Evaluating Trig Ratios for Any Angle Between 0 and 360
April 22, 2015
What's Going On?
CAST
Angle of Mine
Working Backwards
Learning Goal ­ I will be able to evaluate trigonometric ratios
for angles between 0 o and 360 o.
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Evaluating Trig Ratios for Any Angle Between 0 and 360
April 22, 2015
LGL
a. Sketch the given angle in standard position.
b. Use the sketch to determine the exact value of the given trigonometric ratio.
c. State all values of θ that have the same given trigonometric ratio.
tan 300o
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Evaluating Trig Ratios for Any Angle Between 0 and 360
April 22, 2015
LGL
a. Sketch the given angle in standard position.
tan 300o
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Evaluating Trig Ratios for Any Angle Between 0 and 360
April 22, 2015
LGL
b. Use the sketch to determine the exact value of the given trigonometric ratio.
tan 300o
4
Evaluating Trig Ratios for Any Angle Between 0 and 360
April 22, 2015
LGL
c. State all values of θ that have the same given trigonometric ratio.
tan 300o
5
Evaluating Trig Ratios for Any Angle Between 0 and 360
April 22, 2015
The CAST Rule
Use the CAST Rule to determine the sign of each ratio.
sin (150o)
cos (222o)
sec (315o)
cot (95o)
tan (75o)
csc (200o)
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Evaluating Trig Ratios for Any Angle Between 0 and 360
April 22, 2015
Angle of Mine
1) Sketch a circle with its centre at the origin that goes through the point P(3, 4).
a) Determine the radius of the circle
b) Determine the primary trig ratios for the principal angle.
c) Determine the principal angle to the nearest degree.
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Evaluating Trig Ratios for Any Angle Between 0 and 360
April 22, 2015
Angle of Mine
b) Determine the primary trig ratios for the principal angle.
c) Determine the principal angle to the nearest degree.
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Evaluating Trig Ratios for Any Angle Between 0 and 360
April 22, 2015
Angle of Mine
c) Determine the principal angle to the nearest degree.
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Evaluating Trig Ratios for Any Angle Between 0 and 360
April 22, 2015
For any point P(x, y) in the Cartesian plane, the trigonometric ratios for angles in standard position can be expressed in terms of x, y and r.
y
x
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Evaluating Trig Ratios for Any Angle Between 0 and 360
April 22, 2015
Quadrant II
y r
y
x
x is negative
x r
y x
y is positive
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Evaluating Trig Ratios for Any Angle Between 0 and 360
April 22, 2015
Quadrant III
y r
y
x
x r
y x
x is negative
y is negative
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Evaluating Trig Ratios for Any Angle Between 0 and 360
April 22, 2015
Quadrant IV
y r
y
x
x is positive
x r
y x
y is negative
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Evaluating Trig Ratios for Any Angle Between 0 and 360
April 22, 2015
2) Now, choose the point P(‐3, 4) on the circumference of the circle.
a) Determine the primary trig ratios for the principal angle.
b) Determine the principal angle to the nearest degree.
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Evaluating Trig Ratios for Any Angle Between 0 and 360
April 22, 2015
2) Now, choose the point P(3, ‐4) on the circumference of the circle.
a) Determine the primary trig ratios for the principal angle.
b) Determine the principal angle to the nearest degree.
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Evaluating Trig Ratios for Any Angle Between 0 and 360
April 22, 2015
Example 1: Sketch a circle with its centre at the origin that goes through the point P(0, 1).
Determine the radius of the circle.
Determine the primary trig ratios for the principal angle.
Determine the principal angle to the nearest degree.
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Evaluating Trig Ratios for Any Angle Between 0 and 360
April 22, 2015
Working Backwards
Example 2: Determine the values of θ if
and 0o ≤ θ ≤ 360o.
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Evaluating Trig Ratios for Any Angle Between 0 and 360
April 22, 2015
Working Backwards
Example 2: Determine the values of θ if
and 0o ≤ θ ≤ 360o.
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