The Trigonometric Functions

---Mandy

6.1 Trigonometric Functions of Acute Angles

The Trigonometric Ratios

An acute angle is an angle with measure greater than 0° and less than 90°.

Letters that are used to denote an angle : α(alpha), β(beta),γ(gamma), θ(theta), and φ(phi).

Trigonometric Function Values of an Acute Angle θ

sin θ= length of side opposite θ÷ length of hypotenuse

cos θ= length of side adjacent θ ÷ length of hypotenuse

tan θ = side opposite θ ÷ side adjacent θcsc θ = hypotenuse ÷side opposite θ sec θ = hypotenuse ÷ side adjacent to θ cot θ =side adjacent to θ ÷ side opposite θ

6.2 Applications of Right Triangles

Example : Cloud Height

To measure cloud height at night, a vertical beam of light is directed on a spot on the cloud. From a point 135 ft away from the light source, the angle of elevation to the spot is found to be 67.35°. Find the height of the cloud.

From the figure, we have tan 67.35° = h / 135 ft h = 135 ft · tan 67.35° ≈ 324 ft

6.3 Trigonometric Function of Any Angle

Trigonometric Functions of Any Angle θ

Suppose that P(x, y) is any point other than the vertex on the terminal side of any angle θ in standard position, and r is the radius, or distance from the origin to P(x, y). Then the trigonometric functions are defined as follows:

sin θ = y-coordinate / radius = y / r

cos θ = x-coordinate / radius = x / r

tan θ = y-coordinate / x-coordinate = y / r

csc θ = radius / y-coordinate = r / y

sec θ = radius / x-coordinate = r / x

cot θ = x-coordinate / y-coordinate = x / y

6.4 Radians, Arc Length, and Angular Speed

Linear Speed in Terms of Angular Speed

The linear speed v of a point a distance from the center of rotation is given by

v = rω,

where ω is the angular speed in radians per unit of time.

6.5 Circular Functions: Graphs and Properties

Basic Circular Functions

For a real number s that determines a point (x,y) on the unit circle:

sin s = second coordinate = ycos s = first coordinate = xtan s = second coordinate / first coordinate =

y/xcsc s = 1 / second coordinate = 1/y sec s = 1 / first coordinate = 1/xcot s = first coordinate / second coordinate =

x/y

Domain and Range of Sine and Cosine Functions

The domain of the sine function and the cosine function is (-∞, ∞).

The range of the sine function and the cosine function is [-1, 1].

6.6 Graphs of Transformed Sine and Cosine Functions

Graphing a Transformation of Sine and Cosine

AmplitudeThe amplitude of the graphs of y =Asin(Bx-C)

+D and y = Acos(Bx-C)+D is |A|.

PeriodThe period of the graphs of y =Asin(Bx-C)+D

and y = Acos(Bx-C)+D is |2π/B|

Phase ShiftThe phase shift of the graphs is the quantity

C/By = A sin(Bx-C)+D = A sin [B (x - C/B) ] + D andy = A cos(Bx-C)+D = A cos [B (x - C/B) ] + D

Thank You