Slide 8- 1 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

The Trigonometric Functions Chapter 5

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 5.1 Trigonometric Functions of Acute Angles  Determine the six trigonometric ratios for a given acute angle of a right triangle.  Determine the trigonometric function values of 30°, 45°, and 60°.  Using a calculator, find function values for any acute angle, and given a function value of an acute angle, find the angle.  Given the function values of an acute angle, find the function values of its complement.

Slide 8- 4 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Trigonometric Ratios The figure illustrates how a right triangle is labeled with reference to a given acute angle, . The lengths of the sides of the triangle are used to define the six trigonometric ratios: sine (sin)cosecant (csc) cosine (cos)secant (sec) tangent (tan)cotangent (cot)

Slide 8- 5 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Sine and Cosine The sine of  is the length of the side opposite  divided by the length of the hypotenuse: The cosine of  is the length of the side adjacent to  divided by the length of the hypotenuse. Side Adjacent to  Side Opposite  Hypotenuse

Slide 8- 6 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Trigonometric Function Values of an Acute Angle 

Slide 8- 7 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Use the triangle shown to calculate the six trigonometric function values of . Solution:

Slide 8- 8 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Reciprocal Functions Reciprocal Relationships

Slide 8- 9 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Pythagorean Theorem The Pythagorean theorem may be used to find a missing side of a right triangle. This procedure can be combined with the reciprocal relationships to find the six trigonometric function values. 2 5 h

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example If the find the other five trigonometric function values of . Solution: Find the length of the hypotenuse. 2 5

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Function Values of 30  and 60  When the ratio of the opposite side to the hypotenuse is ½,  must have a measure of 30 . Using the Pythagorean theorem the missing side is The missing angle must have a measure of 60 . 2 1

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Function Values of 30  and 60  2 1

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Function Values of 45  The legs of this triangle must be equal, since they are opposite congruent angles. The hypotenuse is found by: 1 1 h

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Function Values of 45  continued 1 1

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Summary of Function Values

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example As a hot-air balloon began to rise, the ground crew drove 1.2 mi to an observation station. The initial observation from the station estimated the angle between the ground and the line of sight to the balloon to be 30 . Approximately how high was the balloon at that point? (We are assuming that the wind velocity was low and that the balloon rose vertically for the first few minutes.)

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example continued Solution: We begin with a drawing of the situation. We know the measure of an acute angle and the length of its adjacent side.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example continued Since we want to determine the length of the opposite side, we can use the tangent ratio, or the cotangent ratio. The balloon is approximately 0.7 mi, or 3696 ft, high.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Cofunctions and Complements The trigonometric function values for pairs of angles that are complements have a special relationship. They are called cofunctions.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Given that sin 40   , cos 40   , and tan 40   , find the six trigonometric function values of 50 .

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 5.2 Applications of Right Triangles  Solve right triangles.  Solve applied problems involving right triangles and trigonometric functions.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solving Right Triangles To solve a right triangle means to find the lengths of all sides and the measures of all angles. This can be done using right triangle trigonometry.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example In, find a, b, and B. Solution: 16.5 a b A B C B = 90   42  = 48 

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Definitions Angle of elevation: angle between the horizontal and a line of sight above the horizontal. Angle of depression: angle between the horizontal and a line of sight below the horizontal.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example To determine the height of a tree, a forester walks 100 feet from the base of the tree. From this point, he measures the angle of elevation to the top of the tree to be 47 . What is the height of the tree? 100 ft h

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Bearing Bearing is a method of giving directions. It involves acute angle measurements with reference to a north- south line.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example An airplane leaves the airport flying at a bearing of N32  W for 200 miles and lands. How far west of its starting point is the plane? The airplane is approximately 106 miles west of its starting point. w 200

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 5.3 Trigonometric Functions of Any Angle  Find angles that are coterminal with a given angle and find the complement and the supplement of a given angle.  Determine the six trigonometric function values for any angle in standard position when the coordinates of a point on the terminal side are given.  Find the function values for any angle whose terminal side lies on an axis.  Find the function values for an angle whose terminal side makes an angle of 30°, 45°, or 60° with the x-axis.  Use a calculator to find function values and angles.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Angle in Standard Position An angle formed by it’s initial side along the positive x-axis, with it’s vertex at the origin, and it’s terminal side placed at the end of the rotation.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Coterminal Angles Two or more angles that have the same terminal side. For example, angles of measure 60  and 420  are coterminal because they share the same terminal side. Example: Find two positive and two negative angles that are coterminal with 30 . 390 , 750 ,  330 , and  690  are coterminal with 30 .

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Trigonometric Functions of Any Angle 

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Find the six trigonometric function values for the angle shown: Solution: First, determine r. 2 4 r (2,4)

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example continued The six trigonometric functions values are:

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Another Example Given that and  is in the first quadrant, find the other function values. Solution: Sketch and label the angle. Find any missing sides. 1 2 r (2,1)

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Reference Angle The reference angle for an angle is the acute angle formed by the terminal side of the angle and the x-axis. The reference angle can be used when trying to find the trigonometric function values for angles that cover more than one quadrant. (ex. 210  )

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Find the sine, cosine, and tangent function values for 210 . Solution: Draw the angle. Note that there is a 30  angle in the third quadrant. Label the sides of the triangle with 1, and 2 as shown  30 

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example continued Notice that both the sine and cosine are negative because the angle measuring 210  is in the third quadrant  30 

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 5.4 Radians, Arc Length, and Angular Speed  Find points on the unit circle determined by real numbers.  Convert between radian and degree measure; find coterminal, complementary, and supplementary angles.  Find the length of an arc of a circle; find the measure of a central angle of a circle.  Convert between linear speed and angular speed.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Radian Measure An angle measures 1 radian when the angle intercepts an arc on a circle equal to the radius of the circle. 1 radian is approximately 57.3 .

