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Mathematics LibreTexts

5.2e: Exercises - Right Angle Trigonometry

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    A: Given two sides of an acute angle, find all six trigonometric ratios

    Exercise \(\PageIndex{A}\)

    \( \bigstar \) Find the values of all six trigonometric functions of angles \(A\) and \(B \) in the right triangle \(\triangle\,ABC \) illustrated below. Simplify!

    alt

    1. \(b = 12 \), \(c = 13\)

    2. \(b = 15 \), \(c = 17\)

    3.  \(a = 7 \), \(c = 25\)

    4.  \(a = 20 \), \(c = 29\)

    5.  \(a = 9 \), \(b = 40 \)

    6.  \(a = 1 \), \(b = 2 \)

    7. \(a = 1 \), \(b = 3\)

    8. \(a = 2 \), \(b = 5\)

    9. \(a = 5 \), \(c = 6\)

    10. \(a = 2 \), \(c = \sqrt{6} \)

    11. \(b = 7 \), \(c = 8\) 

    12. \(b = 3 \), \(c = \sqrt{15} \) 

    \( \bigstar \) In each of the triangles below, find \(\sin \left(A\right), \; \cos \left(A\right),\; \tan \left(A\right),\; \sec \left(A\right),\; \csc \left(A\right),\; \cot \left(A\right)\).

    17.屏幕快照 2019-07-08 上午11.11.26.png     18.屏幕快照 2019-07-08 上午11.11.40.png

    Answers to odd exercises.

    For all triangles #1-12, the trigonometric ratios can be determined using the following formulas: \( \sin A = \cos B = \frac{a}{c}, \)  \( \cos A = \sin B = \frac{b}{c}, \) \( \tan A = \cot B = \frac{a}{b}, \)  \( \csc A = \sec B = \frac{c}{a}, \)  \( \sec A = \csc B = \frac{c}{b}, \) \( \cot A = \tan B = \frac{b}{a}, \)  The missing side ineeded to calculate these ratios is given below:
            1. \(a = 5\)     3. \(b = 24\)     5. \(c = 41 \)     7. \( c = \sqrt{10} \)     9. \(b = \sqrt{11} \)      11. a = \( \sqrt{15} \)
    17. \(\sin \left(A\right) = \frac{5\sqrt{41}}{41}, \; \cos \left(A\right) = \frac{4\sqrt{41}}{41},\; \tan \left(A\right) = \frac{5}{4},\; \sec \left(A\right) = \frac{\sqrt{41}}{4},\; \csc \left(A\right)= \frac{\sqrt{41}}{5},\; \cot \left(A\right) = \frac{4}{5}\)

    B: Given one trig ratio of an acute angle, find the other trigonometric ratios

    Exercise \(\PageIndex{B}\)

    \( \bigstar \) Find the values of the other five trigonometric functions of the acute angle \(A \) given the indicated value of one of the functions. Simplify!

    21. \( \sin A= \dfrac{1}{2} \)

    22. \( \sin A= \dfrac{5}{12} \)

    23.  \(\cos\;A = \dfrac{1}{5}\)

    24.  \(\cos\;A = \dfrac{2}{3}\)

    25.  \(\tan\;A = \dfrac{3}{4}\)

    26.  \(\tan\;A = \dfrac{9}{5}\)

    27.  \(\csc\;A = \dfrac{7}{3}\)

    28. \(\csc A=\dfrac{\sqrt{3}}{1} \)

    29.  \(\sec\;A = 3\)  

    30.  \(\sec\;A = \dfrac{5\sqrt{6}}{3}\)

    31. \( \cot A=100 \)

    32.  \(\cot\;A = 4\)
    Answers to odd exercises.

    21. \( \sin \theta = \frac{1}{2}\),   \( \cos \theta = \frac{\sqrt{3}}{2} \),    \( \tan \theta = \frac{\sqrt{3}}{3}\),    \(\csc \theta = 2\),    \( \sec \theta = \frac{2\sqrt{3}}{3}\),   \( \cot \theta = \sqrt{3} \)
    23. \( \sin \theta = \frac{2\sqrt{6}}{5} \),    \( \cos \theta = \frac{1}{5} \),    \( \tan \theta = 2\sqrt{6} \),    \( \csc \theta = \frac{5\sqrt{6}}{12} \),    \( \sec \theta = 5 \),    \( \cot \theta = \frac{\sqrt{6}}{12}  \)
    25. \( \sin \theta = \frac{3}{5} \),    \( \cos \theta = \frac{4}{5} \),    \( \tan \theta = \frac{3}{4} \),    \( \csc \theta = \frac{5}{3} \),    \( \sec \theta = \frac{5}{4} \),    \( \cot \theta = \frac{4}{3} \)
    27. \( \sin \theta = \frac{3}{7} \),    \( \cos \theta = \frac{2\sqrt{10}}{7} \),    \( \tan \theta = \frac{3\sqrt{10}}{20} \),    \( \csc \theta = \frac{7}{3} \),    \( \sec \theta = \frac{7\sqrt{10}}{20} \),    \( \cot \theta = \frac{2\sqrt{10}}{3} \)
    29. \( \sin \theta = \frac{2\sqrt{2}}{3} \),    \( \cos \theta = \frac{1}{3} \),    \( \tan \theta = 2\sqrt{2} \),    \( \csc \theta = \frac{3\sqrt{2}}{4} \),    \( \sec \theta = 3\),    \( \cot \theta = \frac{\sqrt{2}}{4} \)
    31. \( \sin \theta = \frac{\sqrt{101}}{101} \),    \( \cos \theta = \frac{100\sqrt{101}}{101} \),    \( \tan \theta = \frac{1}{100} \),    \( \csc \theta =\sqrt{101} \),    \( \sec \theta = \frac{\sqrt{101}}{100} \),    \( \cot \theta = 100 \)

    D: Cofunctions

    Exercise \(\PageIndex{D}\)

    Use cofunctions of complementary angles to write an equivalent expression.

