Algebra and Trig. I 4.2 Trigonometric Functions: The Unit...

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Hannah Province – Mathematics Department – Southwest Tennessee Community College

Algebra and Trig. I 4.2 – Trigonometric Functions: The Unit Circle The unit circle is a circle of radius 1, with its center at the origin of a rectangular coordinate system. The equation of this unit circle is

. We can use the following formula for the length of a

circular arc, , to find the length of the intercepted arc.

where 1 is the radius of the circle and t is the radian measure of the central angle.

Thus the length of the intercepted arc is θ, which is also the

radian measure of the central angle. *So in a unit circle, the radian measure of the central angle is equal to the length of the intercepted arc.

θ

(1,0) 0

P(x,y)

(1,0) 0

P(x,y)

θ

When θ is positive the point P is

reached by moving counterclockwise along the unit circle from (1,0)

When θ is negative the point P is

reached by moving clockwise along the unit circle from (1,0)

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The Six Trigonometric Functions –

The inputs of these six functions are θ and the outputs involve the

point on the unit circle corresponding to t and the

coordinates of this point. Trig. Functions have names that are words, rather than single

letters such as . For example, the sine of θ is the y-

coordinate of the point P on the unit circle.

The real number y depends on the letter θ and thus is a function

of t. really means , where sine is the name of the

function and θ, a real number, is an input.

Name Abbreviation Name Abbreviation

sine sin cosecant csc

cosine cos secant sec

tangent tan cotangent cot

Definitions of the Trigonometric Functions in Terms of

Any Circle:

If θ is a real number and is a point on the unit circle

that corresponds to θ, and r is the radius then

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Definitions of the Trigonometric Functions in Terms of a Unit Circle:

If θ is a real number and is a point on the unit circle

that corresponds to θ, where r is the radius which is 1 then

Finding Values of the Trigonometric Functions

If θ is a real number equal to the length of the

intercepted arc of an angle that measures θ

radians and is the point on the

unit circle that corresponds to θ. Find the

values of the six trig. functions at t. Don’t forget to rationalize if needed.

a)

b)

c)

d)

(1,0) 0

P( , )

θ

θ

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e)

f)

Example –

Use the figure to the left to determine the

values of the six trig. functions at .

a)

b)

g)

h)

i)

j)

(1,0) 0

P=

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Domain and Range of Sine and Cosine Functions –

Example – Find

Example – Find

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Trigonometric Functions at

Even and Odd Trigonometric Functions We know that a function is even if and odd if

. We can show that the cosine function is even and

sine is odd.

By definition, the coordinates of the points P and Q are as follows:

In the above figure the x-coordinates of P and Q are the same,

thus thus the cosine function is even.

In the above figure the y-coordinates of P and Q are negatives of

each other, thus thus the sine function is odd

P

Q

(1, 0) 0

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Even and Odd Trigonometric Functions

Only cosine and secant functions are even all other are odd. Example – Find the value of each trigonometric function:

a)

b)

Fundamental Identities Trigonometric identities are equations that are true for all real numbers for which the trigonometric expressions in the equations are defined.

Reciprocal Identities

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Quotient Identities

Example – Given and , find the value of

each of the four remaining trigonometric functions.

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Pythagorean Identities –


using a trigonometric identity.


using a trigonometric identity.

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Periodic Functions A function f is periodic if there exists a positive number p such

that for all in the domain of f. The smallest

positive number p for which f is periodic is called the period of f. Periodic Properties of the Sine and Cosine Functions

The sine and cosine functions are periodic functions and have

period 2π.

The secant and cosecant functions have period 2π. Periodic Properties of the Tangent and Cotangent Functions

The tangent and cotangent functions are periodic functions and

have period π. Example – Find the value of each trigonometric function.

a)

b)

c)

d)

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Repetitive Behavior of the Sine, Cosine and Tangent Functions For any integer n and real number ,

Evaluating Trig. Functions with a Calculator

a)

b)

Algebra and Trig. I 4.2 Trigonometric Functions: The Unit...

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