CRASH COURSE IN PRECALCULUS Shiah-Sen Wang The graphs are prepared by Chien-Lun Lai
Based on : Precalculus: Mathematics for Calculus by J. Stuwart, L. Redin & S. Watson, 6th edition, 2012, Brooks/Cole Chapter 5-7.
LECTURE 6. TRIGONOMETRY: PART I This lecture is the first part of reviewing high school trigonometry: Pythagorean theorem, the definitions of trigonometric and inverse trigonometric functions and some of their properties.
Trigonometry of Right Triangle Angle Measure: If a circle of radius 1 (so its circumference is 2π) with vertex of an angle at its center, then the measure of this angle in radians (abbreviated rad) is the length of the arc that subtends the angle. See where the second graph shows that the length s of an arc that subtends a central angle of θ radians in a circle of radius r is s = rθ
Trigonometry of Right Triangle RELATIONSHIP BETWEEN DEGREES AND RADIANS 180 = π rad, 1 rad = ( 180 π ) 1 = π 180 rad Examples 1. Express 30 in radians. 30 = 30 ( π 180 ) rad = π 6 rad. 2. Express 45 in radians. 45 = 45 ( π 180 ) rad = π 4 rad. 3. Express π 3 rad in degrees. π 3 rad = (π 3 ) (180 π ) = 60. A note on terminology: We often use a phrase such as a 30 degree angle to mean an angle whose measure is 30. Also for an angle θ, we write θ = 30 or θ = π/6 to mean the measure of θ is 30 or π/6rad. When no unit is given, the angle is assume to be in radians.
Area of a Circular Sector In a disc of radius r, the area of a sector with central angle of θ radians is A = πr 2 θ 2π = 1 2 πr 2.
Trigonometry of Right Triangle Pythagorean Theorem: In any right triangle, the square of the length of the hypotenuse equals the sum of the squares of the lengths of the two other sides. If the hypotenuse has length c, and the legs have lengths a and b, then c 2 = a 2 + b 2.
Trigonometric Function: Right Triangle Approach Trigonometric Ratios for 0 < θ < π 2 sin θ = b c cos θ = a c tan θ = b a csc θ = c b sec θ = c a cot θ = a b
Examples 1. Suppose 0 < θ < π 2 and cos θ = 1. Find the values of 3 sin θ, tan θ, cot θ, sec θ and csc θ. By we see that the length of the opposite side is 2 1 1 ( 3 ) = 2 2 2 = sin θ, tan θ = 3 4, cot θ = 2 2, csc θ = 3 2 and sec θ = 3 4
Examples 2. Suppose 0 < θ < π and tan θ = 2. Find the values of 2 sin θ, cos θ, cot θ, sec θ and csc θ. Use We see the length of the hypotenuse side is 5. Hence sin θ = 2 5, cos θ = 1 5, cot θ = 1 2, sec θ = 5 and csc θ = 5 2
Trigonometric Function: Unit Circle Approach The unit circle is the circle of radius 1 centered at the origin in the xy-plane. Its equation is x 2 + y 2 = 1. Suppose θ is a real number. Mark off a distance θ along the unit circle, starting at the point (1, 0) and moving in a counterclockwise direction if θ > 0 or in a clockwise direction if θ < 0. In this way we arrive at a point P(x, y) on the unit circle. The point P(x, y) obtained in this way is called the terminal point determined by the real number θ.
Trigonometric Function: Unit Circle Approach,
Trigonometric Function: Unit Circle Approach Trigonometric Functions for θ R sin θ = y cos θ = x tan θ = y (x 0) x csc θ = 1 y (y 0) sec θ = 1 x (x 0) cot θ = x (y 0) y Note that, when 0 < θ < π 2, a = x, b = y and c = 1, the unit circle approach is identical with the right triangle approach in the defining these trigonometric functions, but the domains of these trigonometric functions are larger using the unit circle approach.
Trigonometric Function: Special Triangles Special Triangles θ in θ in rad sin θ cos θ tan θ csc θ sec θ cot θ 0 0 0 1 0 1 30 π 1 3 3 2 3 2 3 6 2 2 3 3 45 π 2 2 1 2 2 1 4 2 2 60 π 3 1 2 3 3 3 2 3 2 2 3 3 90 π 1 0 1 0 2
Trigonometric Function: Special Triangles
Fundamental Identities of Trigonometric Function csc θ = 1 sin θ Reciprocal Identities sec θ = 1 cos θ cot θ = tan θ = sin θ cot θ = cos θ cos θ sin θ Pythagorean Identities 1 tan θ sin 2 θ + cos 2 θ = 1 tan 2 θ + 1 = sec 2 θ 1 + cot 2 θ = csc 2 θ
Fundamental Identities of Trigonometric Function Even-Odd Identities sin( θ) = sin θ cos( θ) = cos θ tan( θ) = tan θ csc( θ) = csc θ sec( θ) = sec θ cot( θ) = cot θ
Fundamental Identities of Trigonometric Function Cofunction Identities sin ( π 2 θ) = cos θ tan (π 2 θ) = cot θ sec (π θ) = csc θ 2 cos ( π 2 θ) = sin θ cot (π 2 θ) = tan θ csc (π θ) = secθ 2
Examples on Applications of Trigonometric Identities 1. Simplify tan θ + cos θ 1 + sin θ. tan θ + cos θ 1 + sin θ = sin θ cos θ + cos θ 1 + sin θ = sin θ(1 + sin θ) + cos2 θ cos θ(1 + sin θ) = sin θ + sin2 θ + cos 2 θ 1 + sin θ = cos θ(1 + sin θ) cos θ(1 + sin θ) = 1 cos θ = sec θ 2. Prove 1 sin θ 1 + sin θ = (sec θ tan θ)2. 1 sin θ 1 + sin θ = 1 sin θ 1 + sin θ 1 sin θ 2 (1 sin θ)2 sin θ = 1 sin θ 1 sin 2 = (1 θ cos θ ) = ( 1 cos θ sin θ cos θ ) 2 = (sec θ tan θ) 2.
