1. 2.2 Trig Ratios of Any Angle (x, y, r)
Chapter 1 Trigonometry
2.2 Trig Ratios of Any Angle (x, y, r)
30° - 60° - 90° Triangle
45° - 45° - 90° Triangle
300
450 2 1
600
450 1 1
Pre-Calculus 11

2. Finding the Trig Ratios of an Angle in Standard Position
Suppose angle θ is an angle in standard position.
Choose a point (x, y) on the terminal arm, at a distance r from the origin.
P(x, y) r y θ x
As x and y increase r also increases. If r stays the same x and y are inversely proportional.
r2 = x2 + y2
r = √x2 + y2
x, y, and r represent the sides of a right triangle. For angle θ, x = adjacent, y = opposite and r = hypotenuse
Pre-Calculus 11

3. Finding the Trig Ratios of an Angle in Standard Position
Suppose angle θ is an angle in standard position. How are the ratios affected if we choose the point (-x, y) on the terminal arm?
P(-x, y) r y θ 𝜽R -x
r2 = (-x)2 + y2
The horizontal and vertical lengths are considered as directed distances (so they can be positive or negative depending on which quadrant the terminal arm lies in)
Pre-Calculus 11

4. Pre-Calculus 11

5. Sine All Tangent Cosine ( -, + ) ( +, + ) ( -, - ) ( +, - )
How can you tell if the ratios will be positive or negative?
Recall the CAST Rule
Sine
All
( -, + )
( +, + )
Tangent
Cosine
( -, - )
( +, - )
Pre-Calculus 11

6. Finding Trigonometric Ratios of Angles in Standard Position
Sine is positive.
All are positive.
r is always positive
Tan is positive.
Cos is positive.
Pre-Calculus 11

7. Determine the sign of the ratio. 1. sin 1270 2. tan 240
The Cast Rule
Determine the sign of the ratio.
1. sin 1270     tan 240
3. cos tan 1450
5. cos  970     cos 450
7. sin cos 3150
Positive
Positive
Negative
Negative
Negative
Positive
Negative
Positive
Pre-Calculus 11

8. The Cast Rule I II I IV I III II IV
i ) Sine ratios have positive values in quadrants _____ and _____ .
ii) Cosine ratios have positive values in quadrants _____ and _____ .
iii) Tangent ratios have positive values in quadrants _____ and _____ .
iv) sinθ > 0 and cos θ < ______
v) tan θ < 0 and sin θ < ______ I IV I
III II IV
Pre-Calculus 11

9. Finding the Exact Trig Ratios of an Angle in Standard Position
Given point P(5, 12) on the terminal arm calculate the exact values of the primary trig ratios.
P(5, 12) 13 12 θ 5
Pre-Calculus 11

10. Finding the Trig Ratios of an Angle in Standard Position
The point P(-2, 3) is on the terminal arm of θ .in standard position.
Determine the exact value and approximate value of the trigonometric ratios for angle θ.
Exact Values
P(-2, 3) 3 θ
Approximations 𝜽R -2
r2 = x2 + y2
r2 = (-2)2 + (3)2
r2 = 4 + 9
r2 = 13
r = √ 13
← since we did not have to round the decimal here it is technically an exact value. However, we would usually leave as a radical (a/b)
exact values are fractions: ⅓ is an exact value is not
Exact values are more accurate than approximations. However, they may not be as practical. Ex asking someone to cut a piece of paper at cm may be difficult for that piece to visualize that measurement
Pre-Calculus 11

11. x -5 7 Determine the exact value of the trig ratios given
The terminal arm must be in Quadrant III, since sin𝜽 is negative and tan𝜽 is positive x θ -5 7
Use pythagorean theorem to find x (note x < 0 in QIII)
Pre-Calculus 11

12. 1. sin 250 = 2. cos 1210 = 3. tan 3350 = 4. sin 00 = 5. tan 900 =
Calculator: Determine Approximate Trig Ratios (four decimal places)
1. sin 250 =
2. cos =
3. tan =
4. sin 00 =
5. tan 900 =
Why is tan 90° undefined?
Pre-Calculus 11

13. 2.2 B Trig Ratios of Any Angle
Quadrantal Angles and Solve for the Angle
The reference angles for angles in standard position 150° and 210° are equal. Does this imply that ? ? ?
ref 30°
ref 30°
Q II, sine is +ve
Q III, sine is -ve
How does sin 150° compare to sin 30°?
Pre-Calculus 11

14. Quadrantal Angle
If an angle is standard position with the terminal arm on the x or y axis we call the angle a quadrantal angle. Examples include angles at 0o, 900, 180o, 270o, 360o, 450o, … etc.
You should be able to find the trig functions of quadrantal angles and angles with a reference angle of 30°, 45°, and 60° without using a calculator.
Pre-Calculus 11

15. Values of Trigonometric Functions for Quadrantal Angles

16. Quadrantal Angles Example P(0, 3) Q(-4, 0) r2 = x2 + y2 90° 180° 0°
, 360°
Q(-4, 0)
r2 = x2 + y2
270°
Pre-Calculus 11

17. Determine the Measure of an Angle Given a Trig Ratio
Solve for angle θ given
00 ≤ θ < 3600
Sin𝜽 is positive in both quadrant I and II so we will have 2 possible answers 2 2 1 1 θ 𝜽R θ
We know from the previous chapter that sin 300 = 1/2 , so the reference angle must be 300
Sohcahtoa QI
QII
θR = 300
Reference Angle
θR = 300
Reference Angle
Angle in Standard
Position
Angle in Standard
Position
θ = 300
θ = 1500
Pre-Calculus 11

18. Determine the Measure of an Angle Given a Trig Ratio
00 ≤ θ < 3600
nearest degree
QI or QII
Solve for angle θ given 5 5 3 3 θ 𝜽R θ
When the trig ratio cannot be related to the special triangles or quadrantal angle we need to use the Inverse Trig Function on our calculator to determine the angle.
θR = 370
Reference Angle
θR = 370
Reference Angle
Angle in Standard
Position
Angle in Standard
Position
θ = 370
θ = 1430
Pre-Calculus 11

19. 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 two side lengths (ratio) 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

20. 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 plug 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

21. 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°

22. Determine the Measure of the Angle Given the Exact Ratio
Solve for each angle θ given a specific trig ratio.
00 ≤ θ < 3600 QI
QII QI
QIII
θ = 450,
θ = 300
, 2100
1350
RA = 300
RA = 450 QI
QII QI
QIV
θ = 300,
1500
θ = 600
, 3000
RA = 600
RA = 300
QII
QIII
QII
QIV
θ = 1350
, 2250
θ = 1200
, 3000
RA = 450
RA = 600
QII
QIII
QII
QIII
θ = 1200
, 2400
θ = 1500
, 2100
RA = 600
RA = 300
Pre-Calculus 11

23. Determine the Measure of the Angle Given the Approximate Ratio
Determine the measure of angle A, to the nearest degree:
00 ≤ A < 3600
Enter a positive ratio in your calculator
R A
Quadrants
340 I
340 II
1460
sinA =
410 II
1390
III
2210
cosA =
570 II
1230 IV
3030
tanA =
cosA =
530 I
530 IV
3070
3000
III
2400 IV
sinA =
600
tanA =
310 I
310
III
2110
Pre-Calculus 11

24. Assignment Suggested Questions Page 96: 1-9, 14-16, 19, 26-29
Pre-Calculus 11