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Intermediate Trigonometry
Theory Refresher
Click here for "Live" theory.
Radian Measure
In the Cartesian plane, angles are drawn in an anti-clockwise direction
starting from the positive x-axis (which is taken to be 0o).
Angles are identified with the arc they cut on the circle of radius 1 (the unit circle).
The length of this arc is known as the radian measure of
the angle.
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The key:
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(circumference of the unit circle)
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or
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To convert from
radians to degrees:
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Divide by  , multiply by 180o.
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To convert from
degrees to radians:
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Divide by 180o, multiply by .
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"Easy" values to remember:
Trig ratios of ANY Angle
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For the angle t in the diagram, (x,y) are the coordinates of
the point t cuts on the unit circle. From this we get
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These are true for any angle, and are specifically useful for finding trig ratios of 0,
 , and  , the angles that lie on the axes, where the point (x,y) is readily known.
To evaluate trig ratios of an angle, t greater than  rad (90o) and not lying on an axis:
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1.
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Draw the angle t and locate the quadrant of the angle in the Cartesian plane.
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2.
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Determine the algebraic sign of the ratio, depending on the quadrant:
QI: All are positive
QII: Sin is positive, cos and tan are negative
QIII: Tan is positive, sin and cos are negative
QIV: Cos is positive, sin and tan are negative
Remember: "All Stations To Central."
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3.
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Find the reference angle,  , which is the acute angle (less than 90o) that the end of the angle t makes with the closest part of the x-axis.
QI: = t
QII: = - t
QIII: = t -
QIV: = 2 - t
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4.
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The trig ratio of t equals the ratio of with the algebraic sign found in step 2.
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Solving Trig Equations
To evaluate an unknown angle when the trig ratio is known,
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1.
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Using the algebraic sign, determine the two quadrants where the angle could lie.
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2.
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Drop the algebraic sign and find the acute angle with the known trig ratio (use tables or the sin-1, cos-1 or tan-1 button on your calculator; be careful with radians/degrees). This is the reference angle
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3.
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Draw the two possible answers indicating where the reference angle lies.
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4.
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Calculate the two possible answers from your diagram.
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5.
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If the question is asking for all possible solutions between 0 and 2, you have finished.
If the question is asking for all possible solutions, take the two you have found and add 2k, for k = 0,  1, 2,... to each solution.
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Note that these questions always have more than one answer. Your
calculator will only ever give you one answer. You must know how
to use this process to find the other solutions.
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