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10.4 Solving Trig Equations!
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Use the following guidelines when solving the Trigonometric equation f ( x ) = g ( x ) :!
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a. It may be helpful to draw a quick sketch of y = f ( x ) and y = g ( x ) to see roughly !
where the solutions are.!
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b. If the equation involves functions of 2x and x , transform the functions of 2x into !
functions of x by using identities.!
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c. If the equation involves functions of 2x only, it is usually better to solve for 2x !
directly and then solve for x, old school style. !
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d. Be careful not to loose roots when you divide both sides of an equation by a function !
of the variable.!
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Example 1:!
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Solve cos ( 2x ) = 1− sin x for 0 ≤ x < 2#.!
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Example 2:! Solve 3cos ( 2x ) + cos ( x ) = 2 for 0 ≤ x < 2#.!
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Example 3:
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Solve 2sin ( 2x ) = 1 for 0º ≤ x < 360º .!
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Example 4:!
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Find the smallest positive root of sin ( 2.8x ) = cos x .!
10.4 PRACTICE/Homework: (pg 390)!
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Solve each equation $ 0º ≤ x < 360º by using trig identities, where appropriate. Round to the
nearest tenth.!
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17. 2 cos ( x + 45º ) = 1 !
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19. sin ( 60º −x ) = 2sin ( x ) !
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21. sin ( x ) = sin ( 2x ) !
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Solve each equation $ 0 ≤ x < 2π by using trig identities. Round to the nearest hundredth.!
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23. sin ( x ) cos ( x ) =
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25. tan ( 2x ) = 3tan ( x ) !
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27. cos ( 2x ) = 5sin ( x ) − cos ( x ) !
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29. 3sin ( x ) = 1+ cos ( 2x ) !
31. cos ( 2x ) = sec ( x ) !!
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32. sin ( x ) cos ( 2x ) = 1 !
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Solve the equation graphically for −2π ≤ x < 2π by using the graphing calculator.!
36. sin (1.5x ) = 2 cos ( x ) !
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MORE 10.4 PRACTICE. FINISH FOR HW. Solutions online
Solve for 0 ≤ x < 360°.
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7.
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cos(2x) = 1 – sinx
2. " ( cos x + sin x ) = 2sin(2x)
2sin(3x) = 1
4.
3cos (x+45°) = 1
" 8sin 4 x + 2sin 2 x = 3
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2sin(30° + x) = 3cos(x)
sin x = sin(2x)
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tan x =2sin(2x)
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15.
sin (2x)* sec x + 2 cos x = 0
10.
cos(2x) = –2cos2 x
2sin x*cos(2x) = 0
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sin(2x) = 5 cos2 x
–sinx = cos(2x)
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sinx = 1–cos(2x)
3cos(2x) = cosx–1
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" 1− 2sin 2 x =
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17. Challenge!: "
cos(3x) sin(3x)
+
=2 3
sin x
cos x
3
2