TOPIC TRIGONOMETRIC FUNCTIONS

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1 TOPIC. - TRIGONOMETRIC FUNCTIONS..: Angles and Their Measure..: Right Triangle Trigonometry..3: Computing the Values of Trigonometric Functions of Acute Angles..4: Trigonometric Functions of General Angles..5: Unit Circle Approach; Properties of the Trigonometric Functions..6: Graphs of the Sine and Cosine Functions..7: Graphs of the Tangent, Cotangent, Cosecant, and Secant Functions

2 ..: Angles and their measure Angles: An angle is formed by initial side and terminal side The common point for this sides is the vertex of the angle An angle is in standard position if: i) Its vertex is at the origin of a rectangle coordinate system ii) Its initial side lies along the positive x-axis Positive angles generated by counterclockwise rotation Negative angles generated by clockwise rotation An angle is called a quadrantal angle if its terminal side lies on the x-axis or the y-axis

3 Measuring angles using degrees One complete revolution = One quarter of a complete revolution = = one right angle One degree equals 60 minutes, i.e. One minute equals 60 seconds, i.e. = 60 = 60 Angles classified by their degree measurement:. a.acute b. right c.obtuse d.straight angle angle angle angle 3

4 Measuring angles using radians One complete revolution = π radians One radian is the angle subtended at the center of a circle by an arc of the circle equal in length to the radius of the circle. π = 80 c 80 c = π Example: Convert each angle in degrees to radians a. 60 b. 70 c Convert each angle in radians to degrees π radians 4 4π radians 3 a. b. c. 6 radians 4

5 ..: Right Triangle Trigonometry The six trigonometric function For any acute angle θ of a right angled triangle OAB (figure shown) Opposite sinθ = = Hypotenuse b c Adjacent cosθ = = Hypotenuse a c Opposite tanθ = = Adjacent b a cosecθ = secantθ = cot angentθ = sinθ cosθ tanθ 5

6 Fundamental identities Reciprocal identities sin θ = cscθ cos θ = secθ tan θ = cotθ csc θ = secθ = cotθ = sinθ cosθ tanθ Quotient identities sinθ tanθ = cotθ = cosθ cosθ sinθ Pythagorean identities + tan θ = sec θ + cot θ = csc θ sin θ + cos θ = 6

7 Trigonometric functions and complements Cofunction identities The value of a trigonometric function of of the complement of θ sinθ = cos(90 θ) cosθ = sin(90 θis equal to the cofunction θ) tanθ = cot(90 θ) cscθ = sec( 90 θ) secθ = csc( 90 θ) cotθ = tan(90 θ) Example: Find a cofunction with the same value as the given expression Π cot a) sin46 b) 7

8 Solving Right Triangles To solve a right triangle means to find the missing lengths of its sides and the measurements of its angles. Some general guidelines for solving right triangles:. Need to know an angle and a side,. or else two sides.. Then, make use of the Pythagorean Theorem and the fact that the sum of the angles of a triangle is 80 in a right triangle is 90, and the sum of the unknown angles c A b c =a +b A+B=90 a 8

9 Example: If b= and α = 40, find a,c, and β. c β 40 Solution: a 9

10 ..3: COMPUTING THE VALUES OF TRIGONOMETRIC FUNCTIONS OF ACUTE ANGLE We use isosceles triangle and equilateral triangle to find these special angles of 30 60, and 45 sin60 = 3 cos60 = tan60 = 3 sin30 = 3 cos30 = tan 30 = sin45 = = cos45 = tan45 = 3 0

11 ..4: TRIGONOMETRIC FUNCTIONS OF GENERAL ANGLES r x y x nd Quadrant st Quadrant 3 rd Quadrant 4 th Quadrant Definitions of trigonometric functions of any angle Let θ be any angle in standard position, and let P = (x,y) be a point on the terminal side of. If θ r = x + is the distance from (0,0) to (x,y), then the 6 trigonometric functions of are defined by the following ratios: sin θ = y r y x y cos θ = tan θ =, x 0 r x r r x csc θ =, y 0 sec θ =, x 0 cot θ =, y 0 y x y θ

12 Example:.Let P = (4, -3) be a point on the terminal side of θ. Find each of the six trigonometric functions of θ. Evaluate, if possible the cosine function and the cosecant function at the following 4 quadrantal angles a) θ = 0 b) θ = π c) θ = 80º d) θ = 3π **Quadrantal Angles: 0,90,80,70,360

13 The signs of the trigonometric function y sine All (sin, cos, tan) tangent cosine x If θ is not a quadrantal angle, the sign of a trigonometric function depends on the quadrant in which θ lies Example: Given tan θ = -/3 and cos θ < 0, find sin θ and sec θ 3

14 Coterminal Angles Two angles in standard position are said to be coterminal if they have the same terminal side. Example: For example, the angles 60 and 40 are coterminal, as are the angles -40 and 30. Note:. θ is coterminal with θ ± π k, k is any integer.. The trigonometric functions of coterminal angles are equal. Example: sinθ = sin( θ ± πk ) 4

15 Definition of a reference angle Let θ be a nonacute angle in standard position that lies in a quadrant. Its reference angle is the positive acute angle θ formed by the terminal side of θ and the x-axis Example:.Find the reference angle, θ for each of the following angles: a) θ = 0º b) θ = 7π 4 c) θ = -40º d) θ = 3.6. Use the reference angles to find the exact value of the following trigonometric functions: a) sin 300º b) 5 tan π c) 4 π sec 6 5

