trigonometric functions of an acute angle engr. rean navarra
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Trigonometric Functions of an
Acute angle
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 .
Drop a vertical line
from P(x,y) to the
x-axis at Q(x,0) .
Y
r
xO
P(x,y)
Q(x,0)X
…Definition
Y
r
xO
P(x,y)
Q(x,0)X
x
r
y
Q(x,0)
P(x,y)
Definition 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
Notes… 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.
…Notes 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).
Coordinate Signs y
( + , + )
QIVQIII
QII QI x
( - , + )
( - , - ) (+ , - )
The CAST Rule for Positive Trigo
Functionsy
QIVQIII
QII QIx
ALL
COS(& Sec)
TAN(&COT)
Sin(&CSC)
Negative trigo. Functions
QIVQIII
QII QI
NONE
sin, tan,csc,cot
sin,csc,sec, cos
cos, sec,tan, cot
Example: Finding trigonometric functions of
angles
a.cos 135˚b.cos 390˚
Reference Angle
The reference angle associated with is the acute angle formed by the terminal side of and the
x- axis.
Reference Angles:
=
= 180° -
= - 180°
= 360° -
Example: Find the reference Angles
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 16π/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.
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