Graphing Trigonometric Functions: Day 1

Transcription

1 Graphing Trigonometric Functions: Day 1 Pre-Calculus 1. Graph the six parent trigonometric functions.. Apply scale changes to the six parent trigonometric functions. Complete the worksheet Exploration: Introduction to Ch. 4B BEFORE you watch the video for day 1. From the exploration, notice we are graphing the ordered pairs (,sin ) and (,cos ) which gives us the graph of the sine function and the cosine function. You may recall these parent functions from first semester. Sine Function: f ( x) sinx Cosine Function: f ( x) cosx Some new terminology Periodic Functions: Period: Amplitude: Example 1: List the amplitude and period. Then graph at least one period of the function. Be sure to label the scale of each axis. b) y cosx a) y sin x x c) y cos d) 1 x y sin 3 4 4b 1

2 Example : Write two equations of the cosine function whose amplitude is 5 and period is 3. Example 3: Write the equation for the given graph. 1 What about Tangent? Tangent Function: f ( x) tanx Period: Amplitude: Vertical Asymptotes: Example 4: Graph at least one period of x y tan. Be sure to label your axes and clearly identify the asymptotes. We will discuss the graphs of the reciprocal functions together in class! 4b

3 Graphing Trigonometric Functions: Day Pre-Calculus 1. Apply translations to the six parent trigonometric functions.. Given the attributes of a trigonometric function write the equation of a trig. function. 3. Given a graph, identify the attributes of the function and write its equation. To review our translations A number added inside effects the horizontal direction, but it s backwards, and a number added outside effects the vertical direction. When we apply these transformations to the trig functions, we get sin ( ) or cos ( ) y a b x c d y a b xc d Amplitude: Period: Phase Shift: Vertical Shift: Example 1: List the amplitude, period, phase shift and vertical shift. Then graph at least one period of the function. Be sure to label the scale of each axis. a) y 4sin x b) y cosx 1 c) y sin 3 x 3 x d) y tan 5 4 4b 3

4 For this last example, be careful the phase shift is NOT! e) y 3sinx 1 Example : Write two equations of the cosine function whose amplitude is, period is 6, phase shift is 3 and vertical shift is 5. Example 3: Write the equation for the given graph using a) the Sine Function b) the Cosine Function. 4b 4

5 4.7 Inverse Trigonometric Functions Pre-Calculus 1. Use the appropriate notation for inverse trigonometric functions.. Graph the inverse Sine, Cosine and Tangent functions. 3. List the correct Domain and Range of the inverse functions. 4. Find an exact solution to an expression involving an inverse sine, cosine or tangent. 5. Find the composition of trig functions and their inverses. Inverse Sine Inverse Cosine Inverse Tangent y sin 1 x y arcsin x y cos 1 x y arccos x y tan 1 x y arctan x D: [ 1, 1] D: [ 1, 1] D: (, ) R:, 0, R:, R: Example 1: Find the exact value (in radians). a) cos -1 0 b) sin -1 0 c) arcsin 1 d) arctan 1 e) cos f) tan 3 g) arcsin 3 h) arccos 3 4b 5

6 Compositions with Inverse Functions Work from the inside out. Remember domain and range restrictions. Example : Evaluate each expression. a) sin arctan 3 b) cos 1 5 cos 3 Example 3: Find the algebraic expression equivalent to the given expression. a) 1 sin (cos x) b) 1 cot (sin x) 4b 6

7 4.8 Solving Problems with Trigonometry Pre-Calculus 1. Set up and solve application problems involving right trianglee trigonometry.. Use overlapping right triangles to solve word problems including the use of indirect measurement. 3. Solve problems involving simple harmonic motion. Next we will apply what we know about the trigonometric functions, and their inverses, to solve real world application problems. The most important part of these types of questions is an accurate, detailed picture. Angle of Elevation: The acute angle measured from a horizontal line UP to an object. Angle of Depression: The acute angle measured from a horizontal line DOWN to an object. Angle of Depression Angle of Elevation Example 1: From a boat on the lake, the angle of elevation to the top of a cliff is 04. If the base of the cliff is 1394 feet from the boat, how high is the cliff (to the nearest foot)? Example : The bearings of two points on the shore from a boat are 115 and 13. Assume the two points are 855 feet apart. How far is the boat from the nearest point on shore if the shore is straight and runs north-south? 4b 7