Trig Equations

Solving equations involving trigonometric functions and understanding solution domains

Want a deeper conceptual understanding? Try our interactive lesson! (Plus Only)

Exercises

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Key Skills

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)

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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)

Powered by Desmos

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)

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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:

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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.