Trigonometric Functions

Trigonometric Functions Chapter 5 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAA

Angles and Their Measure Section 5.1

Basic Terminology • Ray: A half-line starting at a vertexV • Angle: Two rays with a common vertex

Basic Terminology • Initial side and terminal side: The rays in an angle • Angle shows direction and amount of rotation • Lower-case Greek letters denote angles

Basic Terminology • Positive angle: Counterclockwise rotation • Negative angle: Clockwise rotation • Coterminal angles: Share initial and terminal sides Positive angle Negative angle Positive angle

Basic Terminology • Standard position: • Vertex at origin • Initial side is positive x-axis

Basic Terminology • Quadrantal angle: Angle in standard position that doesn’t lie in any quadrant Lies in quadrant II Quadrantal angle Lies in quadrant IV

Measuring Angles • Two usual ways of measuring • Degrees • 360± in one rotation • Radians • 2¼ radians in one rotation

Measuring Angles • Right angle: A quarter revolution • A right angle contains • 90± • radians

Measuring Angles • Straight angle: A half revolution. • A straight angle contains: • 180± • ¼ radians

Measuring Angles • Negative angles have negative measure • Multiple revolutions are allowed

Degrees, Minutes and Seconds • One complete revolution = 360± • One minute: • One-sixtieth of a degree • One minute is denoted 10 • 1± = 600 • One second: • One-sixtieth of a minute • One second is denoted 100 • 10 = 6000

Degrees, Minutes and Seconds • Example. Convert to a decimal in degrees Problem: 64±3502700 Answer: • Example. Convert to the D±M0S00 form Problem: 73.582± Answer:

Radians • Central angle: An angle whose vertex is at the center of a circle • Central angles subtend an arc on the circle

Radians • One radian is the measure of an angle which subtends an arc with length equal to the radius of the circle

Radians IMPORTANT! • Radians are dimensionless • If an angle appears with no units, it must be assumed to be in radians

Arc Length • Theorem. [Arc Length] For a circle of radius r, a central angle of µ radians subtends an arc whose length s is s = rµ WARNING! • The angle must be given in radians

Arc Length • Example. Problem: Find the length of the arc of a circle of radius 5 centimeters subtended by a central angle of 1.4 radians Answer:

Radians vs. Degrees • 1 revolution = 2¼ radians = 360± • 180± =¼radians • 1± = radians • 1 radian =

Radians vs. Degrees • Example. Convert each angle in degrees to radians and each angle in radians to degrees (a) Problem: 45± Answer: (b) Problem: {270± Answer: (c) Problem: 2 radians Answer:

Radians vs. Degrees • Measurements of common angles

Area of a Sector of a Circle • Theorem. [Area of a Sector] The area A of the sector of a circle of radius r formed by a central angle of µ radians is

Area of a Sector of a Circle • Example. Problem: Find the area of the sector of a circle of radius 3 meters formed by an angle of 45±. Round your answer to two decimal places. Answer: WARNING! • The angle again must be given in radians

Linear and Angular Speed • Object moving around a circle or radius r at a constant speed • Linear speed: Distance traveled divided by elapsed time t = time µ = central angle swept out in time t s = rµ = arc length = distance traveled

Linear and Angular Speed • Object moving around a circle or radius r at a constant speed • Angular speed: Angle swept out divided by elapsed time • Linear and angular speeds are related v = r!

Linear and Angular Speed • Example. A neighborhood carnival has a Ferris wheel whose radius is 50 feet. You measure the time it takes for one revolution to be 90 seconds. (a) Problem: What is the linear speed (in feet per second) of this Ferris wheel? Answer: (b) Problem: What is the angular speed (in radians per second)? Answer:

Key Points • Basic Terminology • Measuring Angles • Degrees, Minutes and Seconds • Radians • Arc Length • Radians vs. Degrees • Area of a Sector of a Circle • Linear and Angular Speed

Trigonometric Functions: Unit Circle Approach Section 5.2

Unit Circle • Unit circle: Circle with radius 1 centered at the origin • Equation: x2 + y2 = 1 • Circumference: 2¼

Unit Circle • Travel t units around circle, starting from the point (1,0), ending at the point P = (x, y) • The point P = (x, y) is used to define the trigonometric functions of t

Trigonometric Functions • Let t be a real number and P = (x, y) the point on the unit circle corresponding to t: • Sine function: y-coordinate of P sin t = y • Cosine function: x-coordinate of P cos t = x • Tangent function: if x 0

Trigonometric Functions • Let t be a real number and P = (x, y) the point on the unit circle corresponding to t: • Cosecant function: if y 0 • Secant function: if x 0 • Cotangent function: if y 0

Exact Values Using Points on the Circle • A point on the unit circle will satisfy the equation x2 + y2 = 1 • Use this information together with the definitions of the trigonometric functions.

Exact Values Using Points on the Circle • Example. Let t be a real number and P = the point on the unit circle that corresponds to t. Problem: Find the values of sin t, cos t, tan t, csc t, sec t and cot t Answer:

Trigonometric Functions of Angles • Convert between arc length and angles on unit circle • Use angle µ to define trigonometric functions of the angleµ

Exact Values for Quadrantal Angles • Quadrantal angles correspond to integer multiples of 90± or of radians

Exact Values for Quadrantal Angles • Example. Find the values of the trigonometric functions of µ Problem:µ = 0 = 0± Answer:

Exact Values for Quadrantal Angles • Example. Find the values of the trigonometric functions of µ Problem:µ = = 90± Answer:

Exact Values for Quadrantal Angles • Example. Find the values of the trigonometric functions of µ Problem:µ = ¼ = 180± Answer:

Exact Values for Quadrantal Angles • Example. Find the values of the trigonometric functions of µ Problem:µ = = 270± Answer:

Exact Values for Quadrantal Angles

Exact Values for Quadrantal Angles • Example. Find the exact values of (a) Problem: sin({90±) Answer: (b) Problem: cos(5¼) Answer:

Exact Values for Standard Angles • Example. Find the values of the trigonometric functions of µ Problem:µ = = 45± Answer:

Exact Values for Standard Angles

Exact Values for Standard Angles • Example. Find the values of the following expressions (a) Problem: sin(315±) Answer: (b) Problem: cos({120±) Answer: (c) Problem: Answer:

Approximating Values Using a Calculator IMPORTANT! • Be sure that your calculator is in the correct mode. • Use the basic trigonometric facts:

Approximating Values Using a Calculator • Example. Use a calculator to find the approximate values of the following. Express your answers rounded to two decimal places. (a) Problem: sin 57± Answer: (b) Problem: cot {153± Answer: (c) Problem: sec 2 Answer:

Circles of Radius r • Theorem. For an angle µ in standard position, let P = (x, y) be the point on the terminal side of µ that is also on the circle x2 + y2 = r2. Then

Trigonometric Functions