Name Honors Advanced Mathematics Test January 21, 2005 Block _____ Sections 4.7, 5.5, 5.6 page 1 Part A (20%) General instructions: Write complete, fully explained solutions, Part B (30%) except where directions say otherwise. If you use your calculator for Part C (20%) a significant step, be sure to tell what you did on the calculator. Part D (30%) Use radians for all angle measures, unless stated otherwise. overall Part A. Inverse trigonometric functions 1. a. The csc x function must be restricted before an inverse function can be formed. Explain why a restriction is necessary in order to get an inverse that is a function. b, Identify the domain and range of the restricted csc x, and the domain and range of csc–1(x). restricted csc x csc–1(x) domain: domain: range: range: c. Sketch the graph of y = csc–1(x). Label any intercepts or asymptotes with their coordinates. Choose suitable scales for the x- and y-axes. 2. a. Evaluate csc–1(–2) b. Given that x is positive, find a non-trigonometric function that is equal to cos(sin–1 x). Name Honors Advanced Mathematics Test January 21, 2005 Block _____ Sections 4.7, 5.5, 5.6 page 2 Part B. Measurements related to triangles π 1. Answer the following questions about ∆XYZ, given that XY = 4, ∠Y = 5 , and YZ = 6. a. What is the area of ∆XYZ? b. What is the length of side XZ? c. What is the length of the angle bisector segment that bisects ∠X ? AB 2. For any triangle ∆ABC, prove that the diameter of the circumscribed circle is equal to . sin C Name Honors Advanced Mathematics Test January 21, 2005 Block _____ Sections 4.7, 5.5, 5.6 page 3 Part C. Your choice Solve one and only one of these two problems. 1. Two circles are tangent to each other and to an angle of measure 2x, as shown in the diagram. Let r = the radius of the smaller circle; R = the radius of the larger circle. r 1 − sin x Prove that = . R 1 + sin x 2a 3 + b3 + c 3 2. Consider ∆ABC with the usual notation for the side lengths. Given that c = , a+b+c find the measure of ∠C. Name Honors Advanced Mathematics Test January 21, 2005 Block _____ Sections 4.7, 5.5, 5.6 page 4 Part D. More geometric problems 1. In ∆XYZ with the usual notation, suppose x = 5, y = 8, and ∠X = 19º. There are two possibilities for the remaining measurements of the triangle. Calculate these measurements, showing your work in an organized way. Then, sketch both possible triangles, labeled with all their side lengths and angle measures. 2. Find the area of quadrilateral ABCD (vertices labeled consecutively) given that AB = 10, ∠B = 60°, BC = 8, ∠C = 135°, and CD = 6.
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