16 ..5: UNIT CIRCLE APPROACH; PROPERTIES OF THE TRIGONOMETRIC FUNCTIONS Definitions of the trigonometric functions in terms of a unit circle If t is a real number and P = (x,y) is a point on the unit circle that corresponds to t, then sin csc t = y y cos t = x tan t =, x 0 x x t =, y 0 sec t =, x 0 cot t =, y 0 y x y Example: Find the values of the trigonometric function at t = π 6

17 The domain and range of the sine and cosine functions The domain of the sine function and the cosine function is the set of all real numbers. The range of these functions is the set of all real numbers from - to, inclusive. Even and odd trigonometric functions The cosine and secant functions are even cos( t) = cos t sec( t) = sec t The sine, cosecant, tangent and cotangent functions are odd sin( t) = sin t csc( t) = csc t tan( t) = tan t cot( t) = cot t Example: Find the exact value of: a) cos( 60 ) b) π tan 6 7

18 Definition of a periodic function A function f is periodic if there exists a positive number p such that f ( t + p) = f ( t) for all t in the domain of f. The smallest number p for which f is periodic is called the period of f Periodic properties of the sine and cosine functions sin( t + π ) = sint and cos( t + π ) = cos t The sine and cosine functions are periodic functions and have period π Periodic properties of tangent and cotangent functions tan( t +π) = tant and cot( t + π ) = cot t The tangent and cotangent functions are periodic functions and have period π Repetitive behavior of the sine, cosine and tangent functions For any integer n and real number t, sin( t + πn ) = sint cos( t + πn) = cos t tan( t + πn) = tan t 8

19 ..6: Graphs of the Sine and Cosine Functions Characteristics of the Sine Function: Domain : all real numbers Range : y Period : π Symmetry through origin : sin( θ ) = sinθ Odd function x - intercepts :..., π, π, 0, π, π, 3π,... y - intercept : 0 max value :, occurs at 3π π 5π x =...,,,,... min value : -, occurs at π 3π 7π x =...,,,,... 9

20 Graphing variations of y=sin x Graph of y=a sin Bx.Identify the amplitude and the period Amplitude = A ; Period = π B. Find the values of x 3. Find the values of y for the one that we find in step 4. Connect all the points and extend to the left or right as desired Graph of y = A sin (Bx C) This graph is obtained by horizontally shifting the graph of y=a sin Bx so that the starting point of the cycle is shifted from x = 0 to C x = B This is called the phase shift C B If > 0 the shift is to the right C B If < 0 the shift is to the left 0

21 Example: - Determine the amplitude of y = 3sin x. Then graph y = sin x and y = 3sin x for 0 x π - Determine the amplitude of y = sin x. Then graph y = sin x and y = sin x for π x 3π 3- Determine the amplitude and period of y = sin x. Then graph the function for 0 x 8π 4- Determine the amplitude, period, and phase shift of y = 3sin( x π 3) Then graph one period of the function

22 Characteristics of the Cosine Function: Domain : all real numbers Range : y π Period : Symmetry about y-axis : Even function cos( θ ) = cosθ 3π π π 3π 5π x - intercepts :...,,,,,,... y - intercept : max value :, occurs at x =..., π, 0, π, 4π,... min value : -, occurs at x =..., π, π, 3π, 5π,...

23 Graphing variations of y=cos x Graph of y=a cos Bx.Identify the amplitude and the period Amplitude = A ; Period = π B. Find the values of x 3. Find the values of y for the one that we find in step 4. Connect all the points and extend to the left or right as desired Graph of y = A cos (Bx C) This graph is obtained by horizontally shifting the graph of y=a cos Bx so C that the starting point of the cycle is shifted from x = 0 to x = B This is called the phase shift C If > 0 the shift is to the right B C B If < 0 the shift is to the left 3

24 Example: Determine the amplitude and period of Then graph the function for x y = 4 cos πx Vertical shifts of sinusoidal graphs For y = A sin (Bx C) + D and y = A cos (Bx C) + D, the constant +D will cause the graph to shift upward while D will cause the graph to move downward. So, the max y is D + A and the min y is D - A Example: Graph one period of the function y = cos x + 4

25 ..7: GRAPHS OF THE TANGENT, COTANGENT, COSECANT, AND SECANT FUNCTIONS Characteristics of the Tangent Function: Domain : all real numbers except odd multiples Range : all real numbers Period : π Symmetry with respect to the origin : tan( θ) = tanθ x - intercepts : y - intercept : 0 Vertical asymptotes : π Odd function x =..., π, π, 0, π, π, 3π,... 3π π π 3π x =...,,,,,... 5

26 Characteristics of the Cotangent Function π Domain : all real numbers except integral multiples of Range : all real numbers Period : π Symmetry with respect to the origin : cot( θ ) = cotθ Odd function x - intercepts : x =..., π, π,0, π, π... y - intercept : none Vertical asymptotes : x =..., π,0, π,π... 6

27 Characteristics of the Cosecant Function: Domain : all real numbers except integral multiples of π Range : all real numbers of y such that y or y Period : π Symmetry with respect to the origin : csc( θ ) = cscθ Odd function x - intercepts : none y - intercept : none Vertical asymptotes : x =..., π,0, π, π... 7

28 Characteristics of the Secant Function: Domain : all real numbers except odd multiples of Range : all real numbers of y such that y or y Period : π Symmetry with respect to y-axis: x - intercepts : none y - intercept : Vertical asymptotes : sec( θ ) = secθ 3π π π 3π x =...,,,,,... π Even function 8

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