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REFERENCE SHEET FOR TRIGONOMETRY Chapters 7, 8, 9, 10 Trigonometric functions of acute angle : opposite sin y adjacent cos x sin opposite y hypotenuse r cos adjacent x hypotenuse r csc hypotenuse 1 r opposite sin y sec hypotenuse 1 r adjacent 1 cos x cot adjacent cos x opposite tan sin y tan Cofunctions of complementary angles are equal. The two acute angles of a right triangle add to 90; thus they are complementary. Given 2 complementary angles, and , the cofunctions are: sin cos cos 90 tan cot cot 90 sec csc csc 90 cos sin sin 90 cot tan tan 90 csc sec sec 90 Identities: (1) sin2 + cos2 = 1; (2) sec2 - tan2 = 1; (3) csc2 - cot2 = 1 Quandrantal angles are angles whose terminal side lies on the x- or y-axis, such as 0=0, 90=/2, 180 = , 270 = 3/2, and 360=2. Their trig function values will always be 0, 1, or undefined. The trigonometric functions are periodic because, if we add an integral multiple of or 2 to , the value of the function is unchanged. All the functions have a period of 2 except the tangent and cotangent, which have a period of . Thus, sin ( + 2) = sin , tan ( + ) = tan , etc. DOMAIN AND RANGE Function Symbol Domain Range Sine [-1, 1] f() = sin (-, ) Cosine [-1, 1] f() = cos (-, ) Tangent f() = tan {xx odd multiples of /2 (90) (-, ) Cosecant f() = csc {xx integral multiples of (180) (-, -1] [1, ) Secant f() = sec {xx odd multiples of /2 (90) (-, -1] [1, ) Cotangent f() = cot {xx integral multiples of (180) (-, ) The sine function, like all the trig functions except cosine and secant, are odd functions, thus symmetric to the origin. It has x-intercepts at integral multiples of . The cosine function is even; thus it is symmetric to the yaxis. It has x-intercepts of multiples of /2. The period of trig functions is affected by ω. The period of cos (ωx) = 2/ω. If ω > 1, the period will be compressed horizontally by a factor of ω (omega). The amplitude (range) of A cos is between -A and A, inclusive. Sinusoidal Graphs: The graph of sin x = cos (x - /2). Because of this similarity, the graphs of the two functions are called sinusoidal. Because the sine function is odd, sin(-ωx) = -sin(ωx). The cosine function is even; therefore, cos(-ωx) = cos (ωx). The tangent function has period , domain = {xx odd multiples of /2}, range= (-, ). It has x-intercepts at integral multiples of ; the y-intercept is 0. The cotangent function also has a period of and its range = (-, ); but its domain = {xx integral multiples of }. It has x-intercepts at odd multiples of /2 and no yintercept. Given a function A sin (ωx – Φ) or A cos (ωx + Φ), the real number Φ/ω represents the phase shift; (ωx – Φ) would be a sift of Φ/ω units to the right and (ωx + Φ) would be a sift of Φ/ω units to the left. The inverse function can be used when we know the function value and want to find the angle . Since these functions are not one-to-one, the domain must restricted in order to find an inverse function. y = sin-1 x means x = sin y for –1 < x < 1 and -/2 < y < /2 y = cos-1 x means x = cos y for –1 < x < 1 and 0 < y < y = tan-1 x means x = tan y for – < x < and -/2 < y < /2 Remember: [1] The domain of f-1(x) is the range of f(x) and vice versa. [2] f-1(f(x) = x for every x in the domain of f and f(f-1(x) = x for every x in the domain of f-1. y = sec-1 x means x = sec y for x > 1 and 0 < y < , y /2 y = csc-1 x means x = csc y for x > 1 and -/2 < y < /2, y 0 y = cot-1 x means x = cot y for – < x < and 0 < y < Basic Trigonometric Identities sin Quotient: tan cos cot sin2 + cos2 = 1; sin(-) = -sin csc(-) = -csc Pythagorean: Even-Odd: cos sin Reciprocal: csc tan2 + 1 = sec2; cos(-) = cos sec(-) = sec 1 sin sec 1 cos cot 1+ cot2 = csc2 tan(-) = -tan cot(-) = -cot 1 tan cos( + ) = cos cos - sin sin [2] cos( - ) = cos cos + sin sin [3] sin( + ) = sin cos + cos sin [4]sin( - ) = sin cos - cos sin tan + tan tan - tan [5] [6] tan( + ) = tan( - ) = 1 - tan tab 1 + tan tab Identities: [7] [8] cos - = sin sin - = cos 2 2 Double-Angle and Half-Angle Formulas [1] sin (2) = 2 sin cos [2] cos(2) = cos2 - sin2 [3] cos(2) = 1 - 2sin2 [4] cos(2) = 2cos2 - 1 [5] [6] 2 1 - cos 2 [7] [8] 1 + cos 2 2tan 2 1 - cos 2 tan( 2 ) = [9] sin 2 = sin = 1 - tan 2 [10] 1 - cos 2 cos = 2 cos = 2 1 + cos 