CIRCULAR FUNCTIONS

Unit Circle Definition of Trig Functions

The Unit Circle

A unit circle is the circle with center at the origin and radius equal to 1 (one unit).

Its equation is (x – 0)2 + (y – 0)2 = 1 or x2 + y2 = 1.

Coordinate Function The coordinate function P is the function

from the set of P(θ) = {(x, y) | x2 + y2 = 1} to the set of real numbers R.

Examples:

1. P( ) is the terminal point of the arc length so that P( ) = (0, -1).

2. If θ = -π, then P(-π) = (-1, 0).

3. If θ = 0, then P(0) = (1, 0).

4. If θ = , then P( ) = (0, 1).

5. If θ = , then P( ) = (0, 1).

Note: The given θ are quadrantal angles.

0, π/2, π, 3π/2, and 2π (and their multiples) are called quadrantal angles.

Note that P(θ) = (x, y).

Circular Functions

Trigonometric functions are defined so that their domains are sets of angles and their ranges are sets of real numbers. Circular functions are defined such that their domains are sets of numbers that correspond to the measures (in radian units) of the angles of analogous trigonometric functions.

Circular Functions

The ranges of these circular functions, like their analogous trigonometric functions, are sets of real numbers.

These functions are called circular functions because radian measures of angles are determined by the lengths of arcs of circles. In particular, trigonometric functions defined using the unit circle lead directly to these circular functions.

Circular Functions The circle below is drawn in a coordinate system where the

circle's center is at the origin and has a radius of 1. This circle is known as a unit circle.

Circular Functions Definition: the sine and the cosine functions in the

set of real numbers are defined by x = cosine θ and y = sine θ, where θ ϵ R and P(θ) = (x, y).

The x and y coordinates for each point along the circle may be ascertained by reading off the values on the x and y axes. If you picture a right triangle with one side along the x-axis:

Definition: the four other circular functions are defined by the following:

tangent θ = , x ≠ 0

cosecant θ = , y ≠ 0

secant θ = , x ≠ 0

cotangent θ = , y ≠ 0where P(θ) = (x, y) is on the unit circle.

Notes: The conventional way of writing the circular

functions is sin θ, cos θ, tan θ, csc θ, sec θ, and cot θ.

The domain of both sine and cosine functions is the set of real numbers R.

The range of both sine and cosine functions is from -1 to 1, or in notations, [-1, 1].

The domain of tan θ and sec θ is the set for which θ ≠ + nπ, n ϵ I and where x ≠ 0.

The domain of cot θ and csc θ is the set for which θ ≠ nπ, n ϵ I and where y ≠ 0.

Signs of the Circular Functions

Note: r is always positive. Thus, the signs of the values of the trigonometric functions of angle θ are determined by the signs of x and y.

The signs of the trigonometric functions depend upon the quadrant in which the terminal side of θ lies.

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Some Fundamental Circular Functions Identities

Any terminal point P(θ) = (x, y) on the unit circle satisfies the equation x2 + y2 = 1, where θ ϵ R and since x = cos θ and y = sin θ, then cos2 θ + sin2 θ = 1.

Note: cos2 θ = (cos θ)2 and cos2 θ ≠ cos θ2 Quotient relations:

tan θ = and cot θ = Reciprocal relations:

sin θ • csc θ = 1

cos θ • sec θ = 1

tan θ • cot θ = 1

Examples

1. Find the circular functions of θ if θ = .

2. Given θ = –π, find the circular functions of –π.

3. P(θ) lies on the line segment joining the origin and N(4, 3). Sketch and find the circular functions of (θ + π).

Special Angles

Special Angles are those angles for which the exact values of the circular functions are easily obtained. We shall study the angles , , , and their multiples.

Triangles with special angle-side relationships, introduced in geometry are useful in evaluating the trigonometric function values of , , , or 30°, 45°, and 60° respectively.

and its odd multiples

and its odd multiples

Circular functions of and its odd multiples

Example:

Find the circular functions of .

and its multiples

and its multiples

Circular functions of and its multiples

Example:

Find the circular functions of .

and its multiples

and its multiples

330°

Circular functions of and its multiples

Example:

Find the circular functions of .

Other illustrations…

Quadrantal Angles An angle in standard position whose terminal

side lies on an axis (either the x-axis or the y-axis) is said to be quadrantal.

Quadrantal Angles

Example:

Find the circular functions of 2π.

The Unit Circle

The Unit Circle

Reference Angles… (a recall)Examples: Find the reference angle for each of the following angles.

1. 460°

2. 165°

3. -40°

4. -283°

Evaluating Trigonometric Functions Using Reference Angles

Procedures:

1. Determine the reference angle θ’ associated with θ.

2. Find the value of the corresponding trigonometric function of θ’. Exact value can be obtained if θ’ is 30°, 45°, or 60°, or it can be an approximate value from a calculator or a table.

3. Affix the proper algebraic sign for the particular function by determining the signs of P(x, y) where the terminal side of θ lies.

Examples

Use reference angles to solve the following.

1. Find tan 120°.

2. Approximate the value of cos (-16°48’) by first expressing it in terms of a function of the associated reference angle.

3. Approximate the value of sin 220°.

Exercises

Evaluate the following using reference angles.

1. cos 210°

2. tan (-45°)

3. sec ( )

4. csc ( )

Do Worksheet 5