Chapter 1 Trigonometric Functions

1.1 Angles

Basic Terms Two distinct points determine a line called line AB. Line segment AB—a portion of the line between A and B, including points A and B. Ray AB—portion of line AB that starts at A and continues through B, and on past B. A B A B A B

Basic Terms continued Angle-formed by rotating a ray around its endpoint. The ray in its initial position is called the initial side of the angle. The ray in its location after the rotation is the terminal side of the angle.

Naming Angles Unless it is ambiguous as to the meaning, angles may be named only by a single letter (English or Greek) displayed at vertex or in area of rotation between initial and terminal sides Angles may also be named by three letters, one representing a point on the initial side, one representing the vertex and one representing a point on the terminal side (vertex letter in the middle, others first or last)

Basic Terms continued Positive angle: The rotation of the terminal side of an angle counterclockwise. Negative angle: The rotation of the terminal side is clockwise.

Angle Measures and Types of Angles The most common unit for measuring angles is the degree. (One rotation = 360o) ¼ rotation = 90o, ½ rotation = 180o, Angle and measure of angle not the same, but it is common to say that an angle = its measure Types of angles named on basis of measure:

Complementary and Supplementary Angles Two positive angles are called complementary if the sum of their measures is 90o The angle that is complementary to 43o = Two positive angles are called supplementary if the sum of their measures is 180o The angle that is supplementary to 68o =

Example: Complementary Angles Find the measure of each angle. Since the two angles form a right angle, they are complementary angles. Thus, k  16 k +20 The two angles have measures of: 43 + 20 = 63 and 43  16 = 27

Example: Supplementary Angles Find the measure of each angle. Since the two angles form a straight angle, they are supplementary angles. Thus, 6x + 7 3x + 2 These angle measures are: 6(19) + 7 = 121 and 3(19) + 2 = 59

Portions of Degree: Minutes, Seconds One minute, 1’, is 1/60 of a degree. One second, 1”, is 1/60 of a minute.

Example: Calculations Perform the calculation. Since 86 = 60 + 26, the sum is written: Perform the calculation. Hint write:

Converting Between Degrees, Minutes and Seconds and Decimal Degrees

Standard Position An angle is in standard position if its vertex is at the origin and its initial side is along the positive x-axis.

Quadrantal Angles Angles in standard position having their terminal sides along the x-axis or y-axis, such as angles with measures 90, 180, 270, and so on, are called quadrantal angles.

Coterminal Angles A complete rotation of a ray results in an angle measuring 360. Given angle A, and continuing the rotation by a multiple of 360 will result in a different angle, A + n360,with the same terminal side: coterminal angles.

Example: Coterminal Angles Find the angles of smallest possible positive measure coterminal with each angle. a) 1115 b) 187 Add or subtract 360 as may times as needed to obtain an angle with measure greater than 0 but less than 360. a) b)

Homework 1.1 Page 6 All: 6 – 9, 14 – 17, 24 – 29, 32 – 35, 38 – 41, 46 – 51, 55 – 58 , 75 – 79 MyMathLab Assignment 1 for practice MyMathLab Homework Quiz 1 will be due for a grade on the date of our next class meeting!!!

1.2 Angle Relationships and Similar Triangles

Vertical Angles When lines intersect, angles opposite each other are called vertical angles Vertical angles in this picture: How do measures of vertical angles compare? Vertical Angles have equal measures. M Q R P N

Parallel Lines Parallel lines are lines that lie in the same plane and do not intersect. When a line q intersects two parallel lines, q, is called a transversal. m n parallel lines q Transversal

Angles and Relationships m n q Angle measures are equal. 2 & 6, 1 & 5, 3 & 7, 4 & 8 Corresponding angles Angle measures add to 180. 4 and 6 3 and 5 Interior angles on the same side of the transversal 1 and 8 2 and 7 Alternate exterior angles Angles measures are equal. 4 and 5 3 and 6 Alternate interior angles Rule Angles Name

Example: Finding Angle Measures Find the measure of each marked angle, given that lines m and n are parallel. What is the relationship between these angles? Alternate exterior with equal measures Measure of each angle? One angle has measure 6x + 4 = 6(21) + 4 = 130 and the other has measure 10x  80 = 10(21)  80 = 130 m n (10x  80) (6x + 4)

Angle Sum of a Triangle The instructor will ask specified students to draw three triangles of distinctly different shapes. All the angles will be cut off each triangle and placed side by side with vertices touching. What do you notice when you sum the three angles? The sum of the measures of the angles of any triangle is 180.