2 [11] tan 2 = tan 2 = 2 1 + cos 2 1 - cos 1 cos sin 1 + cos sin 1 cos PRODUCT-TO-SUM AND SUM-TO-PRODUCT FORMULAS 1 [1] sin sin cos cos 2 1 [3] sin cos sin sin 2 [5] sin sin 2sin 2 cos [6] 2 cos cos 2cos 1 [2] cos cos cos cos 2 [4] sin sin 2sin 2 cos 2 [7] 2 cos cos cos 2sin 2 2 sin 2 Solving Oblique Triangles Oblique triangles are triangles that do not contain a right angle; they have either 3 acute angles or 1 obtuse and 2 acute angles. Solving a triangle means to find the length of all sides and the measure of all angles. To do this we need to have one of the following four sets of information: Case 1: ASA or SAA (one side & 2 angles are known) Case 2: SSA (2 sides and opposite angle are known. Case 3: SAS: (2 sides & the included angle are known) Case 4: SSS (the three sides are known) In cases 1 and 2 for a triangle with sides a, b, c and opposite angles , , , respectively, use Law of Sines. In cases 3 and 4, use law of cosines. Law of sines: sin sin sin a b c c 2 a 2 b 2 2ab cos Law of Cosines: b 2 a 2 c 2 2ac cos a 2 c 2 b 2 2bc cos Converting from polar to rectangular coordinates: x = r cos ; y = r sin . Converting from rectangular to polar coordinates: r x 2 y 2 ; tan 1 y x Tests for symmetry: A polar equation is symmetric to the [1] polar axis (x-axis) if (r, ) when replaced by (r, ) yields the same equation; [2] line = /2 (y-axis) if (r, ) when replaced by (r, -) yields the same equation; [3] pole (origin) if (r, ) when replaced by (-r, ) yields the same equation. Passing these tests proves that the equation has the given symmetry; however, equations failing these tests may still have one of the above symmetries. In the complex plane, the x-axis becomes the real axis (z = x + oi = x) and the y-axis becomes the imaginary axis (z = 0 + yi = yi). The magnitude or modulus of z z is the distance from the origin to the point (x, y); z x 2 y 2 . If z = x + yi is multiplied by its conjugate z = x – yi, the product is x2 + y2 and z zz An equation z = x + yi in rectangular form can be converted to polar coordinates z = r cos + r sini = r(cos + i sin), r > 0 and 0 < < 2. In polar form, the angle is called the argument of z and r is the magnitude of z. Product of complex numbers: z1 z2 r1r2 cos 1 2 i sin 1 2 . z1 r1 cos 1 2 i sin 1 2 . z2 r2 De Moivre’s Theorem: If z = r(cos + i sin) is a complex number, then zn = rn(cos(n)+ sin(n), where n > 1 is a Quotient of complex numbers: positive integer. Complex Roots: Let w = r(cos +i sin) be a complex number and let n > 2 be an integer. If w 0, there are n distinct complex roots of w given by z n cos 0 2k i sin 0 2k , where k = 0, 1, 2,… n-1. k n n n n Two vectors, v and w, are equal if the have the same magnitude and direction, v = w. Vector addition obeys the addition properties of real numbers; ie., commutative, associative, inverse, and identity. For vector v, -v is a vector with the same magnitude but opposite direction; thus, v – w = v + (-w). With vectors, real numbers are scalars, which have only magnitude. If α is a scalar and v is a vector, the scalar product αv obeys the multiplicative properties of real numbers; ie., distributive, associative, and the properties of 0, 1, and -1. The magnitude of vector v is represented by v and has the following properties: v > 0; v = 0 if and only if v = 0; v v ; and v v . An algebraic vector v is shown as v = (a, b), where a and b are real numbers (scalars) called the components of v. If the initial point of v is the origin in a rectangular coordinate system, then v is a position vector with terminal point a , b . Any vector with initial point P1 (a1 , b1 ) and terminal point P2 (a2 , b2 ) is equal to the position vector v x2 x1 , y2 y1 . A vector u for which u = 1 is called a unit vector. Any vector v a , b can be written using the unit vectors i 1, 0 , j 0,1 as follows: v a , b a 1, 0 b 0,1 ai bj ; a and b are the horizontal and vertical components of v, respectively. Addition, subtraction, scalar product, and magnitude are defined in terms of the components of a vector. The vector u = v is a unit vector that has the same direction as a vector v and v v v u . A vector can be written in terms of its magnitude and direction as u cos i sin j and v v cos i sin j , where α is the angle between v and i.