Ch 14 Oblique Triangles and Vectors

1
Ch 14 Oblique Triangles and Vectors

• Section 1 Trigonometric Functions of Any Angle

2
Angles - Terminology
Terminal Side

• Initial Side

3
Angles Coordinate System
Standard Position If initial side begins on the
positive x-axis.
First Quadrant Angle
Second Quadrant Angle
etc
4
Angles Direction of Rotation

• Positive Counterclockwise rotation from the
  initial side.
• Negative Clockwise rotation from the initial
  side.

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Practice

• Draw these angles in standard position.

35? -160? 400? 245? -10?
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Point on the Terminal Side
(x, y)
?
r
?
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Practice

• Determine the values of the trig functions of an
  angle with a terminal side passing through the
  point (3,5).

(3, 5)
?
r
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Practice

• Determine the values of the trig functions of an
  angle with a terminal side passing through the
  point (-2,4).

(-2, 4)
?
r
?
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Determine the Signs of the Trig Functions in
Quadrants I - IV
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Determine the Signs of the Trig Functions

• Quadrant I
• sine ?
• cosine ?
• tangent ?

(x, y)
?
y
r
r
x
r
?
y
x
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Determine the Signs of the Trig Functions
(-x, y)
?

• Quadrant II
• sine ?
• cosine ?
• tangent ?

r
?
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Determine the Signs of the Trig Functions

• Quadrant III
• sine ?
• cosine ?
• tangent ?

?
r
?
(-x, -y)
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Determine the Signs of the Trig Functions

• Quadrant IV
• sine ?
• cosine ?
• tangent ?

?
r
?
(x, -y)
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How to Remember

• In which quadrant(s) is a particular trig
  function positive?
• All Students Take Calculus

All
Sine
Tangent
Cosine
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Reference Angles Quadrant I

• Reference Angle an acute angle drawn from the
  x-axis to the terminal side.

Quadrant I
45?
Quadrant I ? ref. angle
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Reference Angles Quadrant II

• Reference Angle an acute angle drawn from the
  x-axis to the terminal side.

Quadrant II
? 150?
30?
Quadrant II ? 180? - ref. angle
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Reference Angles Quadrant III

• Reference Angle an acute angle drawn from the
  x-axis to the terminal side.

? 220?
40?
Quadrant III
Quadrant III ? 180? ref. angle
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Reference Angles Quadrant IV

• Reference Angle an acute angle drawn from the
  x-axis to the terminal side.

320?
40?
Quadrant IV
Quadrant IV ? 360? - ref. angle
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Using Reference Angles

• An angle lives in Quadrant III. Its reference
  angle is 20?. What is the measure of the angle?
• An angle is known to be in Quadrant IV. Its
  reference angle is 5 ?. What is the measure of
  the angle?

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Using Reference Angles

• An angle lives in Quadrant II. Its reference
  angle is 41?. What is the measure of the angle?
• An angle of 132 ? has a reference angle of _____.
• An angle of 225 ? has a reference angle of _____.

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Use Reference Angles to Determine Possible Value
for ?
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Using the Reference Angles 1 of 4

• Given that sin ? 0.342, find ? for all angles
  from 0? to 360?.

Sine is positive in Quadrants I and II. That
means we have to find two angles
sin-1 (0.342) 19.998772 20.0? (nearest tenth)
The two angles are 20 degrees and 160
degrees sin 160? 0.342 and sin 20 ?
0.342
160?
20?
20?
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Using the Reference Angles 2 of 4

• Given that cos ? 0.4003, find ? for all angles
  from 0? to 360?.

Cosine is positive in Quadrants ___ and ___.
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Using the Reference Angles 3 of 4

• Given that tan ? -1.830, find ? for all angles
  from 0? to 360?.

Tangent is negative in Quadrants ___ and ___.
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Using the Reference Angles 4 of 4

• Given that sin ? -0.4384, and cos ? gt 0 find ?
  for all angles from 0? to 360?.

Sine is negative in Quadrants ___ and ___. and
Cosine is positive in Quadrants ___ and ___.
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Practice

• MathXL Ch 14 Section 1 Homework