Example: Applying the Angle Sum The measures of two of the angles of a triangle are 52 and 65. Find the measure of the third angle, x. Solution? The third angle of the triangle measures 63. 52 65 x

Types of Triangles: Named Based on Angles

Types of Triangles: Named Based on Sides

Similar and Congruent Triangles Triangles that have exactly the same shape, but not necessarily the same size are similar triangles Triangles that have exactly the same shape and the same size are called congruent triangles

Conditions for Similar Triangles Corresponding angles must have the same measure. Corresponding sides must be proportional. (That is, their ratios must be equal.)

Example: Finding Angle Measures on Similar Triangles Triangles ABC and DEF are similar. Find the measures of angles D and E. Since the triangles are similar, corresponding angles have the same measure. Angle D corresponds to angle: Measure of D: Angle E corresponds to angle: Measure of E: A C B F E D 35 112 33

Example: Finding Side Lengths on Similar Triangles Triangles ABC and DEF are similar. Find the lengths of the unknown sides in triangle DEF. To find side DE: To find side FE: A C B F E D 35 112 33 32 48 64 16

Example: Application of Similar Triangles A lighthouse casts a shadow 64 m long. At the same time, the shadow cast by a mailbox 3 m high is 4 m long. Find the height of the lighthouse. The two triangles are similar, so corresponding sides are in proportion, so: The lighthouse is 48 m high. 64 4 3 x

Homework 1.2 Page 14 All: 3 – 7, 9 – 13, 16 – 19, 25 – 36, 41 – 44, 46 – 49, 51 – 54, 57 – 60, 65 – 66, 69 – 70 MyMathLab Assignment 2 for practice MyMathLab Homework Quiz 2 will be due for a grade on the date of our next class meeting!!!

1.3 Trigonometric Functions

Trigonometric Functions Compared with Algebraic Functions Algebraic functions are sets of ordered pairs of real numbers such that every first member, “x”, is paired with exactly one second member, “y” Trigonometric functions are sets of ordered pairs such that every first member, an angle, is paired with exactly one second member, a ratio of real numbers Algebraic functions are given names like f, g or h and in function notation, the second member that is paired with “x” is shown as f(x), g(x) or h(x) Trigonometric functions are given the names, sine, cosine, tangent, cotangent, secant, or cosecant, and in function notation, the second member that is paired with the angle “A” is shown as sin(A), cos(A), tan(A), cot(A), sec(A), or csc(A) – (sometimes parentheses are omitted)

Trigonometric Functions Let (x, y) be a point other the origin on the terminal side of an angle  in standard position. The distance, r, from the point to the origin is: The six trigonometric functions of  are defined as:

Values of Trig Functions Independent of Point Chosen For the given angle, if point (x1,y1) is picked and r1 is calculated, trig functions of that angle will be ratios of the sides of the triangle shown in blue. For the same angle, if point (x2,y2) is picked and r2 is calculated, trig functions of the angle will be ratios of the triangle shown in green Since the triangles are similar, ratios and trig function values will be exactly the same

Example: Finding Function Values The terminal side of angle  in standard position passes through the point (12, 16). Find the values of the six trigonometric functions of angle . (12, 16) 16 12 

Example: Finding Function Values continued x = 12 y = 16 r = 20

Trigonometric Functions of Coterminal Angles Note: To calculate trigonometric functions of an angle in standard position it is only necessary to know one point on the terminal side of that angle, and its distance from the origin In the previous example six trig functions of the given angle were calculated. All angles coterminal with that angle will have identical trig function values ALL COTERMINAL ANGLES HAVE IDENTICAL TRIGONOMETRIC FUNCTION VALUES!!!!

Equations of Rays with Endpoint at Origin: Recall from algebra that the equation of a line is: If a line goes through the origin its equation is: To get the equation of a ray with endpoint at the origin we write an equation of this form with the restriction that:

Example: Finding Function Values Find the six trigonometric function values of the angle  in standard position, if the terminal side of  is defined by x + 2y = 0, x  0. We can use any point on the terminal side of  to find the trigonometric function values.

Example: Finding Function Values continued From previous calculations: Use the definitions of the trig functions:

Finding Trigonometric Functions of Quadrantal Angles A point on the terminal side of a quadrantal angle always has either x = 0 or y = 0 (x = 0 when terminal side is on y axis, y = 0 when terminal side is on x axis) Since any point on the terminal side can be picked, choose x = 0 or y = 0, as appropriate, and choose r = 1 The remaining x or y will then be 1 or -1

Example: Function Values Quadrantal Angles Find the values of the six trigonometric functions for an angle of 270. Which point should be used on the terminal side of a 270 angle? We choose (0, 1). Here x = 0, y = 1 and r = 1. Value of the six trig functions for this angle:

Undefined Function Values If the terminal side of a quadrantal angle lies along the y-axis, then, because x = 0, the tangent and secant functions are undefined: If it lies along the x-axis, then, because y = 0, the cotangent and cosecant functions are undefined.

