Geometry Problem Solving Drill 11: Right Triangle

Transcription

1 Geometry Problem Solving Drill 11: Right Triangle Question No. 1 of 10 Which of the following points lies on the unit circle? Question #01 A. (1/2, 1/2) B. (1/2, 2/2) C. ( 2/2, 2/2) D. ( 2/2, 3/2) The sum of the squares of the coordinates of this point is equal to 2/4 = 1/2 and not equal to 1. For a point to lie on the unit circle, the sum of the squares of the coordinates of the point must be equal to 1. The sum of the squares of the coordinates of this point is equal to 3/4 and not equal to 1. For a point to lie on the unit circle, the sum of the squares of the coordinates of the point must be equal to 1. C. Correct! The sum of the squares of the coordinates of this point is equal to 4/4 = 1. Therefore, this point does lie on the unit circle. The sum of the squares of the coordinates of this point is equal to 5/4 and not equal to 1. For a point to lie on the unit circle, the sum of the squares of the coordinates of the point must be equal to 1. ( 2/2) 2 + ( 2/2) 2 = ( 2) 2 /2 2 + ( 2) 2 /2 2 = 2/4 + 2/4 = (2 + 2)/4 = 4/4 = 1 Therefore, the point ( 2/2, 2/2) does lie on the unit circle because the sum of the squares of its coordinates is equal to 1.

2 Question No. 2 of 10 Calculate the length of the hypotenuse of the right triangle. Question #02 A. 3 B. 7 C. 7 E. 29 Use the Pythagorean Theorem to find the length of the hypotenuse. Use the Pythagorean Theorem to find the length of the hypotenuse. Use the Pythagorean Theorem to find the length of the hypotenuse. D. Correct! This is the number that we obtain by using the Pythagorean Theorem r 2 = a 2 + b 2. Use the Pythagorean Theorem which states that the square of the length of the hypotenuse of a right triangle is equal to the sum of the squares of the lengths of the two remaining sides of the right triangle. r 2 = a 2 + b 2 r 2 = r 2 = r 2 = 29 r = 29

3 Question No. 3 of 10 Find cot θ for the indicated point lying on the unit circle. Question #03 A. - 3 B. 3/2 C. -1/2 D. - 3/3 A. Correct! Find cot θ by dividing the x-coordinate by the y-coordinate. Find cot θ by dividing the x-coordinate by the y-coordinate. Find cot θ by dividing the x-coordinate by the y-coordinate. Find cot θ by dividing the x-coordinate by the y-coordinate. Find cot θ by dividing the x-coordinate by the y-coordinate. Cotθ = (- 3/2)/(1/2) = (- 3/2) * 2 = - 3

4 Question No. 4 of 10 Find the fundamental period of the function shown within the figure. Question #04 A. 2 B. 4 C. 8 D. 16 The fundamental period is equal to the length of the smallest interval on which the function repeats itself. The fundamental period is equal to the length of the smallest interval on which the function repeats itself. C. Correct! The fundamental period is equal to the length of the smallest interval on which the function repeats itself. The fundamental period is equal to the length of the smallest interval on which the function repeats itself. The fundamental period is equal to the length of the smallest interval on which the function repeats itself. The graph of this function starts at x = -8 and ends at x = 8. The graph does not repeat a y-value until x = 0. The length of the interval from - 8 to 0 is 8. Therefore, the fundamental period is equal to 8.

5 Question No. 5 of 10 Which of the following angles does not belong to the domain of sec θ? Question #05 A. 0 B. 45 C. 180 D. 270 Sec θ is defined for θ = 0. Therefore, 0 does belong to the domain of sec θ. Sec θ is defined for θ = 45. Therefore, 45 does belong to the domain of sec θ. C Incorrect! Sec θ is defined for θ = 180. Therefore, 180 does belong to the domain of sec θ. D. Correct! Sec θ is undefined for θ = 270. Therefore, 270 does not belong to the domain of sec θ. An angle will belong to the domain of secθ if the function secant is defined for that particular angle. Conversely, an angle will not the line to the domain of secθ if the function secant is not defined for that particular angle. The angles for which secant is undefined, and hence those angles not belonging to the domain of secθ, are equal to odd multiples of π/2. π/2 is equal to 90. Therefore, in terms of degrees, the only angles not belonging to the domain of secθ are odd multiples of 90. The only degree measurement listed that is equal to an odd multiple of 90 is 270. Thus, the correct answer is D. 270.

