:
2 x 35 20,806... n 180 , n
2 x 55,806... n 180
x 27,90 n 90
Specific Solution: x 27,90 ; 62,10 ; 152,10 ; 117,90 (not arranged in any order)
or x 117,90 ; 27,90 ; 62,10 ; 152,10 (arranged in ascending order)
