4th Periodical Exam Trigonometry Mathematics 9 PDF

**FOURTH QUARTERLY EXAMINATION IN TRIGONOMETRY** **ADDITIONAL SUBJECT IN MATHEMATICS 9** **School Year 2022-2023** **Directions:** Shade the letter of your chosen answer in your answer sheet. Only one shade per item. Please, minimize your erasures on your answer sheet. Show your solution on your...

**FOURTH QUARTERLY EXAMINATION IN TRIGONOMETRY** **ADDITIONAL SUBJECT IN MATHEMATICS 9** **School Year 2022-2023** **Directions:** Shade the letter of your chosen answer in your answer sheet. Only one shade per item. Please, minimize your erasures on your answer sheet. Show your solution on your scratch paper. Submit your scratch paper together with your shaded answer sheet. Good luck! 1. Which of the following statements is/are TRUE? I. All trigonometric functions have inverse over their respective domains. II\. The graph of inverse trigonometric function can be obtained from the graph of their corresponding trigonometric function by interchanging x and y axes. A. I only B. II only C. Both I and II D. Neither I nor II 2. Which of the following is the **domain** of [*y* = sin^ − 1^ x]{.math.inline}? A. [\$- \\frac{\\pi}{2} \\leq x \\leq \\frac{\\pi}{2}\$]{.math.inline} B. [0 ≤ *x* ≤ *π*]{.math.inline} C. [ − 1 ≤ *x* ≤ 1]{.math.inline} D. [\|*x*\| ≥ 1]{.math.inline} 3. What is the **range** of [*y* = 2 sin^ − 1^(2*x*+2)]{.math.inline}? A. [ − *π* ≤ *y* ≤ *π*]{.math.inline} B. [0 ≤ *y* ≤ *π*]{.math.inline} D. [\$- \\frac{\\pi}{2} \\leq y \\leq \\frac{\\pi}{2}\$]{.math.inline} D. [ − 1 ≤ *y* ≤ 1]{.math.inline} 4. Which of the following is NOT TRUE? A. The principal value of [\$\\cos\^{- 1}\\left( - \\frac{1}{2} \\right)\$]{.math.inline} is [\$\\frac{2\\pi}{3}\$]{.math.inline} B. The value of [\$\\sin\^{- 1}\\left( \\sin\\frac{3\\pi}{5} \\right)\$]{.math.inline} is [\$\\frac{2\\pi}{5}\$]{.math.inline} C. If [\$\\cos\^{- 1}\\frac{x}{2} = 0\$]{.math.inline}, then [*x* = 1]{.math.inline} D. The value of [\$\\cos\^{- 1}\\left( \\cos\\frac{14\\pi}{3} \\right)\$]{.math.inline} is [\$\\frac{2\\pi}{3}\$]{.math.inline} 5. What is the value of [\$\\cos\\left( \\sin\^{- 1}\\frac{1}{2} \\right)\$]{.math.inline}? A. [\$\\frac{1}{2}\$]{.math.inline} B. [\$\\frac{\\pi}{6}\$]{.math.inline} C. [\$\\frac{\\pi}{3}\$]{.math.inline} D. [\$\\frac{\\sqrt{3}}{2}\$]{.math.inline} 6. Which of the following set of angles is [arcsin (−1)]{.math.inline}? A. [\$- \\frac{\\pi}{2} + 2n\\pi\$]{.math.inline} B. [\$\\begin{matrix} \\frac{\\pi}{2} + 2n\\pi \\\\ \\frac{3\\pi}{2} + 2n\\pi \\\\ \\end{matrix}\$]{.math.inline} C. [\$\\begin{matrix} - \\frac{\\pi}{3} + 2n\\pi \\\\ \\frac{2\\pi}{3} + 2n\\pi \\\\ \\end{matrix}\$]{.math.inline} D. [2*nπ*]{.math.inline} 7. What is the value of [\$\\cos\\left( \\sin\^{- 1}\\frac{3}{4} + \\cos\^{- 1}\\frac{5}{13} \\right)\$]{.math.inline}? A. [\$\\frac{5\\sqrt{7} + 36}{52}\$]{.math.inline} B. [\$\\frac{5\\sqrt{7} - 36}{52}\$]{.math.inline} C. [\$\\frac{\\sqrt{7} - 36}{52}\$]{.math.inline} D. [\$\\frac{5\\sqrt{7} - 26}{52}\$]{.math.inline} 