2017 FAMAT State Convention. Alpha Trigonometry

Transcription

1 017 FAMAT State Convention Alpha Trigonometry 1

2 On this test, select the best answer choice for each question. If you believe that the correct answer is not among the choices, or that the question is flawed, select NOTA, or None of the Above. Also, the arcfunctions, such as y = arcsin(x), have the traditional restricted ranges. Good luck and have fun! 1. Which angle is co-terminal with 195? A. 45 B. 45 C. 15 D. 5. An angle x where 0 x < π is chosen at random. What is the probability that sin(x) < 0 and cos(x) > 0? A. 1 4 B. 1 C. 4 D. 0. For positive real values a, b, c, the function f (x) = asin bx π c has a period of 1, a maximum of 8 and f (1) = 10. Find a b c. A. π B. 6π C. 8π D. 4π 4. If cos(x) = 4 and tan(x) < 0, find the value of cos(x). A. 9 8 B C. 1 4 D If sin = a cot 1 b for relatively prime positive integers a, b find a + b. A. 51 B. 5 C. 15 D Find the exact value of (sin 7π 8 + cos π 8 ). A. + B. C. 1+ D TRAP is a trapezoid with RA and TP parallel. If TR = 8 and RA = 1, find the area of TRAP. A B C D

3 8. Solve for acute angle θ where cos(θ) + cos(θ)sin (θ) + cos(θ)sin 4 (θ) + cos(θ)sin 6 (θ) +... = A. π B. π 6 C. π 4 D. π 1 9. What is the range for the function f (x) = arcsin(x) + arccos(x)? A. π B. (, ) C. π, π D. 1,1 [ ] 10. Find the amplitude of the function f (x) = sin(x) + cos(x). A. 4 B. 4 C. D For any real value x, cos( tan 1 (x)) is equivalent to which of the following expressions? A. 1 x 1+ x B. 1+ x 1 x 1 x C. 1+ x D. 1 x 1+ x 1. How many positive acute angles x satisfy sin(x) = cos(7x)? A. 1 B. C. D If tan(θ) + cot(θ) = α, express sec(θ)csc(θ) in terms of α. A. α B. α C. 1 α D. α θ 14. Let 0 < θ < π. If cos(θ) = sin 5, find the exact value of sin θ + cos θ. A. B. 1/ C. D. 1/ 15. On[0,π ), what is the difference of the largest and smallest solutions to cos (x) = sin (x)? A. 5π B. 4π C. π D. 11π For any right triangle ABC, find sin ( A) + sin (B) + sin (C). A. 1 B. C. D. 5/

4 17. Find the number of solutions on [0,π ) to cos(4x) + cos(x) = cos(x). A. B. 4 C. 6 D If cos(4x) = acos 4 (x) + bcos (x) + c, find the value of a b c. A. 0 B. -16 C. - D The area of an obtuse triangle is 1. The two side lengths that include the obtuse angle measure 5 and 6. What is the tangent of the angle between these two sides? A. 4 B. 4 C. 4 5 D The sides of a triangle are in the extended ratio 6 : 7 : 8. What is the cosine of the largest angle? A. 1 4 B. 1 4 C. 1 D Equilateral triangles MPH and HUA share a side with the square ALPH. What is the ratio of the area of triangle MHU to the area of square ALPH? A. 1 4 B. 1 C. D. 1. If sec(x) > csc(x), which of the following must be true? I. sin(x) > cos(x) II. cot(x) > 0 III. sin(x) > 0 A. I only B. III only C. I and II only D. II and II only. What is the period of the function f (x) = cos x + tan x 6? A. 6π B. 8π C. 1π D. 4π 4. In a parallelogram the ratio of two consecutive angles is :1. Two adjacent sides of the parallelogram measure 4 and 8. What is the difference of the squares of the lengths of the parallelogram s diagonals? A. 64 B. 64 C. D. 16 4

5 5. What is the measure in radians of the smaller angle between the lines whose equations are 5x + y = 1 and x y = 11? A. π 4 B. π C. π 6 D. π 6. ABCDEFGH is a regular octagon with diagonals AE and AF, as shown. What is the ratio of area AFGH to area AEF? A. : 1 B. 1 : 1 C. 1 : D. :1 7. If A + B = π 4, what is the value of (tan( A) 1)(tan(B) 1) when defined? A. 0 B. 1 C. D. 8. The area of quadrilateral ABCD can be written in the form a + b. Find a + b. A. 59 B. 7 C. 97 D For t (0, π ), solve for y in the system of equations: tan(t) x + sec(t) y = 1 sec(t) x + tan(t) y = sin(t) A. cos(t)sin(t) B. cos(t) C. cos(t) cos(t) D. cos(t) sin(t) 0. Find the value of log(tan(1 )) + log(tan( )) + log(tan( )) log(tan(88 )) + log(tan(89 )). A. 1 B. 0 C. D. 1 5