Trigonometric Review 1 6 Unit Circle The six

Trigonometric Review 1. 6

Unit Circle

The six trigonometric functions of a right triangle, with an acute angle , are defined by ratios of two sides of the triangle. The sides of the right triangle are: hyp the side opposite the acute angle , the side adjacent to the acute angle , θ and the hypotenuse of the right triangle. adj The trigonometric functions are sine, cosine, tangent, cotangent, secant, and cosecant. sin = cos =adj tan =opp hyp adj hyp csc = opp sec =hyp adj cot =adj opp

Calculate the trigonometric functions for . 5 4 3 The six trig ratios are sin = cos = tan = cot = sec = csc =

Geometry of the 45 -45 -90 triangle Consider an isosceles right triangle with two sides of length 1. 45 1 x The Pythagorean Theorem implies that the hypotenuse is of length.

Geometry of the 30 -60 -90 triangle Consider an equilateral triangle with each side of length 2. 30○ The three sides are equal, so the angles are equal; each is 60. 2 The perpendicular bisector of the base bisects the opposite angle. 60○ Use the Pythagorean Theorem to find the length of the altitude, . 2 1 60○ 2 1

Graph of the Sine Function To sketch the graph of y = sin x first locate the key points. These are the maximum points, the minimum points, and the intercepts. x 0 sin x 0 1 0 -1 0 Then, connect the points on the graph with a smooth curve that extends in both directions beyond the five points. A single cycle is called a period. y = sin x y x

Graph of the Cosine Function To sketch the graph of y = cos x first locate the key points. These are the maximum points, the minimum points, and the intercepts. x 0 cos x 1 0 -1 0 1 Then, connect the points on the graph with a smooth curve that extends in both directions beyond the five points. A single cycle is called a period. y = cos x y x

Graph of the Tangent Function To graph y = tan x, use the identity . At values of x for which cos x = 0, the tangent function is undefined and its graph has vertical asymptotes. y Properties of y = tan x 1. domain : all real x 2. range: (– , + ) 3. period: x 4. vertical asymptotes: period:

Graph of the Cotangent Function To graph y = cot x, use the identity. At values of x for which sin x = 0, the cotangent function is undefined and its graph has vertical asymptotes. y Properties of y = cot x 1. domain : all real x 2. range: (– , + ) 3. period: 4. vertical asymptotes: vertical asymptotes x

Graph of the Secant Function The graph y = sec x, use the identity . At values of x for which cos x = 0, the secant function is undefined and its graph has vertical asymptotes. y Properties of y = sec x 1. domain : all real x 2. range: (– , – 1] [1, + ) 3. period: 4. vertical asymptotes: x

Graph of the Cosecant Function To graph y = csc x, use the identity . At values of x for which sin x = 0, the cosecant function is undefined and its graph has vertical asymptotes. y Properties of y = csc x 1. domain : all real x 2. range: (– , – 1] [1, + ) 3. period: 4. vertical asymptotes: where sine is zero. x

Graphing a -> amplitude b -> (2*pi)/b -> period c/b -> phase shift (horizontal shift) d -> vertical shift

Angle of Elevation and Angle of Depression When an observer is looking upward, the angle formed by a horizontal line and the line of sight is called the: angle of elevation. line of sight object angle of elevation horizontal observer When an observer is looking downward, the angle formed by a horizontal line and the line of sight is called the: angle of depression. horizontal angle of depression line of sight object observer

Example 2: A ship at sea is sighted by an observer at the edge of a cliff 42 m high. The angle of depression to the ship is 16. What is the distance from the ship to the base of the cliff? observer cliff 42 m horizontal 16○ angle of depression line of sight 16○ d d= = 146. 47. The ship is 146 m from the base of the cliff. ship

Example 3: A house painter plans to use a 16 foot ladder to reach a spot 14 feet up on the side of a house. A warning sticker on the ladder says it cannot be used safely at more than a 60 angle of inclination. Does the painter’s plan satisfy the safety requirements for the use of the ladder? ladder house 16 sin = = 0. 875 14 θ Next use the inverse sine function to find . = sin 1(0. 875) = 61. 044975 The angle formed by the ladder and the ground is about 61. The painter’s plan is unsafe!

Fundamental Trigonometric Identities for 0 < < 90. Cofunction Identities sin = cos(90 ) cos = sin(90 ) tan = cot(90 ) cot = tan(90 ) sec = csc(90 ) csc = sec(90 ) Reciprocal Identities sin = 1/csc cot = 1/tan cos = 1/sec = 1/cos tan = 1/cot csc = 1/sin Quotient Identities tan = sin /cos cot = cos /sin Pythagorean Identities sin 2 + cos 2 = 1 Pg. 51 & 52 tan 2 + 1 = sec 2 cot 2 + 1 = csc 2

Trig Identities

Homework l READ section 1. 6 – IT WILL HELP!! l Pg. 57 # 1 - 75 odd