Terminal side the ray that moves to form an angle with the initial side. Directed angle angle that can be described as positive or negative

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1 Trigonometry Notes, Definitions, and Formulae Definitions Degree 1/360 of a complete revolution. Minute 1/60 of a degree Second 1/60 of a minute or (1/60 1/60) of a degree Initial side the beginning ray of an angle Terminal side the ray that moves to form an angle with the initial side Directed angle angle that can be described as positive or negative Standard position the position of an angle whose initial side coincides with the positive x axis Quadrantal angle angle whose terminal side lies on a coordinate axis Coterminal used to describe two angles whose terminal sides coincide Trigonometric identity an equation involving trigonometric functions of an angle θ that is true for all values of θ (e.g. tan θ = sin θ/cos θ) Pythagorean identities sin² θ + cos² θ = 1; tan² θ + 1 = sec² θ; cot² θ + 1 = csc² θ Cofunctions any pair of trigonometric functions of complementary acute angles that are equal (e.g. in a triangle with complementary angles A and B, sin A = cos B) Distance formula r = x² + y² Reference angle of θ a unique acute angle α, corresponding to θ, formed by the terminal side of θ and the positive or negative x axis Reference triangle triangle drawn using the reference angle α to be used to find values for trigonometric functions of θ Solving the triangle finding the measurements of all three sides and all three angles Angle of elevation and angle of depression angles ascending or descending from horizontal for application in word problems; they equal each other Radian the ratio of the length of the curve or arc to the radius: θ = _s_ and s = rθ r

2 Circular functions functions that use a circle in their definition but that are used for nongeometric applications like ocean waves or alternating current Periodic repeating. A function is periodic if for some positive constant p f(x+p) = f(x) for every x in the domain of f. The smallest such p is the period of f. Odd function in general f is an odd function if f( x) = (f)x When f is odd, the graph is symmetric with respect to the origin. If the point (x, y) is on the graph of f, then so is the point ( x, y). Even function in general f is an even function if f( x) = f(x) When f is even, the graph is symmetric with respect to the y-axis. If the point (x, y) is on the graph of f, then so is the point ( x, y). Amplitude designated by the letter a it is the difference between the minimum and maximum values of a curve divided by 2. M m = (c + a) (c a) = 2c = a Sine and cosine functions are expressed by the formulas y = c + a sin bx and y = c + a cos bx In these expressions, a is amplitude and c is the number of units the function is shifted above or below the standard position. M + m = (c + a) + (c a) = 2c = c The coefficient b affects the period because the functions have a period of 2π b Trigonometric Functions of Acute Angles sin θ = _y_ = opposite r hypotenuse cos θ = _x_ = adjacent r hypotenuse tan θ = _y_ = opposite x adjacent Reciprocal functions of acute angles cot θ = _x_ = adjacent = _1_ y opposite tan θ sec θ = _r_ = hypotenuse = _1_ x adjacent cos θ csc θ = _r_ = hypotenuse = _1_ y opposite sin θ

3 Cofunctions of complementary angles sin A = cos B cos A = sin B tan A = cot B cot A = tan B sec A = csc B csc A = sec B Trigonometric functions for 30º, 45º, and 60º angles θ sin θ cos θ tan θ csc θ sec θ cot θ 30º _1_ º º 3 _1_ Trigonometric function values for general triangles Function Quadrant of θ value I II III IV sin θ csc θ + + cos θ sec θ + + tan θ cot θ + + Law of Cosines In any triangle ABC, c² = a² + b² 2ab cos C cos C = a² + b² c² b² = a² + c² 2ac cos B 2ab a² = b² + c² 2bc cos A Law of Sines In any triangle ABC, sin A = sin B = sin C and also a = sin A a b c b sin B

4 Circular Functions sin s = y cos s = x tan s = sin s if cos s 0 cot s = cos s if sin s 0 cos s sin s sec s = _1_ if cos s 0 csc s = _1_ if sin s 0 cos s sin s Degree and radian measures of some frequently used angles Degree measure 0º 30º 45º 60º 90º 120º 135º 150º 180º Radian measure _2π_ 3 _3π_ 4 _5π_ 6 π The Reciprocal Identities sin α = _1_ sin α csc α = 1 csc α = _1_ csc α sin α cos α = _1_ cos α sec α = 1 sec α = _1_ sec α cos α _1_ cot α tan α = tan α cot α = 1 cot α = sin α cos α _1_ tan α cos α sin α The Cofunction Identities sin θ = cos (90º θ ) cos θ = sin (90º θ ) tan θ = cot (90º θ ) cot θ = tan (90º θ ) sec θ = csc (90º θ ) csc θ = sec (90º θ ) The Pythagorean Identities sin² α + cos² α = tan² α = sec² α 1 + cot² α = csc² α

5 Notes: Counterclockwise rotations of the terminal side produce positive angles. Clockwise rotations of the terminal side produce negative angles. Angles can be classified according to the quadrant in which their terminal side lies. Use the MODE button on your calculator to switch from degrees to radians or vice versa. If θ is a complete revolution, s = 2πr and θ = 2πr = 2π 180 = π radians R 1º = radians radian = 180º 57.3º π The product of two odd functions is even. The product of an even function and an odd function is odd. The sum of an even function and an odd function is neither. Sine and cosine functions have period 2π. Sine is an odd function. Cosine is an even function. Use that information to derive whether each of the other four functions are odd or even. If a cosine curve were shifted π units to the right, it would coincide with the sine curve. 2 Tangent is an odd function that snakes back and forth between two asymptotes and has period π. General Strategies for Proving Identities 1. Simplify the more complicated side of the identity until it is identical to the other side. 2. Transform both sides of the identity into the same expression. Special Strategies for Proving Identities 1. Express functions in terms of sines and cosines. 2. Look for expressions to which the Pythagorean identities can be applied. 3. Use factoring. 4. Combine terms on each side of the identity into a single fraction. 5. Multiply one side of the equation by an expression equal to one. Addition Formulas for the Sine and Cosine sin (α + β) = sin α cos β + cos α sin β sin (α β) = sin α cos β cos α sin β cos (α + β) = cos α cos β sin α sin β cos (α β) = cos α cos β + sin α sin β

6 Double-Angle Formulas for Sine and Cosine sin 2 α = 2 sin α cos α cos 2 α = cos² α sin² α cos 2 α = 1 2 sin² α cos 2 α = 2 cos² α 1 Half-Angle Formulas for Sine and Cosine sin _θ_ = ± 1 cos θ cos _θ_ = ± 1 + cos θ Addition Formulas for the Tangent tan (α + β) = _tan α + tan β_ 1 tan α tan β tan (α β) = _tan α tan β_ 1 + tan α tan β Double-Angle Formulas for the Tangent tan 2 α = 2 tan α_ 1 tan² α Half-angle Formulas for the Tangent tan _θ_ = ± 1 cos θ tan _θ_ = ± sin θ tan _θ_ = 1 cos θ cos θ cos θ 2 sin θ cot (α + β) = cot α cot β 1 cot α + cot β cot (α β) = cot α cot β + 1 cot β cot α