1. GRAPHS OF THE SINE AND COSINE FUNCTIONS
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1 GRAPHS OF THE CIRCULAR FUNCTIONS 1. GRAPHS OF THE SINE AND COSINE FUNCTIONS PERIODIC FUNCTION A period function is a function f such that f ( x) f ( x np) for every real numer x in the domain of f every integer n and some positive real numer p. The smallest possile value of p is the period of the function. GRAPH OF THE SINE FUNCTION o Recall that for a real numer s the point on the unit circle corresponding to s has coordinates cos ssin s. As s increases from sin s cos s 0 to Increases from 0 to 1 Decreases from 1 to 0 to Decreases from 1 to 0 Decreases from 0 to -1 3 to Decreases from 0 to -1 Increases from -1 to 0 3 Increases from -1 to 0 Increases from 0 to 1 x y (xy) 0 0 (00) The graph is continuous over the entire domain Its x-intercepts are of the form n n. Its period is. The graph is symmetric with respect to the origin so the function is an odd function. For all x in the domain sin( x) sin( x).
2 x y (xy) 0 0 (00) The graph is continuous over the entire domain Its x-intercepts are of the form n1 n. Its period is. The graph is symmetric with respect to the y-axis so the function is an even function. For all x in the domain cos( x) cos( x). AMPLITUDE The graph of y asin x or y acos x with a 0 will have the same shape as the graph of y sin x or y cos x respectively except with range a a. The amplitude is a. PERIOD For 0 the graph of y sin x will resemle that of y sin x ut with period. Also the graph of y cos x will resemle that of y cos x ut with period.
3 GUIDELINES FOR SKETCHING GRAPHS OF THE SINE AND COSINE FUNCTIONS To graph y asin x or y acos x with 0 follow these steps. 1. Find the period. Start at 0 on the x-axis and lay off a distance of.. Divide the interval into equal parts. 3. Evaluate the function for each of the five angle from the x-axis. This will give the x-intercepts maximum points and minimum points.. Plot the points found in (3) and join them with a sinusoidal curve having amplitude a. 5. Draw the graph over additional periods to the right and to the left as needed.. TRANSLATIONS OF THE GRAPHS OF THE SINE AND COSINE HORIZONTAL TRANSLATIONS o The graph of the function defined y y f x d is translated horizontally compared to the graph of f( x ). The translation is d units to the right if d 0 or d units to the left if d 0. With circular functions a horizontal translation is called a phase shift and x d is called the argument. o How to graph: Method 1: Consider ysin x Solve the inequality: 0 x 7 x
4 Divide the resulting interval into equal parts to get the following x values: x x (xy) sin x Method : Consider ysin x y sin x and then graph it again (using a Graph different type of line or color) shifted to the left. VERTICAL TRANSLATIONS o The graph of the function defined y y c f x is translated vertically compared to the graph of f( x ). The translation is c units up if c 0 or c units down if c 0. o My suggestion is to use a comination of methods (1) and () to y c f x d graph functions of the type
5 FURTHER GUIDELINES FOR SKETCHING GRAPHS OF SINE AND y c asin x d or COSINE FUNCTIONS IN THE FORM OF y c acos x d METHOD 1: STEP 1 Find an interval whose length is one period y solving 0 x d. the three-part inequality STEP STEP 3 STEP Divide the interval into four equal parts. Evaluate the function for each of the five x values resulting from step. The points will e maximum points minimum points and points that intersect the line y c ( middle points of the wave ). Plot the points found in step 3 and join them with a sinusoidal curve having amplitude a. STEP 5 Draw the graph over additional periods to the right and to the left as needed. METHOD : First graph the asic circular function. The amplitude of the function is a and the period of the function is. Then use translations to graph the desired function. The vertical translation is if c 0 or c units down if c 0. The translation is d units to the right if d 0 or d units to the left if d 0. The horizontal translation is d units to the right if d 0 or d units to the left if d 0.
6 EXAMPLE: Graph Step 1: 1 y1 cos x 3 0 x d 1 0 x x 0 x Step : 0 3 Step 3: x 1 x 1 1 cos 3 x (xy) Now you try making the graph.
