1. 4.3A Trigonometric Ratios
Point P(x, y) is the point on the terminal arm of angle  ,an angle in standard position, that intersects a circle.
P(x, y)
r2 = x2 + y2 r
r = √x2 + y2 y q x
The three reciprocal ratios are defined as follows:
Math 30-1

2. Finding the Trig Ratios of an Angle in Standard Position
The point P(-2, 3) is on the terminal arm of q in standard position.
Does point P(-2, 3) lie on the unit circle?
No, the radius of a unit circle is 1.
P(-2, 3)
Determine the exact value of the six trigonometric ratios for angle q. 3 q -2
r2 = x2 + y2
r2 = (-2)2 + (3)2
r2 = 4 + 9
r2 = 13
r = √ 13
Math 30-1

3. Review of Exact Trig Ratios
is a point on the terminal arm of angle θ that intersects the unit circle
Math 30-1

4. The point lies at the intersection of the unit circle
and the terminal arm of an angle θ in standard position.
Diagram q
r = 1
Math 30-1

5. The Unit Circle
Exact Values for the trigonometric ratios can be determined using multiples of special angles for points P(θ) on the unit circle.
(1, 0)
(0, 1)
(-1, 0)
(0, -1)
Math 30-1

6. Exact Values For Trigonometric Ratios
1. sin =
2. cos
RA = 300
quadrant IV
RA =
quadrant II
3. tan
4. cos =
RA =
quadrant IV
RA =
quadrant III
Math 30-1

7. Determine the exact value of:
Using the unit circle
Determine the exact value of:
Math 30-1

8. Approximate Values for Trig Ratios (four decimal places)
The mode of calculator must match the domain of the angle, degrees or radians.
1. sin 250 =
2. cos = 3.
4. csc = 5.
6. cot 2700=
Math 30-1

9. 4.3B Trigonometric Ratios
Does point P lie on the unit circle?
Yes
If Point P is the point on the terminal arm of angle  that intersects the unit circle, in which quadrant does P lie?
III
Determine the exact values of the 6 trigonometric ratios.
Math 30-1

10. x -5 7 Determine the exact value of the trig ratios given
Must be in Quad IV x q -5 7
Math 30-1

11. Using Technology sin 30º= Given sin 30º Given sin θº = ½
trig
function
angle
trig ratio
Given sin 30º
Given sin θº = ½
Asking for the sine ratio value
when angle θ is 30°
Asking for the value of angle θ
when the sine ratio is ½
Enter sin 30°, the answer is a ratio
Use the inverse of the
sine ratio which gives the angle.
Enter sin-1 (½), the answer is an angle.
When a positive ratio is used, the calculator will display the reference angle.
The mode of calculator must match the domain of the angle, degrees or radians.
Math 30-1

12. Deteriming the Measure of an Angle Given the Ratio
Determine the measure of θ.
0 ≤ θ < 3600
cos θ =
trig
function
angle
trig ratio
The trig ratio is negative, indicating that the x-coordinate is negative, therefore, the angle θ would be found in Quadrants II or III.
The trig ratio is not an exact value:
use inverse cosine.
Domain is in degrees.
The reference angle is 480.
θ = 1320 or 2280
nSolve(cos(x)=0.6691,x)
Alternate Method
Math 30-1

13. The point lies at the intersection of the unit circle
and the terminal arm of an angle θ in standard position.
Determine the measure of angle θ for each domain.
Degrees
Radians
Point A is in quadrant IV
Point A is in quadrant IV
Reference
Reference
Math 30-1

14. Determine Angle q , Given an Exact Trigonometric Ratio
Determine the value of angle q.
0 ≤ q < 2p
y is positive in
Q1 or Q2
RA =
x is negative in
Q2 or Q3
y/x is negative in
Q2 or Q4
RA =
RA =
Math 30-1

15. Find the Measure of an Angle q , Given an Exact Trigonometric Ratio
0 ≤ q < 2p
x is neg in
Q2 & Q3
y/x is pos in
Q1 & Q3
RA =
RA =
y is pos in
Q1 & Q2
Quadrantal
x value is 0
RA =
Math 30-1

16. What is the relationship between the angles?
Jackie stated that
Is she correct?
What is the relationship between the angles?
Complete each equality.
Math 30-1

17. Assignment
Page 201
1, 2, 3, 5a,b, 6, 7, 8, 10, 11a,c, 12, 13, 14, 16, 19
Math 30-1