1. 5.3 and 5.4 Evaluating Trig Ratios for Angles between 0 and 360
What has to be broken before it can be used?

2. 5.3 and 5.4 Understanding Angles Greater thany
Terminal Arm θ x
Initial Arm
We're going to explore how triangles in a Cartesian plane have trig ratios that relate to each other

3. Angles, Angles, Angles
An angle is formed when a ray is rotated about a fixed point called the vertex
Terminal Arm (the part that is rotated)
Vertex θ
Initial Arm (does not move)

4. Think of Hollywood
Terminal Arm
Depending on how hard the director wants to snap the device, he/she will vary the angle between the initial arm and the terminal arm
Initial Arm

5. The trigonometric ratios have been defined in terms of sides and acute angles of right triangles.
Trigonometric ratios can also be defined for angles in standard position on a coordinate grid. x y
Coordinate grid

6. Standard Position
An angle is in standard position if the vertex of the angle is at the origin and the initial arm lies along the positive x-axis. The terminal arm can be anywhere on the arc of rotation y
Greek Letters such as α,β,γ,δ,θ (alpha, beta, gamma, delta, theta) are often used to define angles!
Terminal Arm θ x
Initial Arm

7. For example…… Standard Form Not Standard Form Terminal Arm
Initial arm
Initial arm
Not Standard Form
Terminal Arm
Initial arm

8. Positive and Negative angles
Positive angles
Negative angles θ θ
A positive angle is formed by a counterclockwise rotation of the terminal arm
A negative angle is formed by a clockwise rotation of the terminal arm

9. The Four Quadrants 0º< θ < 90º 90º < θ < 180º
The x-y plane is divided into four quadrants. If angle θ is a positive angle, then the terminal arm lies in which quadrant?
Quadrant II
Quadrant I
0º< θ < 90º
Quadrant III
Quadrant IV
90º < θ < 180º
180º < θ < 270º
270 º < θ < 360º

10. Principal Angle and Related Acute Angle
The principal angle is the angle between the initial arm and the terminal arm of an angle in standard position. Its angle is between 0º and 360º
The related acute angle is the acute angle between the terminal arm of an angle in standard position (when in quadrants 2, 3, or 4). and the x-axis. The related acute angle is always positive and is between 0º and 90º
Terminal Arm θ
Principal Angle β
Related Acute Angle
Initial Arm
Let’s look at a few examples……

11. No related acute angle because the principal angle is in quadrant 1
In these examples, θ represents the principal angle and β represents the related acute angle θ θ β
Principal angle: 65º
Principal angle: 140 º
Related acute angle: 40º
No related acute angle because the principal angle is in quadrant 1 θ θ β
Principal angle: 225º
Related acute angle: 45º β
Principal angle: 320º
Related acute angle: 40º

12. Notice anything?
In the first quadrant the principal angle and related acute angle are always the same
In the second quadrant we get the principal angle by taking (180º - related acute angle)
In the third quadrant we can get the principal angle by taking (180º + related acute angle)
In the fourth quadrant we can get the principal angle by taking (360º - related acute angle)

13. Let’s work with some numbers!
Angles
Quadrant
Sine
Ratio
Cosine
Tangent
Principal angle
60º 1
0.8660
0.5
1.7320
Related acute angle (none)
Principal angle
135º 2
0.7071 -1
Related acute angle 45º
0.7071
0.7071 1
Principal angle
220º 3
0.8391
Related acute angle 40º
0.6427
0.7760
0.8391
Principal angle
300º 4
0.5
Related acute angle 60º
0.5
0.8660
1.7320

14. Quadrant 1
Sinθ is positive
Cosθ is positive
Tanθ is positive θ

15. Quadrant 2 Sinθ = sin (180° - θ) -Cosθ = cos (180° - θ)
-Tanθ = tan (180° - θ)
(180° - θ) θ

16. Quadrant 3 -Sinθ = sin (180° + θ) -Cosθ = cos (180° + θ)
Tanθ = tan (180° + θ)
(180° + θ) θ

17. Quadrant 4 -Sinθ = sin (360° - θ) Cosθ = cos (360° - θ)
-Tanθ = tan (360° - θ)
(360° - θ) θ

18. S A T C Only sine is positive All ratios are positive
Summary
Only sine is positive
All ratios are positive S A
Only tangent is positive
Only cosine is positive T C

19. Summary
For any principal angle greater than 90 , the values of the primary trig ratios are either the same as, or the negatives of, the ratios for the related acute angle
When solving for angles greater than 90 , the related acute angle is used to find the related trigonometric ratio. The CAST rule is used to determine the sign of the ratio
CAST Rule
Quadrant II
Quadrant I
Sine
All q q q q
Tangent
Cosine
Quadrant III
Quadrant IV

20. Example1.
Point P(-3,4) is on the terminal arm of an angle in standard position.
a)Sketch the principal angle θ
b) Determine the value of the related acute angle to the nearest degree
c) What is the measure of θ to the nearest degree?

21. Solution
a) Point P(-3,4) is in quadrant 2, so the principal angle θ terminates in quadrant 2.
b) The related acute angle β can be used as part of a right triangle with sides of 3 and 4. We can figure out β using SOHCAHTOA.
P(-3,4) θ -3 4 β
Note…..Whenever we make a triangle such as the one above there is something important to remember…
THE HYPOTENUSE will always be expressed as a positive value, regardless of the quadrant in which it occurs!! Lets look at an example….

22. Example 2
Point (3,-4) is on the terminal arm of an angle in standard position
What are the values of the primary trigonometric functions?
What is the measure of the principal angle θ to the nearest degree?

23. Solution
Assuming that you can draw a circle around the x-y axis, with your point lying somewhere on the perimeter, then it would follow that the hypotenuse of our right angled triangle would be the same as the radius of the circle. 3 -4
r =5 θ
Using pythagorean theorem, we find that r = 5 (note it is positive regardless of the quadrant. Using these values, then
To evaluate B, select cosine and solve for B. Using cos B gives us
……….

24. From the sketch, clearly θ is not 53°
From the sketch, clearly θ is not 53°. This angle is the related acute angle. In this case θ = 360°-53° = 307°
Just as a side note….once again notice that if you take the cos of 307° you get and if you take the cos of 53° you also get

25. Ok, hmk
Wait for it, wait for it…
Well Ross, what is it?
WOOOOW!