7 a. GRAPHS OF THE OTHER CIRCULAR FUNCTIONS GRAPHS OF THE COSECANT AND SECANT FUNCTIONS o Cosecant Since cosecant values are reciprocals of the corresponding sine values the period of the function y csc x is. When sin x 1 csc x 1 and when 0 sin x 1 csc x 1. Similarly when 1 sin x 0 csc x 1. As x 0 sin x 0 and csc x. Therefore the vertical lines at x n are all vertical asymptotes. o Secant Since secant values are reciprocals of the corresponding cosine values the period of the function y sec x is. When cos x 1 sec x 1 and when 0 cos x 1 sec x 1. Similarly when 1 cos x 0 sec x 1. As x cos x 0 and sec x. Therefore the vertical lines at x n are all vertical asymptotes.
8 GUIDELINES FOR SKETCHING GRAPHS OF COSECANT AND SECANT FUNCTIONS To graph y acsc x or y asec x with 0 follow these steps. Step 1 Step Step 3 Graph the corresponding reciprocal function as a guide using a dashed curve. Sketch the vertical asymptotes. Sketch the graph of the desired function y drawing the typical U-shaped ranches etween the adjacent asymptotes. The ranches will e aove the graph of the guide function when the guide function values are positive and elow the graph of the guide function when the guide function values are negative. GRAPHS OF THE TANGENT AND COTANGENT FUNCTIONS o Tangent Period is. sin x Since tan x tangent values are zero when the sine cos x values are zero and undefined when the cosine values are zero. As x-values go from to tangent values go from to and increase throughout the interval.
9 o Cotangent Period is. cos x Since cot x tangent values are zero when the sin x cosine values are zero and undefined when the sine values are zero. As x-values go from 0 to tangent values go from to and decrease throughout the interval. GUIDELINES FOR SKETCHING GRAPHS OF TANGENT AND COTANGENT FUNCTIONS To graph y a tan x or y acot x with 0 follow these steps: Step 1 Determine the period. To locate two adjacent vertical asymptotes solve the following equations for x : For y a tan x : x and x. For y a cot x : x 0 and x. Step Sketch the two vertical asymptotes found in Step 1. Step 3 Divide the interval formed y the vertical asymptotes into four equal parts. Step Evaluate the function for the first-quarter point midpoint and third-quarter point using the x-values found in Step 3. Step 5 Join the points with a smooth curve approaching the vertical asymptotes. Indicate additional asymptotes and periods of the graph as necessary.
10 . HARMONIC MOTION SIMPLE HARMONIC MOTION o A spring with a weight attached to its free end is in equilirium (or rest) position. o If the weight is pulled down a units and released the spring s elasticity causes the weight to rise a units a 0 aove the equilirium position and then oscillate aout the equilirium position. This motion can e descried mathematically y a sinusoid if friction is neglected. The position of a point oscillating aout an equilirium position at time t is modeled y either s( t) acos t or s( t) asin t where a and are constants with 0. The amplitude of the motion is a the period is and the frequency is. Example: An oject is attached to a coiled spring. It is pulled down a distance of 6 units from its equilirium position and then released. The time for one complete oscillation is seconds. a) Give an equation that models the position of the oject at time t. Solution: When the oject is released at t 0 the distance of the oject from the equilirium position is 6 units elow equilirium. If st () is to model the motion then s (0) must equal 6. We use s( t) acos t with a 6. We choose the cosine function since cos (0) cos 0 1 and Had we chosen the sine function a phase shift would have een required. The period is so. So we have s( t) 6cos t.
11 ) Determine the position at t 1.5 seconds. Solution: 5 s(1.5) 6cos 1.5 6cos units. c) Find the frequency. 1 1 The frequency is. DAMPED OSCILLATORY MOTION o Previously we did not consider the effect of friction. Friction causes the amplitude of the motion to diminish gradually until the weight comes to rest. This is referred to as the motion has een damped y the force of friction. o A typical example of damped oscillatory motion is provided y t the function defined y s( t) e sin t.
Notice there are vertical asymptotes whenever y = sin x = 0 (such as x = 0).
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