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Trigonometric Review

Trigonometric Review. 1.6. Unit Circle. θ. adj. opp. sin  = cos  = tan  = csc  = sec  = cot  =. hyp. adj. hyp. hyp. adj. opp. adj. opp.

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Trigonometric Review

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  1. Trigonometric Review 1.6

  2. Unit Circle

  3. θ adj opp sin  = cos  = tan  = csc  = sec  = cot  = hyp adj hyp hyp adj opp adj opp 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 , opp  the side adjacent to the acute angle , adj  and the hypotenuse of the right triangle. The trigonometric functions are sine, cosine, tangent, cotangent, secant, and cosecant. Trigonometric Functions

  4. 5 4  3 sin  = cos  = tan  = cot  = sec  = csc  = Calculate the trigonometric functions for  . The six trig ratios are Example: Six Trig Ratios

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

  6. 2 2 60○ 60○ 2 Use the Pythagorean Theorem to find the length of the altitude, . Geometry of the 30-60-90 triangle Consider an equilateral triangle with each side of length 2. 30○ 30○ The three sides are equal, so the angles are equal; each is 60. The perpendicular bisector of the base bisects the opposite angle. 1 1

  7. x 0 sin x 0 1 0 -1 0 y = sin x y x 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. 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.

  8. x 0 cos x 1 0 -1 0 1 y = cos 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. 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.

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

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

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

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

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

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

  15. d = = 146.47. 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 horizontal 16○ angle of depression cliff42 m line of sight 16○ ship d The ship is 146 m from the base of the cliff. Example 2: Application

  16. sin  = = 0.875 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 angleof inclination. Does the painter’s plan satisfy the safetyrequirements for the use of the ladder? ladder house 16 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! Example 3: Application

  17. 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 cos = 1/sec tan = 1/cotcot = 1/tan sec = 1/cos csc = 1/sin Quotient Identities tan  = sin  /cos  cot  = cos  /sin  Pythagorean Identities sin2  + cos2  = 1 tan2 + 1 = sec2 cot2  + 1 = csc2  Pg. 51 & 52

  18. Trig Identities

  19. Homework • READ section 1.6 – IT WILL HELP!! • Pg. 57 # 1 - 75 odd

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