Download 5.4 – Trigonometric Ratios for All Angles An angle in the Cartesian

5.4 – Trigonometric Ratios for All Angles
An angle in the Cartesian plane is in standard position if its vertex lies at the origin and its
initial arm lies on the positive x-axis.
Principal Angle (θ) – the counter-clockwise angle between the initial arm and the terminal arm of
an angle in standard position. Its value is between 0° and 360°.
Related Acute Angle (β) – the acute angle between the terminal arm of an angle in standard
position and the x-axis when the terminal arm lies in quadrants 2, 3, or 4.
An angle in standard position is determined by a counter-clockwise rotation and is always
positive. An angle determined by a clockwise rotation is always negative.
If the terminal arm of an angle in standard position lies in quadrants 2, 3, or 4, there exists a
related acute angle (β) AND a principal angle (θ).

If the terminal arm of the principal angle lies in quadrant 2 then the related acute angle
is calculated as β = 180° - θ.

If the terminal arm of the principal angle lies in quadrant 3 then the related acute angle
is calculated as β = θ - 180°.

If the terminal arm of the principal angle lies in quadrant 4 then the related acute angle
is calculated as β = 360° - θ.
Any point can be represented as a point on a circle of radius r, and has the formula x2 + y2 = r2.
Use the image below to help you write an expression for each of the primary trigonometric
functions in terms of x, y, and r for each quadrant.
Quadrant 1
Trig Function
Sign
Quadrant 2
Trig Function
Sign
Quadrant 3
Trig Function
Sign
Quadrant 4
Trig Function
Sign
Note: For any principal angle (θ) 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 (β).
The CAST rule is an easy way to remember which primary trigonometric ratios of the principal
angle (θ) are positive in which quadrant.
Given that all trigonometric ratios are each positive and negative in two quadrants, each ratio
will result in two possible principal angles.
Sine
When the ratio of sin θ is positive, the principal angle is located in either quadrant
or
The principal angle is either: θ1 = θ or θ2 = 180° - β, when 0° < θ < 180°.
When the ratio of sin θ is negative, the principal angle is located in either quadrant
or
The principal angle is either: θ3 = 180° + β or θ4 = 360° - β, when 180° < θ < 360°.
**The trig inverse of most negative ratios will yield a negative angle. The related acute angle
(β) is equal to the absolute (positive) value of this calculated (negative) angle.**
Cosine
When the ratio of cos θ is positive, the angle is either in quadrant
or
The principal angle is either: θ1 = θ or θ2 = 360° - β, when 0° < θ < 90° or 270° < θ < 360°.
When the ratio of cos θ is negative, the principal angle is located in either quadrant
The principal angle is either: θ3 = 180° - β or θ4 = 180° + β, when 90° < θ < 270°.
or
Tangent
When the ratio of tan θ is positive, the angle is either in quadrant
or
The principal angle is either: θ1 = θ or θ2 = 180° + β, when 0° < θ < 90° or 180° < θ < 270°.
When the ratio of tan θ is negative, the angle is either in quadrant
or
The principal angle is either: θ3= 180° - β or θ4= 360° - β, when 90° < θ < 180° or 270° < θ < 360°.
Example One
Determine all possible principal angles for θ where 0° ≤ θ ≤ 360°.
a) tan θ = -0.67
b) sin θ = 0.8
Example Two
Angle θ is a principal angle that lies in quadrant 2 such that 0° ≤ θ ≤ 360°. Given each
trigonometric ratio,
i.
determine the exact values of x, y, and r
ii.
sketch angle θ in standard position
iii.
determine the principal angle θ and the related acute angle β to the nearest degree
a) sin θ =
2
3
b) cos θ = -0.8
Example Three
Given that sin θ =
−7
25
, where 0° ≤ θ ≤ 360°,
a) in which quadrant could the terminal arm of θ lie?
b) Determine all possible primary trigonometric ratios for θ.
c) Evaluate all possible values of θ to the nearest degree.
Complete: p. 299 – 301 #2, 4 – 6, 8, 12.