Practice Exam 2 Solution - Plane Trigonometry | MATH 111, Study notes of Trigonometry

Download Practice Exam 2 Solution - Plane Trigonometry | MATH 111 and more Study notes Trigonometry in PDF only on Docsity! Math 111 - Trigonometry Practice Exam 2 Warning: This study guide is not all inclusive- there may be material on the test which is covered in the book but not here. This is simply meant to be a supplemental study aid to the homework and class notes 1. Graph two periods of y = −2 + 4 cos(π(x − 2)). Find the amplitude, period, and average value. Label the axes in a way to reflect the important characteristics of the graph (i.e. when the function reaches it maximum, minimum, and average value). -5 -4 -3 -2 -1 0 1 2 3 4 5 -6 -5 -4 -3 -2 -1 1 2 2. Below is a data table for a function involving a trigonometric function. x -1 0 1 2 3 4 y 3 7 11 7 3 7 Determine a possible formula for the trigonometric function. Solution: Here are two possibilities. There are others: y = 7 + 4 sin( π 2 x) y = 7 + 4 cos( π 2 (x− 1)) 3. Consider a function with the below graph: -5 -4 -3 -2 -1 0 1 2 3 4 5 -1 1 2 3 (a) Find a formula for this function involving sine. Solution: y = 1 + 2 sin (π 2 (x− 1) ) (b) Find a formula for this function involving cosine. Solution: y = 1− 2 cos (π 2 x ) 4. Find the following exact values (a) The period of the function y = tan(4(x− 3)) Solution: π/4 (b) The horizontal shift of the function f(x) = 1− 2 sin(3(x− 4)) Solution: 4 units to the right (c) The average value of the function g(x) = 2− 2 cos(π(x+ π)) Solution: 2 (d) tan(cos−1(− 5 13 )) Solution: -12/5 (e) cos(arctan( √ 3) + sin−1(1 3 )) (Hint: Use the Cosine of a Sum identity) Solution: x = { π 3 + 2πn 5π 3 + 2πn The only solution that is in the interval [0, π] (the range of arccosx) is x = π/3, so cos−1(1/2) = π/3. 11. Consider the following two statements: (a) x = sin−1 y always means y = sinx (b) y = sinx always means x = sin−1 y Is statement (a) true or false? Is statement (b) true or false? Explain. Solution: Recall the definition of inverse sine is that x = sin−1 y means y = sinx AND x ∈ [−π/2, π/2]. So statement a) is true; the definition tells us that if x = sin−1 y, then it must be true that y = sinx. However statement b) is false. As an example to see when this is not true, consider x = π. Then y = sinπ = 0, but sin−1 y = sin−1 0 = 0, and 0 6= π = x. In fact any number x that is outside the restricted domain [−π/2, π/2] of y = sinx that we used to define inverse sine will show that statement b) is false, exactly because these x cannot satisfy the second part of the definition of inverse sine. 12. Is it true that if x = tan 5, then 5 = arctan x? Solution: The range of arctanx is (−π/2, π/2), and 5 is not in this interval, so the statement cannot possible be true. In general, arctan(tanx) = x only if x ∈ (−π/2, π/2). 13. For what values of x will sin(arcsinx) = x? (a) all x (b) [−1, 1] (c) [ −π 2 , π 2 ] (d) [0, π] (e) none of the above Solution: This statement is true for all x in the domain of arcsinx, which is [−1, 1]. MISS LENHART COULDNT BE HERE TODAY, 50 SHE ASKED MR. MUNROE, Miss LENHART ME TO SUBSTITUTE. NEVER TAUGHT US THIS. LCM ‘ 1‘VE PUT OUT YOUR TESTS. PLEASE GET STARTED. THATS BECAUSE MIS LENHART DOESN'T UNDERSTAND How IMPORTANT CERTAIN / KINDS OF NATH ARE. BUT THE. JUST LOOKS ~~ ~ THK MATER $ MORE VIAL THAN ANTTING YOUVE EYER LEARNED BUT-- NO BUTS. ——— THS |S A MATTER OF LIFE AND DEATH. 1. The velociraptor spots you 40 meters away and attacks, accelerating at 4 m/s*2 up to its top speed of 25 m/s. When it spots you, you begin te flee, quickly reaching your top speed of 6 m/s. How far can you get before you're caught and devoured? You are at the center of a 20m equilateral triangle with a raptor at each corner, The top raptor has a wounded leg and is limited to a top speed of 10 m/s. ! (Not to scale) k xo ‘The raptors will run toward you. At what angle should you run to maximize the time you stay alive? . Raptors can open doors, but they are slowed by them, Using the floor plan on the next page, plot a route through the building, assuming raptors take 5 minutes to open the first door and halve the time for each subsequent door. Remeber, raptors run at 10 m/s and they do not know fear.