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Converting between Degree and Radian Measure To convert from degree to radian measure, multiply by To convert from radian to degree measure, multiply by

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Convert each of the following to either radians or degrees. a) 150  b)  75  radians c) radiansd) 3 radians

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Radian Measure

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Find the length of an arc of a circle of radius 10 cm associated with an angle of radians.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Definitions Linear Speed: the distance traveled per unit of time, where s is the distance and t is the time. Angular Speed: the amount of rotation per unit of time, where  is the angle of rotation and t is the time.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Linear Speed in Terms of Angular Speed The linear speed v of a point a distance r from the center of rotation is given by v = r , where  is the angular speed in radians per unit of time.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Find the angle of revolution of a point on a circle of diameter 30 in. if the point moves 4 in. per second for 11 seconds. Since t = 11,  must be determined before we can solve for . The angle of revolution of the point is approximately 3 radians.

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 5.5 Circular Functions: Graphs and Properties  Given the coordinates of a point on the unit circle, find its reflections across the x-axis, the y-axis, and the origin.  Determine the six trigonometric function values for a real number when the coordinates of the point on the unit circle determined by that real number are given.  Find function values for any real number using a calculator.  Graph the six circular functions and state their properties.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Basic Circular Functions

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Find each of the following function values. a)b) c)d) Solutions: a) The coordinates of the point determined by are

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example continued b) The coordinates of the point determined by are c) The coordinates of the point determined by are d) The coordinates of the point determined by are

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Graphs of the Sine and Cosine Functions 1. Make a table of values. 2. Plot the points. 3. Connect the points with a smooth curve.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 1. Make a table of values.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example continued 2. Plot the points. 3. Connect the points with a smooth curve.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Domain and Range of Sine and Cosine Functions The domain of the sine and cosine functions is ( ,  ). The range of the sine and cosine functions is [  1, 1]. Periodic Function A function f is said to be periodic if there exists a positive constant p such that f(s + p) = f(s) for all s in the domain of f. The smallest such positive number p is called the period of the function.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Amplitude The amplitude of a periodic function is defined as one half of the distance between its maximum and minimum function values. The amplitude is always positive. The amplitude of y = sin x and y = cos x is 1.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Graph of y = tan s

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Graph of y = cot s

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Graph of y = csc s

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Graph of y = sec s

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 5.6 Graphs of Transformed Sine and Cosine Functions  Graph transformations of y = sin x and y = cos x in the form y = A sin (Bx – C) + D and y = A cos (Bx – C) + D and determine the amplitude, the period, and the phase shift.  Graph sums of functions.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Graphs of Transformed Sine and Cosine Functions: Vertical Translation y = sin x + D and y = cos x + D The constant D translates the graphs D units up if D > 0 or |D| units down if D < 0. Example: Sketch a graph of y = sin x  2.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Graphs of Transformed Sine and Cosine Functions: Amplitude y = A sin x and y = A cos x If |A| > 1, then there will be a vertical stretching by a factor of |A|. If |A| < 1, then there will be a vertical shrinking by a factor of |A|. If A < 0, the graph is also reflected across the x-axis.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Sketch a graph of y = 3 sin x. The sine graph (y = sin x) is stretched vertically by a factor of 3.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Graphs of Transformed Sine and Cosine Functions: Period y = sin Bx and y = cos Bx If |B| < 1, then there will be a horizontal stretching. If |B| > 1, then there will be a horizontal shrinking. If B < 0, the graph is also reflected across the y-axis. The period will be

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Sketch a graph of y = sin 2x. The sine graph (y = sin x) is shrunk horizontally. The period is

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Graphs of Transformed Sine and Cosine Functions: Horizontal Translation or Phase Shift y = sin (x  C) and y = cos (x  C) The constant C translates the graph horizontally |C| units to the right if C > 0 and |C| units to the left if C < 0.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Sketch the graph of y = sin (x +  ). The sine graph (y = sin x) is translated  units to the left.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Combined Transformations y = A sin (Bx  C) + D and y = A cos (Bx  C) + D The amplitude is |A|. The period is. The graph is translated vertically D units. The graph is translated horizontally C units.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Find the vertical shift, amplitude, period, and phase shift for the following function: y = 2 sin (4x  2  )  3. Solution: Write the function in standard form. |A| = |2| = 2 means the amplitude is 2 B = 4 means the period is C/B = means the phase shift is units to the right. D =  3 means the vertical shift is 3 units down.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example continued Sketch Amplitude = 2 Vertical shift = 3 down Phase shift = right Period = First, sketch y = sin 4x.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example continued Second, sketch y = 2 sin 4x. Third, sketch.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example continued Finally, sketch.