6 Question No. 6 of 10 A calculator does not have a button for csc θ. Which of the following sequences of keystrokes can be used to calculate csc θ? Question #06 A. First press cos and then press 1/x. B. First press sin and then press 1/x. C. First press tan and then press 1/x. D. First press 1/x and then press cos. To calculate csc θ we must first press the key that corresponds to the reciprocal of csc θ and then press the reciprocal key (1/x). B. Correct! This is correct because sin θ is the reciprocal of csc θ which must be pressed prior to reciprocal key (i.e., 1/x). To calculate csc θ we must first press the key that corresponds to the reciprocal of csc θ and then press the reciprocal key (1/x). To calculate csc θ we must first press the key that corresponds to the reciprocal of csc θ and then press the reciprocal key (1/x). Csc θ can be calculated by first pressing the key that corresponds to the reciprocal of csc θ followed by pressing the reciprocal key (1/x). Sin θ is the reciprocal of csc θ. Therefore, the correct answer is B. First press sin and then press 1/x.

7 Question No. 7 of 10 What is cos (π/8) equal to given that sin (π/8) = ? Question #07 A. ± B. ± C. ± D. ± We may use the Pythagorean identity, sin 2 (π/8) + cos 2 (π/8) = 1, to find cos(π/8). We may use the Pythagorean identity, sin 2 (π/8) + cos 2 (π/8) = 1, to find cos(π/8). We may use the Pythagorean identity, sin 2 (π/8) + cos 2 (π/8) = 1, to find cos(π/8). D. Correct! This is the pair of numbers that we obtain by making use of the Pythagorean identity, sin 2 (π/8) + cos 2 (π/8) = 1. We may use the Pythagorean identity sin 2 (π/8) + cos 2 (π/8) = 1 to find cos(π/8). sin 2 (π/8) + (0.3825) 2 = 1 sin 2 (π/8) = 1 sin 2 (π/8) = sin 2 (π/8) = sin(π/8) = ± sin(π/8) = ±0.9240

8 Question No. 8 of 10 Which of the following set of numbers simultaneously belongs to both the range of cosine and the range of secant? Question #08 A. {-1, 1} B. {0} C. All real numbers between -1 and 1 D. The set of all real numbers A. Correct! -1 and 1 are the only two numbers that simultaneously belongs to both the range of cosine and the range of secant. 0 belongs to the range of cosine, but does not belong to the range of secant. Any number that is simultaneously strictly less than 1 and strictly greater than -1 belongs to the range of cosine, but not to the range of secant. Neither the range of cosine nor the range of secant consists of the entire set of real numbers. The range of cosine consists of all real numbers less than or equal to one and greater than or equal to negative one. Therefore, both -1 and 1 belong to the domain of cosine. The range of secant consists of all real numbers either greater than or equal to one or less than or equal to negative one. Therefore, both -1 and 1 belong to the domain of secant. Therefore, the correct answer is A. Both negative and positive one belong simultaneously to both the range of cosine and to the range of secant.

9 Question No. 9 of 10 Calculate the length of the side opposite angle with measure 25 of the given right triangle. Question #09 A B C D A. Correct! This is the number that we obtain by making use of the formula opp = 5 * sin 25. Use the trigonometric function that relates the opposite side and the hypotenuse. Use the trigonometric function that relates the opposite side and the hypotenuse. Use the trigonometric function that relates the opposite side and the hypotenuse. We are asked to find the length of the opposite side of the right triangle given the length of the hypotenuse. The formula that will yield the length of the opposite side, given the length of the hypotenuse, is opp = 5 * sin 25 opp = 5 * opp = 2.113

10 Question No. 10 of 10 Calculate the length of the side adjacent to the angle with measure 2π/5 of the right triangle shown. Question #10 A B C D Use the trigonometric function that relates the opposite leg to the adjacent leg. B. Correct! This is the number that we obtain by making use of the formula adj = 7/tan(2π/5). Use the trigonometric function that relates the opposite leg to the adjacent leg. Use the trigonometric function that relates the opposite leg to the adjacent leg. We are asked to find the length of the adjacent side of the right triangle given the length of the opposite side of the right triangle. The formula that will yield the length of the adjacent side given the length of the opposite side is adj = 7 * cot(2π/5) If the calculator that we are using does not have a key for cotangent (i.e., cot), then we may use the fact that the reciprocal of cotangent is tangent (i.e., we may use the fact that cotangent is equal to the reciprocal of tangent). cot(2π/5) = 1/tan(2π/5) Therefore, adj = 7/tan(2π/5) = 7/3.071 = 2.279