    36. \( \cos (34°)= \sin (\_\_°)\) 37. \( \cos (\dfrac{π}{3})= \sin (\_\_\_) \) 38. \( \csc (21°) = \sec (\_\_\_°)\) 39. \( \tan (\dfrac{π}{4})= \cot (\_\_)\)
    Answers to odd exercises.

    37. \(\dfrac{π}{6}\)       39. \(\dfrac{π}{4}\) 

    C: Given one side and and an acute angle for a right triangle, find the other sides and angles

    Exercise \(\PageIndex{A}\)

    \( \bigstar \) In each of the following triangles, solve for the unknown sides and angles. Give answers to 4 decimal digits.

    41.屏幕快照 2019-07-08 上午11.12.00.png                42.屏幕快照 2019-07-08 上午11.12.58.png             43.屏幕快照 2019-07-08 上午11.12.39.png        

    44.屏幕快照 2019-07-08 上午11.12.17.png         45. 屏幕快照 2019-07-08 上午11.13.11.png         46. A right triangle with corners labeled A, B, and C. Sides labeled b, c, and 16.5. Angle of 81 degrees also labeled.  

    47. A right triangle with corners labeled A, B, and C. Hypotenuse has length of 15 times square root of 2. Angle B is 45 degrees.                  48.屏幕快照 2019-07-08 上午11.13.29.png 

    49. \(a=5,\) angle opposite side \(a\) is \( ∡ A=60^∘\)

    50. Hypotenuse \(c=12,\) and one acute angle is \( ∡ A=45^∘\)

    \( \bigstar \) Find \(x\).

    51.  A triangle with angles of 36 degrees and 50 degrees and side x. Bisector in triangle with length of 85.        52.  A triangle with angles of 63 degrees and 39 degrees and side x. Bisector in triangle with length of 82.     

    53. A right triangle with side of 119 and angle of 26 degrees. Within right triangle there is another right triangle with angle of 70 degrees instead of 26 degrees. Difference in side length between two triangles is x.        54. A right triangle with side of 115 and angle of 35 degrees. Within right triangle there is another right triangle with angle of 56 degrees. Side length difference between two triangles is x.     

    Answers to odd exercises.

    41. \(b \approx 12.1244\), \( c =14\), \( B = 60^{\circ} \)    43. \(a \approx 5.3171\), \( c \approx 11.3257\), \( A = 28^{\circ} \)    45. \(a \approx 9.0631\), \( b \approx 4.2262\), \( B = 25^{\circ} \)
    47. \(a = 15\), \( b = 15\), \( A = 45^{\circ} \)    49. \(b \approx 2.8868\), \( c \approx 5.7735\), \( B = 30^{\circ} \)    51. \(188.3159\)     53. \(200.6737\) 

    E: Applications

    Exercise \(\PageIndex{E}\)

    60. A radio tower is located \(400\) feet from a building. From a window in the building, a person determines that the angle of elevation to the top of the tower is \(36°\), and that the angle of depression to the bottom of the tower is \(23°\). How tall is the tower?

    61. A radio tower is located \(325\) feet from a building. From a window in the building, a person determines that the angle of elevation to the top of the tower is \(43°\), and that the angle of depression to the bottom of the tower is \(31°\). How tall is the tower?

    62. A \(200\)-foot tall monument is located in the distance. From a window in a building, a person determines that the angle of elevation to the top of the monument is \(15°\), and that the angle of depression to the bottom of the tower is \(2°\). How far is the person from the monument?

    63. A \(400\)-foot tall monument is located in the distance. From a window in a building, a person determines that the angle of elevation to the top of the monument is \(18°\), and that the angle of depression to the bottom of the monument is \(3°\). How far is the person from the monument?

    64. There is an antenna on the top of a building. From a location \(300\) feet from the base of the building, the angle of elevation to the top of the building is measured to be \(40°\). From the same location, the angle of elevation to the top of the antenna is measured to be \(43°\). Find the height of the antenna.

    65. There is lightning rod on the top of a building. From a location \(500\) feet from the base of the building, the angle of elevation to the top of the building is measured to be \(36°\). From the same location, the angle of elevation to the top of the lightning rod is measured to be \(38°\). Find the height of the lightning rod.

    66. A \(33\)-ft ladder leans against a building so that the angle between the ground and the ladder is \(80°\). How high does the ladder reach up the side of the building?

    67. A \(23\)-ft ladder leans against a building so that the angle between the ground and the ladder is \(80°\). How high does the ladder reach up the side of the building?

    68. The angle of elevation to the top of a building in New York is found to be \(9\) degrees from the ground at a distance of \(1\) mile from the base of the building. Using this information, find the height of the building.

    69. The angle of elevation to the top of a building in Seattle is found to be \(2\) degrees from the ground at a distance of \(2\) miles from the base of the building. Using this information, find the height of the building.

    70. Assuming that a \(370\)-foot tall giant redwood grows vertically, if I walk a certain distance from the tree and measure the angle of elevation to the top of the tree to be \(60°\), how far from the base of the tree am I? 

    Answers to odd exercises.

    61. \(498.3471\) ft     63. \(1060.09\) ft     65. \(27.372\) ft     67. \(22.6506\) ft     69. \(368.7633\) ft 

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