Math 107 Study Guide for Chapters 5 and Sections 6.1, 6.2 & 6.5

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1 Math 07 Study Guide for Chapters 5 and Sections.,. &.5 PRACTICE EXERCISES. Answer the following. 5 Sketch and label the angle θ = in the coordinate plane. Determine the quadrant and reference angle for the angle θ =. 5 Find one positive and one negative angle coterminal with the angle α = 580. Convert 5 radians to degrees. (e) Convert 00 to radians. (f) Give the point on the unit circle corresponding to θ =. (g) What is the sign of csc( θ ) if the terminal side of θ is in Quadrant III? (h) What is the period of f( t) = cos( t)? (i) What is the period of f( t) = sec( t)? (j) State the domain and range of the inverse tangent function f(t) = arctan(t).. The central angle θ = 0 forms an arc of length 9 cm. What is the radius of the circle? Draw a picture (neatly) representing this problem, labeling the radius, central angle, and arc length.. Name the reference angle for the angles given: θ = 5 ; θ = 5 ; θ = 0.. Given the reference angle θ r = 50, find and sketch an angle θ (that has the given reference angle) in each of the following quadrants. θ in Quadrant II θ in Quadrant III θ in Quadrant IV

2 5. Solve the right triangle and find the specified trigonometric functions. Give eact values. 7 β c α = β = c = sin β = cos β = tan β = α 7 cscα = secα = cotα =. Find the value of the si trigonometric functions given that P =, 5 is on the terminal side of the angle θ with θ in standard position. Sketch θ in standard position. 7. Find two angles coterminal to Find two angles in the interval [0, ] that satisfy sin(θ) =. 9. Given that tan( θ ) = and csc( θ ) < 0, then θ is in Quadrant. Determine the value of the other five trigonometric functions. 0. Find the values of, y, and r using the information given, and state the quadrant of the terminal side of θ. Then state the value of the si trig functions of θ. cos θ = ; sin θ < 0 tan θ = ; cos θ > Determine an angle v such that 0 < v < 0, v 5 and cos v = cos 5.. Evaluate the following. Give eact answers. (Hint: Use reference angles and terminal points on the unit circle.) sin = csc = tan ( 80 ) = sec = ( ) cos 00 = tan = sec csc = = ( )

3 . At a certain time of day, the Washington Monument casts a shadow 700 feet long. From the tip of the shadow, the angle from the horizontal to the top of the monument is 0 o. Use this information to find the height of the monument.. A ladder leans against the side of a barn forming an angle of elevation of 0. The bottom of the ladder is.5 feet out from the base of the barn. Find the length of the ladder and the height at which the ladder reaches the barn. 5. From the observation deck of a sea side building 0 meters high, Armando sees two fishing boats in the distance. The angle of depression to the nearer boat is 0, while for the boat farther away the angle is 5. How far out to the sea is the nearer boat? How far apart are the two boats?. From a hotel room window, a tourist sees some window washers high above on the building across the street. The tourist estimates that the buildings are 50 ft. apart, the angle of elevation to the workers is about 5, and that the angle of depression to the base of the building across the street is 0. How high above the ground is the window of the tourist s hotel room? How high above ground are the workers? 7. For the function g() = 7 cos +, determine the following information then sketch two complete periods. Label the tick marks on the -ais. Identify and label three points on the graph. Note that the primary interval for sine and cosine is found by solving the inequality 0 B + C (tet p.90). Amplitude: Period: Horizontal Shift: Vertical Shift: Domain: Range: Primary interval: y-intercept:

4 8. Eercises # 7-8 in page 7 in your tetbook. 9. Determine the amplitude, period, horizontal shift, vertical shift, domain and range of the function H(t) = sin t + 9. Sketch the graph of H(t) over one period. 0. The graph below represents a mathematical model of the form y = A sin(bt + C) + D. Use the graph to determine the values of the parameters A, B, C, and D.. Sketch the graph of the function given over two periods: ht ( ) = sec( t) k(t) =.5csc(t). State the domain, range and period of the given function, then sketch its graph over the interval [, ]. y = tan(t) y = cot t. Determine the equation for the graph given, and state the domain of the function. f ( t) = Domain:

5 . A mass hangs from a spring attached to the ceiling. It moves according to the model, y(t) = A cos(bt), where y is the vertical position of the object and t is time in seconds. The position of the object at rest is y = 0. Suppose at time t = 0 the object is pulled down cm (to y = ) and released. After.5 seconds the object first returns to the starting position. Determine A and B. 5. At a particular point in the ocean, the vertical change of a floater changes due to wave action. Assume that the vertical change, y in meters, can be modeled by a sinusoidal function y = f(t) where the time, t, is given in seconds. If the floater bobs up and down through a distance of meters every 0 sec. What is the equation for this motion? Assume that initially the floater is at its lowest point.. The number of hours of daylight in a particular day of the year is modeled by the formula K Dt ( ) = sin ( t 79) + 5 where Dt () is the number of daylight hours on day t of the year and K is a constant related to the total variation of daylight hours, latitude of the location, and other factors. What does a vertical shift of means in this contet? What does a horizontal shift of 79 means in this contet? 7. Determine an epression for cot θ and sec θ given that csc θ = u +9. u 8. Write cos( θ ) in terms of sin( θ ) given that θ is in quadrant II or III. 9. Verify the following identities. Show all steps. cot ( )(sec ( ) ) = ( tan( )) = sec ( ) tan( ) tan (csc + cot ) = + sec cot cot + tan sin = 0. Verify the identities: sin( ) tan( ) + sin( ) = cos( ) tan( ) + tan ( ) sec( ) csc( ) = tan( ) sin( ) sec( ) cos( ) sec( ) sec( ) = sin ( ) cos ( ) tan ( ) = cos ( ). Given f() = cos ( ) which of the following are valid inputs? Circle your answers. =, =, =, = 0

6 . Evaluate each of the following if possible. Give eact values. Eplain your reasoning. sin csc () arccos sin sin (e) cos(cos ()) (f) tan tan (g) sin cos (h) sec arcsin. Evaluate each epression by drawing a right triangle and labeling the sides. Assume > 0. 5 tan arcsin 9 sec arctan cot arccos 9 sin arcsec. Evaluate each of the following eactly. Give angles in radians. arctan arcsin arctan cos arccos 7 (e) arccos cos (f) arccos cos (g) arccos sin (h) csc arcsin 5 (i) 5 cot arctan 5. Evaluate each epression by drawing a right triangle and labeling the sides. Assume > 0. sin cos 7 8 tan sin 7 7 tan arcsec cot sin 8 +. Given f() = arccos( + ) The domain of f in interval notation is: The range of f in interval notation is: