1. 2.1 Angles in Standard Position
Chapter 2 Trigonometry
2.1 Angles in Standard Position
In geometry, an angle is formed by two rays with a common endpoint. In trigonometry, angles may be interpreted as rotations of a ray.
Vertex
Initial Arm
Terminal Arm
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2. Angles in Standard Position
Yes Examples
No Examples
Where would these go? No
Yes No
How would you describe an angle in standard position?
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3. Angles in Standard Position
Chapter
Identify the angles sketched in standard position.
Check answer
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4. Angles in Standard Position
An angle θ is said to be in standard position if its vertex is at the origin of a rectangular coordinate system and its initial arm coincides with the positive x-axis. The rotation of angle θ is measured in degrees.
Initial arm
Vertex
Terminal arm x y
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5. Example Sketch each angle in standard position.
State the quadrant in which the terminal arm lies.
170°
50°
200°
300°
QII QI
QIV
QIII
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6. Reference Angles, 𝜃R
An acute angle formed between the terminal arm and x-axis. This angle is always positive and measures between 0o and 90o
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7. Determining Reference Angles by Quadrant

8. Reflections of Reference Angles
Determine the Angle in Standard Position w/ a Reference angle of 30°
300
300
1500
300
Angle in Standard Position_______
Angle in Standard Position_______
300
300
Angle in Standard Position_______
2100
3300
Angle in Standard Position_______
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9. Reflections of Reference Angles
Determine the Angle in Standard Position w/ a Reference angle of 60°
600
600
600
1200
Angle in Standard Position _______
Angle in Standard Position _______
600
600
3000
Angle in Standard Position _______
2400
Angle in Standard Position _______
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10. Reflections of Reference Angles
Determine the Angle in Standard Position w/ a Reference angle of 45°
45°
45°
135°
45°
Angle in Standard Position _______
Angle in Standard Position _______
45°
45°
315°
225°
Angle in Standard Position _______
Angle in Standard Position _______
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11. Sketching Angles in Standard Position
Sketch the following angles in standard position. State the size of the reference angle.
In which quadrant does the terminal arm lie?
a) 1500
b) 2900
c) 2150
Quadrant II
Quadrant IV
III
Quadrant
Reference Angle
300
Reference Angle
700
Reference Angle
350
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12. Drawing Angles in Standard Position Given a Point on the Terminal Arm
a) Draw an angle, θ, in standard position such that the point P(-4, 3) lies on the terminal arm of an angle θ
P(-4, 3) 3 θR θ -4
b) Suppose the point P(-4, 3) was reflected about the y - axis
P(-4, 3) θ
Q(4, 3) θR θR
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13. Drawing Angles in Standard Position Given a Point on the Terminal Arm
c) Suppose the point P(-4, 3) was reflected about the x-axis
P(-4, 3) θ θR θR
R(-4, -3)
d) Suppose the point P(-4, 3) was reflected about the x-axis and y-axis
P(-4, 3) θ θR θR
S(4, -3)
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14. Torso Angle - Fast Torso Angle - Touring
Torso angle is very dependent upon the cyclists choice of performance and comfort. A lower position is more aerodynamic as frontal surface area is reduced. 30° to 40° is a good compromise of performance and comfort but does rely on reasonably good flexibility to lower back and hamstrings.
Torso Angle - Touring
A more relaxed torso angle will take the pressure off the lower back, hamstrings and the neck and distribute loads from hands to seat. 40° to 50° is a suitable angle for longer distances where comfort is the priority over speed.
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15. Reference Angles Determine the measure of the reference angle.
Angle in Standard Position (θ)
Quadrant
Reference Angle (θR)
165°
320°
250°
60° II
15° IV
40°
III
70° I
60°
Reference Angle (θR)
Quadrant
Angle in Standard Position (θ)
85°
III
46° I
37° IV
52° II
Determine the measure of the angle in standard position.
265°
46°
323°
128°
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16. Chapter 1 Sequences and Series
2.1 B Angles in Standard Position - Exact Values
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17. The Primary Trigonometric Ratios
Trigonometry compares the ratios of the sides in a right triangle.
The Primary Trigonometric Ratios
There are three primary trig ratios:
sine cosine tangent
Opposite the right-angle
hypotenuse
Opposite the angle.
opposite
adjacent
Next to the angle
30° 1 2
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18. Trig Equations
sin 30º=
trig
function
angle
trig ratio
Knowing the measure of the reference angle, can you label the triangle?
300 2 1
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19. Recall the Special Right Triangles and Pythagorean Theorem
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20. Exact Values for Trig Ratios of Special Right Angles
Example
For the following triangle to determine the three trig ratios for 45°
Use what you know about the ratio of sides in a triangle or pythagorean theorem to determine the value of c.
Example
For the following triangle to determine the three trig ratios for 30° and 60°