8. What is the value of [\$\\sin\\left( \\arcsin\\frac{1}{2} - \\arccos\\frac{1}{3} \\right)\$]{.math.inline}? A. [\$\\frac{2\\sqrt{5} - 1}{6}\$]{.math.inline} B. [\$\\frac{2 + \\sqrt{6}}{6}\$]{.math.inline} C. [\$\\frac{253}{204}\\ \$]{.math.inline} D. [\$\\frac{1 - 2\\sqrt{6}}{6}\$]{.math.inline} 9. If [\$\\sin\^{- 1}\\left( 1 - x \\right) - 2\\sin\^{- 1}x = \\frac{\\pi}{2}\$]{.math.inline}, then what is the value of x? A. {.math.inline} B. [\$\\frac{1}{2}\$]{.math.inline} C. [\$1,\\ \\frac{1}{2}\$]{.math.inline} D. [\$0,\\frac{1}{2}\$]{.math.inline} 10. If [cos^ − 1^*x* + sin^ − 1^*x* = *π*]{.math.inline}, then what is the value of x? A. [\$\\frac{3}{2}\$]{.math.inline} B. [\$\\frac{3\\sqrt{2}}{2}\$]{.math.inline} C. [\$\\frac{\\sqrt{3}}{2}\$]{.math.inline} D. no solution 11. Which of the following is the range of [tan^ − 1^]{.math.inline}? 12. Which of the following corresponds to the principal value branch of [sec^ − 1^]{.math.inline}? A. [\$\\left( - \\frac{\\pi}{2},\\frac{\\pi}{2} \\right)\$]{.math.inline} B. [(0,*π*)]{.math.inline} C. [\$\\left\\lbrack 0,\\left. \\ \\frac{\\pi}{2} \\right) \\cup \\left( \\frac{\\pi}{2},\\left. \\ \\pi \\right\\rbrack \\right.\\ \\right.\\ \$]{.math.inline} D. [\$\\left\\lbrack - \\frac{\\pi}{2},\\left. \\ 0 \\right) \\cup \\left( 0,\\left. \\ \\frac{\\pi}{2} \\right\\rbrack \\right.\\ \\right.\\ \$]{.math.inline} 13. What is the **range** of the function [*y* = arctan 2*x*]{.math.inline}? 14. What is the value of [cot (sin^ − 1^*x*)]{.math.inline}? A. [\$\\frac{\\sqrt{1 + x\^{2}}}{x}\$]{.math.inline} B. [\$\\frac{x}{\\sqrt{1 + x\^{2}}}\$]{.math.inline} C. [\$\\frac{1}{x}\$]{.math.inline} D. [\$\\frac{\\sqrt{1 - x\^{2}}}{x}\$]{.math.inline} 15. If [\$\\theta = \\arctan\\left( \\cot\\frac{\\pi}{4} \\right)\$]{.math.inline} what is [*θ*]{.math.inline}? A. [\$\\frac{\\pi}{6}\$]{.math.inline} B. [\$\\frac{\\pi}{3}\$]{.math.inline} C. [\$\\frac{\\pi}{4}\$]{.math.inline} D. [\$- \\frac{\\pi}{3}\$]{.math.inline} 16. Which of the following statements is TRUE? I. [\$\\cos\^{- 1}\\left( \\csc\\frac{\\pi}{3} \\right) = \\frac{\\pi}{6}\$]{.math.inline} II. [tan (2tan^ − 1^(−1)) = 0]{.math.inline} A. I only B. II only C. Both I and II D. Neither I nor II 17. Which of the following has NO value? A. [arctan 0]{.math.inline} B. [arcsin (−2)]{.math.inline} C. [*tan*(arcsin 1)]{.math.inline} D. [*tan*(arcsin 0)]{.math.inline} 18. What is the exact value of [\$\\cos\\left( 2\\arctan\\frac{4}{3} \\right)\$]{.math.inline}? A. [\$- \\frac{7}{25}\$]{.math.inline} B. [\$\\frac{7}{25}\$]{.math.inline} C. [\$\\frac{24}{25}\$]{.math.inline} D. [\$\\frac{8}{3}\$]{.math.inline} 19. What is the exact value of [\$\\tan\\left( 2\\sin\^{- 1}\\frac{\\sqrt{3}}{2} \\right)\$]{.math.inline}? A. [\$\\sqrt{3}\$]{.math.inline} B. [\$\\frac{\\sqrt{3}}{3}\$]{.math.inline} C. [\$\\frac{3}{5}\$]{.math.inline} D. [\$- \\sqrt{3}\$]{.math.inline} 20. Which of the following is equivalent to [\$\\sec\^{- 1}\\left( - \\frac{2\\sqrt{3}}{3} \\right)\$]{.math.inline}? A. [\$\\sin\^{- 