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.