Commonly Used Function Values undefined 1 360 1 270 180 90 0 csc  sec  cot  tan  cos  sin  

Finding Trigonometric Functions of Specific Angles Until discussing trigonometric functions of specific quadrantal angles such as 90o, 180o, etc., we have found trigonometric functions of angles by knowing or finding some point on the terminal side of the angle without knowing the measure of the angle At the present time, we know how to find exact trigonometric values of specific angles only if they are quadrantal angles In the next chapter we will learn to find exact trigonometric values of 30o, 45o, and 60o angles In the meantime, we can find approximate trigonometric values of specific angles by using a scientific calculator set in degree mode

Finding Approximate Trigonometric Function Values of Sine, Cosine and Tangent Make sure your calculator is set in degree mode Depending on your calculator, Enter the angle measure first then press the appropriate sin, cos or tan key to get the value Press the sin, cos, or tan key first, then enter the angle measure Practice on these:

Exponential Notation and Trigonometric Functions A trigonometric function defines a real number ratio for a specific angle, for example “sin A” is the real number ratio assigned by the sine function to the angle “A” Since “sin A” is a real number it can be raised to any rational number power, such as “2” in which case we would have “(sin A)2” However, this value is more commonly written as “sin2 A” sin2 A = (sin A)2 Using this reasoning then if “tan A = 3”, then: tan4 A =

Homework 1.3 Page 24 All: 5 – 8, 17 – 28, 33 – 40 MyMathLab Assignment 3 for practice MyMathLab Homework Quiz 3 will be due for a grade on the date of our next class meeting!!!

1.4 Using Definitions of the Trigonometric Functions

Identities Recall from algebra that an identity is an equation that is true for all values of the variable for which the expression is defined Examples:

Relationships Between Trigonometric Functions In reviewing the definitions of the six trigonometric functions what relationship do you observe between each function and the one directly beneath it? They are reciprocals of each other

Reciprocal Identities This relationship can be summarized: Each identity is true for angles except those that that make a denominator equal to zero These reciprocal identities must be memorized

Example: Find each function value. cos  if sec  = Since cos  is the reciprocal of sec  : sin  if csc 

Signs of Trig Functions by Quadrant of Angle Considering the following three functions and the sign of x, y and r in each quadrant, which functions are positive in each quadrant?

Signs of Other Trig Functions by Quadrant of Angle Reciprocal functions will always have the same sign All functions have positive values for angles in Quadrant I Sine and Cosecant have positive values for angles in Quadrant II Tangent and Cotangent have positive values for angles in Quadrant III Cosine and Secant have positive values for angles in Quadrant IV

Memorizing Signs of Trig Functions by Quadrant It will help to memorize by learning these words in Quadrants I - IV: “All students take calculus” And remembering reciprocal identities Trig functions are negative in quadrants where they are not positive

Example: Identify Quadrant Identify the quadrant (or quadrants) of any angle  that satisfies tan  > 0, sin  < 0. tan  > 0 in quadrants: I and III sin  < 0 in quadrants: III and IV so, the answer satisfying both is quadrant: III

Domain and Range of Sine Function Given an angle A in standard position, and (x,y) a point on the terminal side a distance of r > 0 from the origin, sin A = y/r Domain of sine function is the set of all A for which y/r is a real number. Since r can’t be zero, y/r is always a real number and domain is “any angle” Range of sine function is the set of all y/r, but since y is less than or equal to r, this ratio will always be equal to 1 or will be a proper fraction, positive or negative:

Domain and Range of Cosine Function Given an angle A in standard position, and (x,y) a point on the terminal side a distance of r > 0 from the origin, cos A = x/r Domain of cosine function is the set of all A for which x/r is a real number. Since r can’t be zero, x/r is always a real number and domain is “any angle” Range of cosine function is the set of all x/r, but since x is less than or equal to r, this ratio will always be equal to 1, -1 or will be a proper fraction, positive or negative:

Domain and Range of Sine & Cosine What relationship do you notice between the domain and range of the sine and cosine functions? They are exactly the same: Domain: Range:

Domain and Range of Tangent Function Given an angle A in standard position, and (x,y) a point on the terminal side a distance of r > 0 from the origin, tan A = y/x Domain of tangent function is the set of all A for which y/x is a real number. Tangent will be undefined when x = 0, therefore domain is all angles except for odd multiples of 90o Range of tangent function is the set of all y/x, but since all of these are possible: x=y, x<y, x>y, this ratio can be any positive or negative real number:

Domain and Range of Cosecant Function Given an angle A in standard position, and (x,y) a point on the terminal side a distance of r > 0 from the origin, csc A = r/y Domain of cosecant function is the set of all A for which r/y is a real number. Cosecant will be undefined when y = 0, therefore domain is all angles except for integer multiples of 180o Range of cosecant function is the reciprocal of the range of the sine function. Reciprocals of numbers between -1 and 1 are:

Domain and Range of Secant Function Given an angle A in standard position, and (x,y) a point on the terminal side a distance of r > 0 from the origin, sec A = r/x Domain of secant function is the set of all A for which r/x is a real number. Secant will be undefined when x = 0, therefore domain is all angles except for odd multiples of 90o Range of secant function is the reciprocal of the range of the cosine function. Reciprocals of numbers between -1 and 1 are:

Domain and Range of Cotangent Function Given an angle A in standard position, and (x,y) a point on the terminal side a distance of r > 0 from the origin, cot A = x/y Domain of cotangent function is the set of all A for which x/y is a real number. Cotangent will be undefined when y = 0, therefore domain is all angles except for integer multiples of 180o Range of cotangent function is the reciprocal of the range of the tangent function. The reciprocal of the set of numbers between negative infinity and positive infinity is:

Ranges of Trigonometric Functions For any angle  for which the indicated functions exist: 1  sin   1 and 1  cos   1 tan  and cot  can equal any real number; sec   1 or sec   1 csc   1 or csc   1. (Notice that sec  and csc  are never between 1 and 1.)

Deciding Whether a Value is in the Range of a Trigonometric Function Tell which of the following is in the range of the trig function: sin A = 1.332 cos A = ¼ tan A = 1,998,214 sec A = ½ csc A = 0.2485 cot A = 0 sin A = - 0.3359 cos A = -3 tan A = -3

Development of Pythagorean Identities For every point (x,y) on the terminal side of an angle A at a distance of r > 0 from the origin, we have the following relationship based on the Pythagorean Theorem: Dividing both sides by r2 gives:

Development of Pythagorean Identities For every point (x,y) on the terminal side of an angle A at a distance of r > 0 from the origin, we have the following relationship based on the Pythagorean Theorem: Dividing both sides by x2 gives:

Development of Pythagorean Identities For every point (x,y) on the terminal side of an angle A at a distance of r > 0 from the origin, we have the following relationship based on the Pythagorean Theorem: Dividing both sides by y2 gives:

Pythagorean Identities MUST MEMORIZE!!!

Development of Quotient Identities Based on x, y, r definitions of sine and cosine functions:

Development of Quotient Identities Based on x, y, r definitions of sine and cosine functions:

Quotient Identities MUST MEMORIZE!!!

Using Identities to Find Missing Function Values Given the quadrant of the angle and the value of one trig function, the other five trig function values can be found using various identities Examples that follow will illustrate the approach

Example: Other Function Values Find sin and cos given that tan  = 4/3 and  is in quadrant III. Since  is in quadrant III, sin and cos will both be negative. sin and cos must be in the interval [1, 1].

Example: Other Function Values continued There is no identity that directly gives sin or cos from tan, but which one will give a reciprocal of sin or cos from tan?

Solving Trigonometric Equations In algebra there are many types of equations that involve a variable that are either true or false depending on the value of the variable This equation is true only if x = 10, so we say that 10 is the solution to the equation In trig we likewise have many types of equations that involve a variable representing an unknown angle that are true or false depending on the value of the variable In this course we will develop methods for solving various types of trigonometric equations

Using Identities to Find a Value of an Angle that Solves a Trigonometric Equation Given a trigonometric equation with an unknown angle, one solution (not all) can be found by using identities to convert both sides to the same trig function and then setting the unknown angles equal to each other as shown in the following example:

Find One Solution:

Homework 1.4 Page 33 All: 3 – 6, 9 – 10, 15 – 18, 21 – 24, 27 – 40, 47 – 54, 56 – 61, 65 – 70 MyMathLab Assignment 4 for practice MyMathLab Homework Quiz 4 will be due for a grade on the date of our next class meeting!!!