Section. (Introduction) Graphs of Trig Functions Objectives:. To graph basic trig functions using t-bar method. A. Sine and Cosecant. y = sinθ y y y y 0 --- --- 80 --- --- 30 0 0 300 5 35 5 35 60 50 0 330 90 --- --- 70 360 360 35 70 5 80 35 90 5 5 90 35 80 5 70 35 360. y = cscθ 6 360 35 70 5 80 35 90 5 5 90 35 80 5 70 35 360 6 B. Cosine and Secant. y = cos x x y x y x y x y 0 --- --- --- --- 6 3 3 5 3 3 5 7 3 5 6 7 6 6 --- --- 3 3/ / / 3/. y = secθ 3/ / / 3/ Honors Algebra Chapter Page
C. Tangent and Cotangent. y = tan x x y x y x y x y 0 --- --- --- --- 6 3 3 5 3 3 5 7 3 5 6 7 6 6 --- --- 3 3/ / / 3/ D. y = cot x 3/ / / 3/ Honors Algebra Chapter Page
Section. Graphing Trigonometric Functions Objectives:. To find the amplitude and period for a trigonometric function.. To write equations of trigonometric functions given amplitude and period. I. Graphing Equations A. Forms:.. B. What does A affect?. Graph: a) y = 3sinθ b) y = 5cosθ. Amplitude ( A ) Distance from. 3. Example: a) Graph manually: y =.5sinθ 360 80 80 360 b) What is the amplitude of y = 7 cos x?. How does A affect the graph? a) Graph: y = 3cos x and y = 3cos x b) Graph: y = 3sin x and y = 3sin x c) Graph manually: y = 7 cos x 8 6 6 8 Honors Algebra Chapter Page 3
C. What does B affect?. Graph: a) y = sin 3x b) y = cos( x). ( B ) -- of in 360 or. 3. Period to complete cycle period = or (Note: tangent is in 80 or.). Examples: a) Graph: y = cos x (Note: When graphing, break your cycle into equal parts) b) Graph: y = tan θ 5. How does B affect the graph? a) Graph: y = cos( 3 x) and y = cos 3x b) Graph: y = sin( 3 x) and y = sin 3x 6. Graph manually: y = cos θ Honors Algebra Chapter Page
D. Doing Both Amplitude and Period. Graph manually: y = 5cos3x. Graph manually: y = sin x 3. Graph manually: y = sin θ 3 II. Writing Equations: Examples:. Amplitude = and Period = 90. The motion of a prong of a tuning fork can be described by a modified trig function. Represent displacement of a prong from the rest positions to the right as positive and displacement to the left as negative. At time zero, a prong reaches its maximum displacement, 0.03-cm to the right. The fork has a frequency of 00 cycles per second. a) Write a function that represents the position of the prong, y, in reference to its rest position, in terms of the time, t, measured from t = 0 seconds. b) Find y, when t = 0.0 seconds Homework: p.766-3 all, 5-33 odds, 35-37 all, 39- all, 3-50 all Honors Algebra Chapter Page 5
Section. Translations of Trigonometric Graphs Objectives:. To find the phase shift and translations for a trigonometric function.. To graph trig functions with phase shifts and translations 3. To write equations of trigonometric functions given phase shifts and translations. I. Graphing Equations A. Forms:.. B. What does D affect?. Graph: a) y = sin( θ 60 ) b) y = cos( x + ). Phase Shift (D) _ the vertical axis or. 3. How does -D affect the graph? Graph: y = cosx and cos y = x + a) b). Examples: a) Graph manually: y = sin( θ 5 ) -630-50 -50-360 -70-80 -90 90 80 70 360 50 50 630 7 - b) What is the phase shift of y = cos( θ + 80 )? C. What does C affect?. Graph: a) y = 3+ sin x b) y = + cos x. Translation (C) translating or. 3. How does -C affect the graph? Graph: y = 3+ cos x and y = 3+ cos x a) b) - Honors Algebra Chapter Page 6
. Examples: a) Graph manually: y = + cos x -630-50 -50-50 -50-360 -70-360 -80-70 -90-8090-90 80 70 360 90 5080 50 70 630 7360 50 50 630 7 - - - - b) What is the translation of y = 5 sinθ? D. Doing Both Phase Shift and Translation. Graph manually: y = 5 + cos( x ). Graph manually: y = + sin( θ + 5 ) E. Basic Type of Transformations Review. Original Graph: y = C + Asin B( x D) Rewrite using r. Horizontal shift r units to the right: 3. Horizontal shift r units to the left:. Vertical shift r units to the down: 5. Vertical shift r units to the up: 6. Reflection (about the x-axis): 7. Reflection (about the y-axis): F. Doing it all. Graph manually: y = 5 3cos ( x ). Graph manually: y = + 6sin θ + 5 Honors Algebra Chapter Page 7
3. Graph manually: y = + 6csc θ + 5. Graph manually: y = 5 3tan ( x ) II. Writing Equations: Examples:. Amplitude =, Period = 90, phase shift = 5 and translation =. As you ride a Ferris wheel, the height that you are above the ground varies periodically. Consider the height of the center of the wheel to be the equilibrium point. A particular wheel has a diameter of 38-ft., travels at a rate of revolutions per minute, and the seats of the Ferris wheel clear the ground by 3-ft. a) Write an equation to describe the change in height, h, of the seat that was filled last before the ride began in terms of time, t, in seconds. b) Find the height of the seat after -s. Homework: worksheets and p. 775-6 all Honors Algebra Chapter Page 8
