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Trig equations & identities
Track your progress across all skills in your objective. Mark your confidence level and identify areas to focus on.
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The key idea with triangles who's hypotenuse lie on the unit circle (that form an angle of θ with the x-axis) is that cosθ represents length of the base, and sinθ represents the height.
Take a look at the graph below and notice the following relationships always hold:
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The following table shows the values of sinθ and cosθ for the so called critical angles θ. These are angles that give "nice" values for sin and cos:
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The unit circle can be divided into quadrants based on the sign of cosθ and sinθ. These correspond to the 4 quadrants produced by the intersection of the x and y axes. The quadrants are denoted Q1, Q2, Q3 and Q4.
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Since a full circle is 2π radians, adding 2π to any angle θ gives the same point on the unit circle. In fact, adding any integer multiple of 2π gives the same point:
Once the values of sin and cos are memorized for critical angles in the first quadrant, you can use the rules of symmetry and periodicity to determine the values of critical angles in arbitrary quadrants.
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Notice that both sinx and cosx have a domain of x∈R and a range of (−1,1).
A sinusoidal function is a generalization of sin and cos to the form
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or
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The tan function is defined by tanx=cosxsinx.
The domain is thus x=22k+1π (there are vertical asymptotes at those x′s), and the range is all real numbers R.
The function has roots at x=0,±π,±2π… (ie x=kπ where k∈Z)
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The reciprocal trig functions are functions of the form f(x)1, where f is sin,cos or tan. The three reciprocal trig functions are:
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Domain: x=2(2k+1)π=±2π,±23π…
Range: secx≤−1 or secx≥1.
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Domain: x=kπ=0,±π,±2π…
Range: cosecx≤−1 or cosecx≥1.
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Domain: x=kπ=0,±π,±2π…
Range: cotx∈R.
Since sinθ represents the y-coordinate of a point on the unit circle, solving the equation
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
the solutions are
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Since cosθ represents the x-coordinate of a point on the unit circle, solving the equation
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
the solutions are
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Since tanθ represents the angle between the line y=xtanθ and the x-axis, solving
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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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
therefore
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.
The inverse of sin is arcsin, also written sin−1. Its domain is the range of sin: (−1,1), and its range is the domain of sinx, restricted so that the inverse function passes the vertical line test: [−2π,2π].
The graph of y=arcsinx is the mirror image of sinx (restricted to −2π<x<2π) in the line y=x, giving an increasing function:
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The inverse of cos is arccos, also written cos−1. Its domain is the range of cos: (−1,1), and its range is the domain of cosx, restricted so that the inverse function passes the vertical line test: [0,π].
The graph of y=arccosx is the mirror image of cosx (restricted to 0<x<π) in the line y=x, giving a decreasing function:
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The inverse of tan is arctan, also written tan−1. Its domain is the range of tan: all real numbers, and its range is the domain of tanx, restricted so that the inverse function passes the vertical line test: (−2π,2π).
The graph of y=arctanx is the mirror image of sinx (restricted to −2π<x<2π) in the line y=x, giving an increasing function:
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The double angle identity for sin states that
The double angle identity for cosine states that
These three different forms come from leveraging sin2θ+cos2θ=1.
If we take the identity sin2θ+cos2θ=1 and divide through by cos2θ we find
If we take the identity sin2θ+cos2θ=1 and divide through by sin2θ we find
The double angle identity for tan states that
The compound angle identity for tan states that