8-2 Trigonometric Functions

Unit 8 – Trigonometric and Circular Functions

Concepts and Objectives

� Definitions of Trigonometric and Circular Functions

(Obj. #27)

� Find the values of the six trigonometric functions of

angle θ.

� Find the function values of quadrantal angles.� Find the function values of quadrantal angles.

� Identify the quadrant of a given angle.

� Find the other function values given one value and

the quadrant

Trigonometric Ratio Review

� In Geometry, we learned that for any given right triangle,

there are special ratios between the sides.

=opposite

sinhypotenuse

A

A

op

po

site

adjacent

=sinhypotenuse

A

=adjacent

coshypotenuse

A

=opposite

tanadjacent

A

Trigonometric Functions

� Consider a circle centered at the origin with radius r:

� The equation for this circle is x2 + y2 = r2

� A point (x, y) on the circle creates a right triangle whose

sides are x, y, and r.

� The trig ratios are now� The trig ratios are now(x, y)

r

x

yθ=siny

rθ

=cosx

rθ

=tany

xθ

Trigonometric Functions

� There are three other ratios in addition to the three we

already know : cosecant, secant, and cotangent.

� These ratios are the inverses of the original three:

= =1

cscr

θ(x, y)

r

x

yθ

= =1

cscsin

r

yθ

θ

= =1

seccos

r

xθ

θ

= =1

cottan

x

yθ

θ

Finding Function Values

� Example: The terminal side of an angle θ in standard

position passes through the point (15, 8). Find the

values of the six trigonometric functions of angle θ.

We know that x = 15 and y = 8, but

8

15

(15, 8)

θ

We know that x = 15 and y = 8, but

we still have to calculate r:

Now, we can calculate the values.

= +2 2r x y

= + =2 215 8 17 17

Finding Function Values

� Example: The terminal side of an angle θ in standard

position passes through the point (15, 8). Find the

values of the six trigonometric functions of angle θ.

= =8

sin17

y

rθ = =

17csc

8

r

yθ

8

15

(15, 8)

θ

17

= =sin17r

θ

= =15

cos17

x

rθ

= =8

tan15

y

xθ

= =csc8y

θ

= =17

sec15

r

xθ

= =15

cot8

x

yθ

The Unit Circle

� Angles in standard position whose terminal sides lie on

the x-axis or y-axis (90°, 180°, 270°, etc.) are called

quadrantal angles.

� To easily find function values of quandrantal angles, we use a circle with a radius of 1, which

� Notice what the different x, y,

and r values are at the quadrantal

angle points (x and y are either 0,

1, or –1; r is always 1).

use a circle with a radius of 1, which

is called a unit circle.

Values of Quadrantal Angles

� Example: Find the values of the six trigonometric

functions for an angle of 270°.

At 270°, x = 0, y = –1, r = 1.

−° = = −

1sin270 1

−° = = −

1sin270 1

1

° = =0

cos270 01

−° = =

1tan270 undefined

0

Values of Quadrantal Angles

� Example: Find the values of the six trigonometric

functions for an angle of 270°.

At 270°, x = 0, y = –1, r = 1.

° = = −1

csc270 1° = = −−

1csc270 1

1

° = =1

sec270 undefined0

° = =−

0cot270 0

1

Identifying an Angle’s Quadrant

� To identify the quadrant of an angle given certain

conditions, note the following:

� In the first quadrant, x and y are both positive.

� In QII, x is negative and y is positive.

In QIII, both are negative.� In QIII, both are negative.

� In QIV, x is positive and y is

IVIII

II I

(+,+)(+,–)

(–,–)

negative.

(+,–)

Identifying an Angle’s Quadrant

� Example: Identify the quadrant (or possible quadrants)

of an angle θ that satisfies the given conditions.

a) sin θ > 0, tan θ < 0 b) cos θ < 0, sec θ < 0

I, II II, IV

II

II, III II, III

II, III

Homework

� College Algebra (brown book)

� Page 512: 30-102 (×3s, omit 63)

� Turn in: 42, 54, 72, 78, 96, 102

Classwork: Algebra & Trigonometry (green book)� Classwork: Algebra & Trigonometry (green book)

� Page 728: 77-80