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Trig Equations
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Solving equations involving trigonometric functions and understanding solution domains
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Solving sinθ=a
SL 3.8
Since sinθ represents the y-coordinate of a point on the unit circle, solving the equation
sinθ=a
is equivalent to drawing the line y=a, finding the intersections with the unit circle, and determining the angle of these intersections relative to the positive x-axis.
This helps visualize all the possible solutions. For
sinθ=a,0≤θ<2π
the solutions are
θ=sin−1(a),π−sin−1(a)
Solving cosθ=a
SL 3.8
Since cosθ represents the x-coordinate of a point on the unit circle, solving the equation
cosθ=a
is equivalent to drawing the line x=a, finding the intersections with the unit circle, and determining the angle of these intersections relative to the positive x-axis.
This helps visualize all the possible solutions. For
cosθ=a,0≤θ<2π
the solutions are
θ=cos−1(a),2π−cos−1(a)
Solving tan(x)=a
SL 3.8
Since tanθ represents the angle between the line y=xtanθ and the x-axis, solving
tanθ=a
is equivalent to drawing the line y=a, and measuring the minor and major angles it forms with the x-axis:
Solving trig equations algebraically in specific domain
SL 3.8
When we have a trig equation where the argument to the trig function is of the form ax+b, we need to find the domain of ax+b using the domain of x. For example, if 0≤x<2π and we have sin(2x+2π)=1, then
2⋅0+2π≤2x+2π<2⋅2π+2π
therefore
2π≤2x+2π<29π
Trigonometric Quadratics
SL 3.8
Trigonometric functions can also show up in pseudo-quadratics - a quadratic where the variable being squared is not x but a trig function.
On exams, these equations often require using the Pythagorean identity sin2θ+cos2θ=1.
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