1}\\left( - \\frac{\\sqrt{3}}{2} \\right)\$]{.math.inline} B. [\$\\cos\^{- 1}\\left( - \\frac{\\sqrt{3}}{2} \\right)\$]{.math.inline} C. [\$\\csc\^{- 1}\\left( - \\frac{\\sqrt{3}}{2} \\right)\$]{.math.inline} D. [\$\\cot\^{- 1}\\left( - \\frac{\\sqrt{3}}{2} \\right)\$]{.math.inline} 21. Which of the following is FALSE? A. [\$\\sin\\left( \\arctan 2 \\right) = \\frac{2\\sqrt{5}}{5}\$]{.math.inline} B. [tan (arcsin0) = 1]{.math.inline} C. [tan (arctan(−1)) =  − 1]{.math.inline} D.[\$\\csc\^{- 1}\\left( \\tan\\frac{\\pi}{6} \\right) = undefined\$]{.math.inline} 22. What is the value of [\$\\tan\\left( \\cos\^{- 1}\\ \\frac{3}{5} + \\tan\^{- 1}\\frac{1}{4} \\right)\$]{.math.inline}? A. [\$\\frac{19}{8}\$]{.math.inline} B. [\$\\frac{8}{19}\$]{.math.inline} C. [\$\\frac{19}{12}\$]{.math.inline} D. [\$\\frac{3}{4}\$]{.math.inline} 23. What is the value of [sin \[cot^ − 1^(cos(tan^ − 1^1))\]]{.math.inline}? A. {.math.inline} B. {.math.inline} C. [\$\\frac{\\sqrt{3}}{3}\$]{.math.inline} D. [\$\\frac{\\sqrt{6}}{3}\$]{.math.inline} 24. If [*x* = 2tan (*t*+1)]{.math.inline} which of the following is CORRECT in solving *t* in terms of *x*? A. [*t* = 2tan^ − 1^*x* − 1]{.math.inline} B. [*t* = tan^ − 1^2*x* − 1]{.math.inline} C. [\$t = \\tan\^{- 1}\\frac{x}{2} - 1\$]{.math.inline} D. [\$t = \\tan\^{- 1}\\frac{x}{2} + 1\$]{.math.inline} 25. What is the value of x in the equation [\$\\tan\^{- 1}x = \\sin\^{- 1}\\frac{24}{25}\$]{.math.inline}? A. [\$\\frac{25}{7}\$]{.math.inline} B. [\$\\frac{7}{25}\$]{.math.inline} C. [\$\\frac{7}{24}\$]{.math.inline} D. [\$\\frac{24}{7}\$]{.math.inline} 26. If [tan^ − 1^3 + tan^ − 1^*x* = tan^ − 1^8]{.math.inline}, then what is the value of *x*? A. {.math.inline} B. [\$\\frac{1}{5}\$]{.math.inline} C. [\$\\frac{5}{14}\$]{.math.inline} D. [\$\\frac{14}{5}\$]{.math.inline} 27. If [tan^ − 1^*x* − tan^ − 1^*y* = tan^ − 1^*A*]{.math.inline}, then what is *A*? A. [*x* − *y*]{.math.inline} B. [*x* + *y*]{.math.inline} C. [\$\\frac{x - y}{1 + xy}\$]{.math.inline} D. [\$\\frac{x + y}{1 - xy}\$]{.math.inline} 28. Which of the following is the solution of [\$\\tan\^{- 1}\\left( x + \\frac{\\sqrt{2}}{2} \\right) = \\frac{\\pi}{4}\$]{.math.inline} A.[\$\\frac{2 - \\sqrt{2}}{2}\$]{.math.inline} B. [\$\\frac{\\sqrt{5}}{2}\$]{.math.inline} C.[\$\\frac{\\sqrt{2}}{2}\$]{.math.inline} D. [\$\\frac{1 + \\sqrt{2}}{2}\$]{.math.inline} **For numbers 29-30**. You are playing a game of darts with your friend. The darts you threw landed on the dartboard in the given pattern as shown in the diagram below. 29. Which point represents the polar coordinates of ? A. [(6.5, 90^∘^)]{.math.inline} B. [(6.5, 300^∘^) ]{.math.inline} C. [(7.8, 180^∘^)]{.math.inline} D. [(7.8, 0^∘^) ]{.math.inline}  30. Which point represents the polar coordinates of ? A. [(6.5, 90^∘^)]{.math.inline} B. [(6.5, 300^∘^) ]{.math.inline} C. [(7.8, 180^∘^)]{.math.inline} D. [(7.8, 0^∘^)]{.math.inline} 31. What is the other polar coordinate of the point [\$\\left( 5,\\frac{2\\pi}{3} \\right)\$]{.math.inline} for which [*r* \> 0,  − 2*π* \