1. Trig Ratios of Any Angles
Review

2. Review of Understanding Angles
In grade 11 you learned how to find the trigonometric ratios of any angle
Before we can do this we must first define some key features of angles x y
Terminal Arm θ
Initial Arm
Initial Arm: the ray that defines the beginning of the angle.
Standard Position: when the initial arm lies on the positive x-axis and the vertex of the angle is at the origin (0,0).
Terminal Arm: the ray that defines the end of the angle

3. Review of Understanding Angles
Angles can either be positive or negative
If the terminal arm rotates Counter clockwise = POSTIVE, Clockwise = NEGATIVE
NEGATIVE ANGLE θ
POSITIVE ANGLE θ
π/2 rad
90o
Terminal Arm
π/4 rad
π rad
180o θ
0 rad 0o
Terminal Arm
Initial Arm θ
-3π/4 rad
3π/2 rad
270o

4. Review of Understanding Angles
To understand angles we also need to know the terms: Principal Angle and Acute Angle x y
θ=7π/4
Principal Angle: the angle between 0° and 360°
Related Acute Angle: the angle formed between the terminal arm and the x-axis, and has a measure of between 0° and 90°
θ=π/4

5. Review of Understanding Angles
Finally, let us review the trigonometric ratios:

6. Review of Understanding Angles
Let us use trigonometry to calculate angles in standard position
Find the value of angle θ in radians x y
Label the triangle using positive values for x and y: x=4 and y=3 and label the hypotenuse as r
Find the principal angle θ:
θ = 2π –
θ = 5.64 rads
Solve for β:
tan β=3/4
β = tan-1(3/4)
β = rads
Since you have x and y you must use the tangent ratio:
tan β = y/x θ
Create an acute right triangle at x=3 4 β -3 r
(4, -3)

7. Review of Understanding Angles
Try this example:
Find the primary trig ratios and the value of angle θ in radians
ANGLE
sin β = y/r
sin β = 2/(2√5)=1/√5
β = sin-1(1/√5)
β = rad OR 26.57o
.: θ = π – = 2.678rad OR
= 1800 – 26.57o 153.4o
RATIO
sinθ= 2/(2√5)=1/ √5
ANGLE
cos β = x/r
Use positive values for x, y, and r when finding the acute angle
cos β = 4/(2√5)=2/√5
β = rad OR 26.57o
.: θ = π – = 2.678rad OR
= 180o – 26.57o =153.4o
RATIO
cosθ= -4/(2√5)=-2/√5
r2 = x2 + y2
r2 = (-4)2 + (2)2
r2 = (16) + (4)
r2 = 20
r = 2√5
r ≈ 4.47
ANGLE
tan β = y/x
Use positive values for x, y, and r when finding the acute angle
tan β = 2/(4)=1/2
β = rad OR 26.57o
.: θ = π – = 2.678rad OR
= 180o – 26.57o =153.4o
RATIO
tanθ= 2/(-4)=-1/2
To find the trig ratios find the value of r x y
(-4, 2) θ
r=2√5 r 2 β -4

8. Review of Understanding Angles
To summarize: x y
In relation to your diagram, if the angle IS in the SECOND QUADRANT:
Your angle is (π – acute angle) and the SINE ratio is only POSITIVE ratio.
When finding the acute angle use the positive values for x and y
In relation to your diagram, if the angle IS in the FIRST QUADRANT:
The acute angle is your angle and ALL the trig ratios are POSITIVE
ALWAYS draw your angle using the terminal and initial arm
In relation to your diagram, if the angle IS in the THIRD QUADRANT:
Your angle is (π + acute angle) and the TANGENT ratio is only POSITIVE ratio.
In relation to your diagram, if the angle IS in the FOURTH QUADRANT:
Your angle is (2π – acute angle) and the COSINE ratio is only POSITIVE ratio.
QUADRANT 1:
ALL RATIOS ARE POSITIVE
QUADRANT 2:
SINE RATIO IS POSITIVE θ θ θ θ
QUADRANT 4:
COSINE RATIO IS POSITIVE
QUADRANT 3:
TANGENT RATIO IS POSITIVE

9. Trigonometric Ratios and Reciprocal Trigonometric Ratios
The reciprocal trigonometric ratios are the reciprocal of the primary ratios:
Primary Ratio
Reciprocal Ratio
Sine Ratio
sinθ = opposite/hypotenuse
sinθ = y/r
Cosecant Ratio
cscθ =hypotenuse/opposite
cscθ = r/y
Cosine Ratio
cosθ = adjacent/hypotenuse
cosθ = x/r
Secant Ratio
secθ =hypotenuse/adjacent
secθ = r/x
Tangent Ratio
tanθ = opposite/adjacent
tanθ = y/x
Cotangent Ratio
cotθ =opposite/adjacent
cotθ = x/y

10. Trigonometric Ratios and Reciprocal Trigonometric Ratios
Find the trigonometric and the reciprocal trigonometric ratios for θ = 7π/6
Step 1: Draw the angle on your unit circle.
Since the denominator is 6, every π (180o) is broken into 6 parts.
3π/6 x y
4π/6
2π/6
QUADRANT 2
QUADRANT 1
5π/6
π/6 θ
Step 2: Find your ratios. Remember that the angle is in the THIRD QUADRANT therefore, only TANGENT and COTENGENT are POSITIVE
6π/6
7π/6
7π/6
QUADRANT 3
TANGENT POSITIVE
Step 2: Remember to change your calculator to radians. REMEMBER, your calculator may not give the proper sign. Use your drawn angle.
QUADRANT 4
sin (7π/6) = -0.5
csc (7π/6) = 1/-0.5 = 2
cos (7π/6) =
sec (7π/6) = 1/ = -1.15
tan (7π/6) = 0.577
cot (7π/6) = 1.73

11. Trigonometric Ratios and Reciprocal Trigonometric Ratios
Example 1: Find the trigonometric and reciprocal trigonometric ratio for the following angles.
a) 2 rads
b) 3π/4
c) 11π/6 x y x y x y
sin (2 rads) = 0.909
csc (2 rads) = 1.10
cos(2 rads) =
sec(2 rads) = -2.40
tan (2 rads) = -2.19
cot (2 rads) =
sin (3π/4 rads) = 0.707
csc (3π/4 rads) = 1.414
cos(3π/4 rads) =
sec(3π/4 rads) =
tan (3π/4 rads) = -1
cot (3π/4 rads) = -1
sin (11π/6 rads) = -0.5
csc (11π/6 rads) = -2
cos(11π/6 rads) = 0.866
sec(11π/6 rads) = 1.155
tan (11π/6 rads) =
cot (11π/6 rads) =