Honors Algebra Chapter Worksheet Graphing Sinusoidal Functions in Radian Mode Find the amplitude, period, phase (horizontal) displacement and translation (vertical displacement). Then use the information to find the critical points and sketch two cycles for the graph (one to the right and one to the left of the center point)... y = 3+ cos ( x ) 5 Amplitude: Translation: y = + 5sin x + 3 Amplitude: Translation: 3. y = 6cos3x + 6 Amplitude: Translation:. + 6sin ( ) x Amplitude: Translation: 5. y = 5 sin ( x + ) 3 Amplitude: Translation: Honors Algebra Chapter Page 9
Honors Algebra Chapter Worksheet Graphing Sinusoidal Functions in Degree Mode Find the amplitude, period, phase (horizontal) displacement and translation (vertical displacement). Then use the information to find the critical points and sketch two cycles for the graph (one to the right and one to the left of the center point). 6. y = 7 + cos3( θ 0 ) Amplitude: Translation: 7. y = 0 + 0sin ( θ + 30 ) 8. 9. Amplitude: Translation: y = 3 5cos ( θ + 90 ) Amplitude: Translation: 000 + 3000 sin ( 60 ) 3 θ + Amplitude: Translation: 0. y = 6 sin( θ 7 ) Amplitude: Translation: Honors Algebra Chapter Page 0
Advanced Algebra Chapter Worksheet Graphing Tangent and Reciprocal Functions Graph the following: Assume the function is circular (use radians) if the independent variable is x and trigonometric (use degrees) if the independent variable is θ. Graph two cycles of each (one to the right and left of the center point).. y = tan θ 9. y = + 3tan x. y = cot x 3 0. y = 5 + 3cot θ 3. y = csc x. y = 6 + csc5θ. y = sec3θ. y = + sec x 0 5. y = cot x 3. y = + 3cot ( θ 30 ) 6. y = tan θ. y = + 5 tan ( x 3) 8 7. y = sec θ 3 5. y = + 6 sec ( x + ) 8. y = cscθ 6. 3+ csc ( θ + 0 ) Honors Algebra Chapter Page
Section.3 Basic Trig Identities Objectives:. To identify and use reciprocal identities, quotient identities, and Pythagorean identities I. Reciprocal Identities A. Identities. sin A = or csc A = or _ = Note: sin sin A A. cos A = or sec A = or = 3. tan A = or cot A = or = II. B. Example: If sec x =, find cos x. 3 Quotient Identities Identities. tan A = Proof:. cot A = III. Pythagorean Identities A. Identities. Proof:. 3. B. Examples: 3. If cos A =, find csc A. 5. Simplify: csc x cos x cot x 3. Simplify the gravitational force formula: F = sec sec x tan x mg x mg Homework: p.779 3-35 odds, 37- all, 6-5 all Honors Algebra Chapter Page
Section. Verifying Trig Identities Objectives:. To use the basic trig identities to verify other identities.. To find the numeric values of trig functions A. Method. : Transform the more side of the into the of the side.. : one or more basic trig to simplify the expression. 3. : or to simplify the expression.. Always repeat the and step until goal is met. 5. Tricks: Multiply both and by the. Note: When trying to verify possible trig identities, you can only work with one side. If you perform an operation with both sides, then you are assuming they are equal B. Examples tan x sin x cos x. tan x sin x. + tan x sin x + cos x sec x 3. Find the numeric value of one trig function: cot x 3 csc x = 5 Homework: p.78-7 odds, 3, 3, 35, 36, 3-9 all Honors Algebra Chapter Page 3
Section.5 Sum and Difference Identities Objectives:. To use the sum and difference identities to find exact values and prove identities. A. Identities. cos( A ± B) =. sin( A ± B) = 3. tan( A ± B) = B. Examples. Find the exact value for cos 75. Prove: cos( A + 80 ) cos A 3. Find the exact value for sin75 Hint: Find the exact value for sin5. Prove: tan x tan( + x) 5 5. If sin A = such that 0 A and 3 cos( A + B). 35 cos B 3 = such that B, find 37 Homework: p.788 5-39 odds, 0,, 8-60 all Honors Algebra Chapter Page
Section.6 Double Angle and Half-Angle Identities Objectives:. To use the double angle and half-angle identities to find exact values and prove identities. A. Identities. Double Angle Identities a) sin A = Proof: b) cos A = _ or _ or _ c) tan A =. Half-Angle Identities A a) sin = ± Proof: A b) cos = ± A c) tan = ±, cos A B. Examples 3. If sin x = and x terminates in quadrant IV. Find the exact values for: a) sin x b) cos x. Prove: cos x sec x tan x 3. Use the half-angle identity to find the exact value for sin5 Homework: p.79 3-37 odds, 8-57 all Honors Algebra Chapter Page 5
Section.7 Solving Trig Equations Objectives:. To solve trig equations. A. Examples. Solve sin x sin x 0 + = for the principle values of x.. Solve sin x = for all values of x. 3. Solve cos x sin x sin x 3 cos x + 3 = 0 for all values of x.. Solve cos x sin x = 0 for 0 x. 5. Solve using a calculator: sin 3x cos x = 0, x Homework: p.80 (5-39)/3, 7-55 all Honors Algebra Chapter Page 6