Trigonometric Graphs A function f is periodic if there is a positive p such that f (θ + p) = f (θ) for every θ. The least such positive number is the period of f. If f has period p, then the graph of f on any interval of length p is called one complete period of f. Periodic Properties of Sine and Cosine The functions sine and cosine have period 2π: sin(θ + 2π) = sin θ cos(θ + 2π) = cos θ
Trigonometric Graphs Periodic Properties of tangent and Cotangent The functions tangent and cotangent have period π: tan(θ + π) = tan θ cot(θ + π) = cot θ
Trigonometric Graphs Periodic Properties of Secant and Cosecant The functions secant and cosecant have period 2π: sec(θ + 2π) = sec θ csc(θ + 2π) = csc θ
Examples 1. sin 2π 3 = sin (π 2 + π 6 ) = cos ( π 6 ) = cos π 6 = 3 2. 2. cos π = cos ( π 2 + π 2 ) = sin ( π 2 ) = sin π 2 = 1. 3. cot 5π 4 = cot [π 2 + (π 2 + π 4 )] = tan [ (π 2 + π 4 )] = tan ( π 2 + π 4 ) = cot ( π 4 ) = cot π 4 = 1. 4. tan 3π 2 = tan (2π π 2 ) = tan ( π 2 ) = tan π 2 = 0. 5. sec 5π 6 = sec (2π π 6 ) = sec ( π 6 ) = sec π 6 = 2 3 3. 6. csc 31π 6 = csc (5π + π 6 ) = csc (π + π 6 ) = sec [ (π 2 + π 6 )] = sec ( π 2 + π 6 ) = csc ( π 6 ) = csc π 6 = 1 sin π 6 = 1 1 2 = 2.
Domains and Ranges of Trigonometric Functions sin R [ 1, 1] cos R [ 1, 1] tan R {kπ + π 2 k Z} R cot R {kπ k Z} R sec R {kπ k Z} (, 1] [1, ) csc R {kπ + π k Z} (, 1] [1, ) 2
Signs of Trigonometric Functions Quadrant Positive Functions Negative Functions I all none II sin, csc cos, sec, tan, cot III tan, cot sin, csc, cos, sec IV cos, sec sin, csc, tan, cot
Inverse Trigonometric Functions The Inverse Sine Function is the function defined by sin 1 [ 1, 1] [ π 2, π 2 ] sin 1 x = y sin y = x The inverse sine function is also called arcsine, denoted by arcsin.
Inverse Trigonometric Functions Examples i sin 1 1 2 = π 6. 3 ii arcsin ( 2 ) = π 3. iii sin 1 2 =? Because 2 > 1, 2 is not in the domain of arcsin, so sin 1 2 is not defined. iv arcsin (sin π 4 ) = π 4. v sin 1 (sin 7π 6 ) = sin 1 ( 1 2 ) = π 6.
Inverse Trigonometric Functions The Inverse Cosine Function is the function defined by cos 1 [ 1, 1] [0, π] cos 1 x = y cos y = x The inverse cosine function is also called arccosine, denoted by arccos.
Inverse Trigonometric Functions The Inverse Tangent Function is the function defined by tan 1 R ( π 2, π 2 ) tan 1 x = y tan y = x The inverse tangent function is also called arctangent, denoted by arctan.
Inverse Trigonometric Functions The Inverse Cotangent Function is the function defined by cot 1 R (0, π) cot 1 x = y cot y = x The inverse cotangent function is also called arccotangent, denoted by arccot.
Inverse Trigonometric Functions The Inverse Secant Function is the function defined by sec 1 {x R x 1} [0, π 2 ) (π 2, π] sec 1 x = y sec y = x The inverse cosecant function is also called arcsecant, denoted by arcsec.
Inverse Trigonometric Functions The Inverse Cosecant Function is the function defined by csc 1 {x R x 1} [ π 2, 0) (0, π 2 ] csc 1 x = y csc y = x The inverse cosecant function is also called arccosecant, denoted by arccsc.
Inverse Trigonometric Functions Examples 1. sin 1 cos 2π 3 = sin 1 1 2 = π 6. 2. cos 1 tan ( π 4 ) = cos 1 ( 1) = π. 3. tan arcsin 1 2 = tan π 6 = 3 3. 4. csc cos 1 2 2 == csc π 4 = 2.