§5.4 – Evaluating Trig Ratios for Angles 0°-360°
November 11, 2013
Evaluating Trigonometric Ratios for Any Angle Between 0° and 360°
In the previous section, we discovered that for a principal angle greater than 90°, the values of
the primary trigonometric ratios are either equal to, or the negatives of, the corresponding ratios
of the related acute angle.
For a principal angle θ in quadrant 2, the related acute angle is
β = 180° – θ, and
• sin θ = sin β
• cos θ = –cos β
Note: only the sine is positive
• tan θ = –tan β
For a principal angle θ in quadrant 3, the related acute angle is
β = θ – 180°, and
• sin θ = –sin β
• cos θ = –cos β
Note: only the tangent is positive
• tan θ = tan β
For a principal angle θ in quadrant 4, the related acute angle is
β = 360° – θ, and
• sin θ = –sin β
• cos θ = cos β
Note: only the cosine is positive
• tan θ = –tan β
So a procedure to find the value of a trigonometric ratio of any angle θ having a measure
between 0° and 360° would be:
1. Determine the quadrant in which the angle θ lies:
• If 0° < θ < 90°, the angle is in quadrant 1, and the desired ratio can be obtained
immediately using a calculator.
• If 90° < θ < 180°, the angle is in quadrant 2.
• If 180° < θ < 270°, the angle is in quadrant 3.
• If 270° < θ < 360°, the angle is in quadrant 4.
2. For angles in quadrants 2-4, determine the related acute angle β. Using a calculator,
determine what the value of the desired ratio would be for angle β.
3. The desired ratio for angle θ will be the value obtained from step 2, but with its sign
adjusted as follows:
• if θ is in quadrant 2, only the sine ratio is positive; all others are negative
• if θ is in quadrant 3, only the tangent ratio is positive; all others are negative
• if θ is in quadrant 4, only the cosine ratio is positive; all others are negative
MCR3U—S. Inrig
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§5.4 – Evaluating Trig Ratios for Angles 0°-360°
November 11, 2013
Trigonometric Ratios Defined in Terms of a Point on the Cartesian Plane
So far, we have used definitions for trigonometric ratios that were based on right triangles
involving either the principal angle or its related acute angle. However, there is an alternate set
of definitions for these ratios which uses the coordinates of a point on the Cartesian plane.
Consider a point P(x, y) on the Cartesian plane, and a circle which passes through this point and
has its centre at the origin. We know that the radius of this circle, r, is the distance from the
origin to P.
Note the right triangle formed by point P, the
origin (O), and C, the point of intersection of a
vertical line through P with the x-axis. In this
triangle, the angle POC is the related acute
angle for the principal angle θ.
Note also that the length of OC is equal to the
x-coordinate of P, and the length of CP is equal
to the y-coordinate of P. Therefore,
‫ݕ ܲܥ‬
sin ߠ =
=
ܱܲ ‫ݎ‬
ܱ‫ݔ ܥ‬
cos ߠ =
=
ܱܲ ‫ݎ‬
‫ݕ ܲܥ‬
tan ߠ =
=
ܱ‫ݔ ܥ‬
Note that the values of the three ratios depend only on the value of θ; if P moves along the
terminal arm of angle θ, the values of x, y, and r will all change, but they will change
proportionally, so the ratios will remain constant.
It follows that if r is 1 (in which case we would have a unit circle, i.e. one with centre at the
origin and radius 1), the above ratios simplify to:
sin ߠ = ‫ݕ‬
cos ߠ = ‫ݔ‬
௬
tan ߠ = ௫
So for any angle in standard position having a point P(x, y) on its terminal arm, we can determine
all three primary trigonometric ratios directly from the coordinates of P. Note that these
definitions automatically give us the correct signs for each of the ratios; we don’t need to
consider which quadrant the terminal arm is in.
Note that when θ is either 90° or 270°, the value of x will be 0, and since x is the denominator in
the definition of tan θ, the tangent ratio is undefined for angles of 90° and 270°.
MCR3U—S. Inrig
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