21. Example Exact Value - Approximation - Exact Values of Trig Ratios
Exact Value Vs Decimal Approximation
Example
Exact Value -
Approximation -
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22. Comparing Trigonometric Ratios using Common angles
Quadrant
Sin
Cos
Tan
30°
150°
210°
330°
Note: I II
III IV
What do the angles have in common?
The reference angle is θR = 30°
What do you notice about the ratios of the lengths of sides?
They all have the same reference angle
For each trig function the ratio is the same only the sign changes
Q I: all +ve, QII sin +ve, QIII tan +ve, QIV cos +ve
The side length have the same ratios as a triangle,
Make a conjecture to determine the sign of the trig ratio for each quadrant
QI: all trig ratios are positive
QII: only sin is positive,
QIII: only tan is positive
QIV: only cos is positive
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23. Comparing Trigonometric Ratios using Common angles
Quadrant
Sin
Cos
Tan
60°
120°
240°
300°
θR = 60° I II
III IV
Angle
Quadrant
Sin
Cos
Tan
45°
135°
225°
315°
θR = 45° I
Note: II
III IV
How can we use the quadrant and trig function to determine the sign of the trig ratio?
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24. The Cast Rule
The CAST diagram helps us to see which quadrants the trig ratios are positive

25. McGraw-Hill Ryerson Precalculus 11 Page 82 Example 4
Calculate the horizontal distance to the midline, labeled a. a
Which trig ratio would you use to determine the length of side a?
The exact horizontal
distance is 10
10 cm.
60° a
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26. Using Exact Values Homework
2. cos 450 =
3. tan 450 =
4. sin 600 =
5. sin 1500 =
6. cos 1200 =
RA = 300
RA = 600
7. tan 1350 =
RA = reference angle
8. tan 1200 =
RA = 450
RA = 600
9. sin 1350 =
10. cos 1500 =
RA = 450
RA = 300
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27. Inverse Trig Functions
The inverse trigonometric functions sin-1(x), cos-1(x), and tan-1(x), are used to find the unknown measure of an angle of a right triangle when the side lengths are known
Note the “-1” in this function is not an exponent. It is simply a notation that we use to tell us we are dealing with an inverse trig function.
If we wanted a function to denote 1 over cosine we would use:
We can also use the notation arcsin(x), arccos(x), and arctan(x) to denote an inverse trig function

28. Inverse Trig Functions
To evaluate inverse trig remember that the following statements are equivalent:
So, when we evaluate an inverse trig function we are asking “what angle, 𝜽, do we plug into the original trig function to get x ? “
Example
In other words, what angle, 𝜽, do we need to input into cosine to get ?
We can use our knowledge of special triangle or a calculator to solve this problem
When using the inverse function on your calcular the output value (answer) is treated as the reference angle. This is because there is 2 possible answers, the calculator can only give us 1.
Cosine is positive in quadrant I and IV so,
and

29. Inverse Trig Functions
Example
Given , determine the measure of x, to the nearest tenth of a degree
When evaluating a trig function in which the ratio is negative we will ignore the negative sign when determining the reference angle. The negative value will be accounted for when we determine the angle in standard position.
← use only positive numbers in this formula
Note: ≈ means “approximately”
Cosine is negative in quadrants II and III
In Quadrant II:
In Quadrant III:
Therefor, x = 169.9° and
x = 190.1°

30. Assignment Suggested Questions Page 83:
1-7, 9, 10, 13, 14, 16, 18, 23, 24
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