Engr. Rean Navarra

Trigonometric functions on an acute angle
 Let be an acute angle in standard position in a rectangular coordinate system, and let
P(x,y) be any point other than the point O on the terminal side of .
Y r
P(x,y)
 Drop a vertical line from P(x,y) to the x-axis at Q(x,0) .
O x
Q(x,0)
X

Y r
P(x,y)
O x
Q(x,0)
X

Q(x,0)
P(x,y) r y x

 Let be an acute angle in standard position in a rectangular coordinate system, and let
P(x,y) be any point other than the point O on the terminal side of .
 If d (O, P) = then ….

Trigonometric functions of an any angle
• tan and sec are undefined if x = 0
• csc and cot are zero if y = 0

 The trigonometric formulas does not depend on the point
P(x, y) that is chosen on the terminal of .
 The fundamental identities are true for trigonometric functions of any angle.

 The domains of trigonometric functions consists of all angles for which the functions is defined
(where zero denominators does not occur).

…because the denominator r > 0 for any angle.

The tangent and secant are undefined if x = 0 ( if the terminal side of the angle is on y – axis).

The cotangent and cosecant are undefined if y = 0 ( if the terminal side of the angle is on x – axis).

y
( - , + ) ( + , + )
( - , - )
QII QI
QIII QIV
(+ , - ) x

The CAST Rule for Positive
Trigo Functions y
S in A LL
(&CSC)
QII QI x
QIII QIV
T AN C OS
(&COT) (& Sec)

cos, sec,tan, cot
QII QI
NONE sin,csc,
QIII sec, cos
QIV sin, tan,csc, cot

Example: Finding trigonometric functions of angles a.
cos 135˚ b.c
os 390˚

 The reference angle associated with is the acute angle formed by the terminal side of and the x- axis.

Reference Angles:
=
= 180 ° -
= 360 ° -
= - 180 °

a. Θ = 5π/3 b. Θ= 870°
Ans. a. 30 ° b. 20°

Evaluating Trigonometric
Functions for any angle
 Find the reference angle associated with the angle .
 Determine the sign of the trigonometric function of by the quadrant in which lies.
 The value of trigonometric function of is the same, except possibly for sign, as the value of the trigonometric function .

Example: Using the reference Angle to evaluate trigonometric functions.
 Find sin 240 °
 Find cot 495 °
 Sin 16 π/3
 Sec (π/4)

Example: Using the reference Angle to evaluate trigonometric functions.
 If tan θ = 2/3 and θ is in Q-III, find cos θ.
 If sec θ = 2 and θ is in Q-IV, find the other five trigonometric functions of θ.