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Chapter 4 Vocabulary Section 4.1 vocabulary An angle is determined by a rotating ray (half-line) about its endpoint. The starting point of the ray is the initial side of the angle. The position of the ray after the rotation is the terminal side of the angle. The endpoint of the ray is the vertex of the angle. When an angle fits a coordinate system in which the origin is the vertex of the angle, and the initial side coincides with the positive x-axis that angle is in standard position. Positive angles are generated by counterclockwise rotation. Negative angles are generated by a clockwise rotation. Two angles that have the same initial and terminal sides are called coterminal angles. The measure of an angle is determined by the amount of rotation from the initial side to the terminal side. A central angle is an angle whose vertex is the center of the circle. One radian is the measure of a central angle Ѳ that intercepts an arc s equal in length to the radius r of the circle. Ѳ = s / r , where Ѳ is measured in radians Angles between 0 and ∏ / 2 are called acute angles. Angles between ∏/2 and ∏ are called obtuse angles. A way to measure angles is in degrees where 1 degree is equivalent to a rotation of 1/360 of a complete revolution about the vertex. Complementary angles add to be 90 degrees or∏/2. Supplementary angles sum to equal 180 degrees or ∏. Linear speed Linear speed = Arc length / time Angular speed Angular speed = central angle/ time The unit circle Given by the equation : 2 2 X +y =1 Definitions of Trigonometric functions • • • • • • Sin (t) = y Cos(t) = x Tan(t) = y/x Csc(t) = 1/y Sec(t) = 1/x Cot(t) = x/y • A function f is periodic if there exists a positive real number c such that : f(t + c) = f(t) For all t in the domain of f. The least number c for which f is periodic is called the period of f. Even/ odd trig functions Even • cos(-t) = cos(t) • sec(-t) = sec(t) odd • Sin(-t) = -sin(t) • tan(-t) = -tan(t) • csc(-t) = -csc(t) • cot(-t) = -cot(t) Section 4.3 Vocabulary Right triangle def. of Trig Functions • • • • • • Sin(Ѳ) = opp/hyp Cos(Ѳ)= adj/hyp Tan(Ѳ) = opp/adj Csc(Ѳ) = hyp/opp Sec(Ѳ) = hyp/adj Cot(Ѳ) = adj/opp Sines of special angles •Sin(30) =sin(∏/6) = ½ •Sin (45) = sin(∏/4) = √2/2 •Sin(60) = sin(∏/3) = √3/2 Cosines of special angles • Cos(30) = cos(∏/6) = √3/2 • Cos(45) = cos(∏/4) = √2/2 • Cos(60) = cos(∏/3) = ½ Tangents of Special angles • Tan(30) = tan(∏/6) = √3/3 • Tan(45) = tan(∏/4) = 1 • Tan(60) = tan(∏/3) = √3 Reciprocal Identities • • • • • • Sin(Ѳ) = 1/csc(Ѳ) Cos(Ѳ) = 1/ sec(Ѳ) Tan(Ѳ) = 1/cot(Ѳ) Csc(Ѳ) = 1/sin(Ѳ) Sec(Ѳ) = 1/cos(Ѳ) Cot(Ѳ) = 1/tan(Ѳ) Quotient identities • Tan(Ѳ) = sin(Ѳ) / cos(Ѳ) • Cot(Ѳ) = cos(Ѳ) / sin(Ѳ) Pythagorean Identities 2 •Sin (Ѳ) 2 cos (Ѳ) + =1 2 2 •1 + tan (Ѳ) = sec (Ѳ) 2 2 •1 + cot (Ѳ) = csc (Ѳ) Angle of elevation •The angle from the horizontal up to the object Angle of Depression •The angle from the horizontal downward to the object. Section 4.4 Vocabulary Definitions of Trig Functions Sin Ѳ = y/r cos Ѳ = x/r Tan Ѳ = y/x Cot Ѳ = x/y Sec Ѳ = r/x Csc Ѳ = r/y Reference Angle • Let Ѳ be an angle in standard position. Its reference angle is the acute angle Ѳ’ formed by the terminal side of V and the horizontal axis. Section 4.6 Vocabulary Amplitude • The amplitude of y = a sin(x) And y = a cos(x) Represents half of the distance between the max and the min values of the function, and is given by Amplitude = |a| Period • The b be a positive real number. The period of y = a sin(bx) and t = a cos(bx) is given by Period = 2∏/b Damping factor •In the function f(x) = x sin(x), the factor x is called the damping factor. Section 4.7 Vocabulary4 Inverse sine function y = sin (x) has a unique inverse function called inverse sine function. It is denoted by Y =arcsin(x) or y = -1 sin (x) Inverse cosine function y = cos (x) has a unique inverse function called inverse cosine function. It is denoted by Y =arccos(x) or y = -1 cos (x) Inverse tangent function y = tan (x) has a unique inverse function called inverse tangent function. It is denoted by Y =arctan(x) or y = tan-1 (x)