Trigonometry Basic trigonometric ratios, graphs, identities, and equations are covered in this section. BASIC TRIGONOMETRIC RATIOS sin A = ᎏ hy o p p o p t o e s n it u e se ᎏ opposite refers to the length of the leg opposite angle A. cos A = ᎏ hy a p d o ja te c n en u t se ᎏ adjacent refers to the length of the leg adjacent to angle A. tan A = ᎏ a o d p j p a o c s e i n te t ᎏ TRIGONOMETRIC IDENTITIES sin 2 x + cos 2 x = 1 tan x = ᎏ c si o n s x x ᎏ sin 2x = 2sin x cos x cos 2x = cos 2 x − sin 2 x tan 2x = ᎏ 1 2 − ta ta n n x 2 x ᎏ Practice Questions Directions After reading each question, solve each problem, and then choose the best answer from the choices given. (Remember, the ACT Math Test is different from all the other tests in that each math question contains five answer choices.) When you are taking the official ACT, make sure you carefully fill in the appropriate bub- ble on the answer document. You may use a calculator for any problem, but many problems are done more quickly and easily with- out one. Unless directions tell you otherwise, assume the following: ■ figures may not be drawn to scale ■ geometric figures lie in a plane ■ “line” refers to a straight line ■ “average” refers to the arithmetic mean Remember, the questions get harder as the test goes on. You may want to consider this fact as you pace yourself. – ACT MATH TEST PRACTICE– 164 1. How is five hundred twelve and sixteen thousandths written in decimal form? a. 512.016 b. 512.16 c. 512,160 d. 51.216 e. 512.0016 2. 4 ᎏ 1 3 ᎏ − 1 ᎏ 3 4 ᎏ = ? f. 2 ᎏ 1 7 2 ᎏ g. 3 ᎏ 1 5 2 ᎏ h. 3 ᎏ 2 3 ᎏ i. 2 ᎏ 1 5 2 ᎏ j. 1 ᎏ 1 8 ᎏ 3. Simplify |3 − 11| + 4 × 2 3 . a. 24 b. 40 c. 96 d. 520 e. 32 4. The ratio of boys to girls in a kindergarten class is 4 to 5. If there are 18 students in the class, how many are boys? f. 9 g. 8 h. 10 i. 7 j. 12 5. What is the median of 0.024, 0.008, 0.1, 0.024, 0.095, and 0.3? a. 0.119 b. 0.095 c. 0.0595 d. 0.024 e. 0.092 – ACT MATH TEST PRACTICE– 165 6. Which of the following is NOT the graph of a function? f. g. h. i. j. – ACT MATH TEST PRACTICE– 166 7. 4.6 × 10 5 = ? a. 4.60000 b. 0.000046 c. 4,600,000 d. 460,000 e. 0.0000046 8. What is the value of x 5 for x = 3? f. 15 g. 243 h. 125 i. ᎏ 5 3 ᎏ j. 1.6 9. What is the next number in the pattern below? 0,3,8,15,24, a. 35 b. 33 c. 36 d. 41 e. 37 10. What is the prime factorization of 84? f. 42 × 2 g. 7 × 2 × 3 h. 2 2 × 3 × 7 i. 2 × 6 × 7 j. 2 3 × 7 11. Find the slope of the line 7x = 3y − 9. a. 3 b. −9 c. ᎏ 7 3 ᎏ d. −3 e. ᎏ 3 7 ᎏ – ACT MATH TEST PRACTICE– 167 12. The perimeter of a rectangle is 20 cm. If the width is 4 cm, find the length of the rectangle. f. 6 cm g. 16 cm h. 5 cm i. 12 cm j. 24 cm 13. Find the area of the figure below. a. 79 square inches b. 91 square inches c. 70 square inches d. 64 square inches e. 58 square inches 14. Five cans of tomatoes cost $6.50. At this rate, how much will 9 cans of tomatoes cost? f. $13.00 g. $11.70 h. $1.30 i. $11.90 j. $12.40 15. For all x ≠ 0, ᎏ 3 2 x ᎏ + ᎏ 1 5 ᎏ = ? a. ᎏ 1 2 5x ᎏ b. ᎏ 1 1 0 5 + + 3 x x ᎏ c. ᎏ 10 1 + 5x 3x ᎏ d. ᎏ 15 2 + x ᎏ e. ᎏ 5 1 x ᎏ 10 in 7 in 7 in 3 in – ACT MATH TEST PRACTICE– 168 16. Which inequality best represents the graph below? f. −1.5 > x > −1 g. x ≤ 0 h. −0.5 > x > 0 i. −1.5 < x < 0 j. −1.5 ≤ x ≤ 0 17. Simplify −(6x 4 y 3 ) 2 . a. −36x 6 y 5 b. 36x 2 y c. −36x 8 y 6 d. 36x 8 y 4 e. −36xy 18. If 2x + 3y = 55 and 4x = y + 47, find x − y. f. 28 g. 16 h. 5 i. 12 j. 24 19. Which inequality represents the graph below? a. −4x < 0 b. −20x > 5 c. x < −4 d. −x ≤−4 e. −x < 4 20. Simplify ͙ 3 16x 5 y 4 ෆ . f. 2xy͙ 3 2x 2 y ෆ g. 8x 2 y h. 8xy͙ 3 2 ෆ i. 2xy͙ 3 xy ෆ j. 4x 2 y 2 ͙ 3 x ෆ −40 4 −2−3 −10 1 – ACT MATH TEST PRACTICE– 169 . 0.024, 0.0 95, and 0.3? a. 0.119 b. 0.0 95 c. 0. 059 5 d. 0.024 e. 0.092 – ACT MATH TEST PRACTICE– 1 65 6. Which of the following is NOT the graph of a function? f. g. h. i. j. – ACT MATH TEST PRACTICE– 166 7 For all x ≠ 0, ᎏ 3 2 x ᎏ + ᎏ 1 5 ᎏ = ? a. ᎏ 1 2 5x ᎏ b. ᎏ 1 1 0 5 + + 3 x x ᎏ c. ᎏ 10 1 + 5x 3x ᎏ d. ᎏ 15 2 + x ᎏ e. ᎏ 5 1 x ᎏ 10 in 7 in 7 in 3 in – ACT MATH TEST PRACTICE– 168 16. Which inequality. written in decimal form? a. 51 2.016 b. 51 2.16 c. 51 2,160 d. 51 .216 e. 51 2.0016 2. 4 ᎏ 1 3 ᎏ − 1 ᎏ 3 4 ᎏ = ? f. 2 ᎏ 1 7 2 ᎏ g. 3 ᎏ 1 5 2 ᎏ h. 3 ᎏ 2 3 ᎏ i. 2 ᎏ 1 5 2 ᎏ j. 1 ᎏ 1 8 ᎏ 3. Simplify |3 −