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Mathematics LibreTexts

5.4E: Exercises

  • Page ID
    18824
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    5.4: Right Triangle Trigonometry

    Verbal

    1) For the given right triangle, label the adjacent side, opposite side, and hypotenuse for the indicated angle.

    Answer

    2) When a right triangle with a hypotenuse of \(1\) is placed in the unit circle, which sides of the triangle correspond to the \(x\)- and \(y\)-coordinates?

    3) The tangent of an angle compares which sides of the right triangle?

    Answer

    The tangent of an angle is the ratio of the opposite side to the adjacent side.

    4) What is the relationship between the two acute angles in a right triangle?

    5) Explain the cofunction identity.

    Answer

    For example, the sine of an angle is equal to the cosine of its complement; the cosine of an angle is equal to the sine of its complement.

    Algebraic

    For the exercises 6-9, use cofunctions of complementary angles.

    6) \( \cos (34°)= \sin (\_\_°)\)

    7) \( \cos (\dfrac{π}{3})= \sin (\_\_\_) \)

    Answer

    \(\dfrac{π}{6}\)

    8) \( \csc (21°) = \sec (\_\_\_°)\)

    9) \( \tan (\dfrac{π}{4})= \cot (\_\_)\)

    Answer

    \(\dfrac{π}{4}\)

    For the exercises 10-16, find the lengths of the missing sides if side \(a\) is opposite angle \(A\), side \(b\) is opposite angle \(B\), and side \(c\) is the hypotenuse.

    10) \( \cos B= \dfrac{4}{5},a=10\)

    11) \( \sin B= \dfrac{1}{2}, a=20\)

    Answer

    \(b= \dfrac{20\sqrt{3}}{3},c= \dfrac{40\sqrt{3}}{3}\)

    12) \( \tan A= \dfrac{5}{12},b=6\)

    13) \( \tan A=100,b=100\)

    Answer

    \(a=10,000,c=10,000.5\)

    14) \(\sin B=\dfrac{1}{\sqrt{3}}, a=2 \)

    15) \(a=5, ∡ A=60^∘\)

    Answer

    \(b=\dfrac{5\sqrt{3}}{3},c=\dfrac{10\sqrt{3}}{3}\)

    16) \(c=12, ∡ A=45^∘\)

    Graphical

    For the exercises 17-22, use Figure below to evaluate each trigonometric function of angle \(A\).

    17) \(\sin A\)

    Answer

    \(\dfrac{5\sqrt{29}}{29}\)

    18) \( \cos A \)

    19) \( \tan A \)

    Answer

    \(\dfrac{5}{2}\)

    20) \(\csc A \)

    21) \( \sec A \)

    Answer

    \(\dfrac{\sqrt{29}}{2}\)

    22) \( \cot A \)

    For the exercises 23-,28 use Figure below to evaluate each trigonometric function of angle \(A\).

    23) \( \sin A\)

    Answer

    \(\dfrac{5\sqrt{41}}{41}\)

    24) \( \cos A\)

    25) \( \tan A \)

    Answer

    \(\dfrac{5}{4}\)

    26) \( \csc A\)

    27) \( \sec A\)

    Answer

    \(\dfrac{\sqrt{41}}{4}\)

    28) \(\cot A\)

    For the exercises 29-31, solve for the unknown sides of the given triangle.

    29)

    Answer

    \(c=14, b=7\sqrt{3}\)

    30)

    31)

    Answer

    \(a=15, b=15 \)

    Technology

    For the exercises 32-41, use a calculator to find the length of each side to four decimal places.

    32)

    33)

    Answer

    \(b=9.9970, c=12.2041\)

    34)

    35)

    Answer

    \(a=2.0838, b=11.8177\)

    36)

    37) \(b=15, ∡B=15^∘\)

    Answer

    \(a=55.9808,c=57.9555\)

    38) \(c=200, ∡B=5^∘\)

    39) \(c=50, ∡B=21^∘\)

    Answer

    \(a=46.6790,b=17.9184\)

    40) \(a=30, ∡A=27^∘\)

    41) \(b=3.5, ∡A=78^∘\)

    Answer

    \(a=16.4662,c=16.8341\)

    Extensions

    42) Find \(x\).

    43) Find \(x\).

    Answer

    \(188.3159\)

    44) Find \(x\).

    45) Find \(x\).

    Answer

    \(200.6737\)

    46) 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?

    47) 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?

    Answer

    \(498.3471\) ft

    48) 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?

    49) 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?

    Answer

    \(1060.09\) ft

    50) 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.

    51) 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.

    Answer

    \(27.372\) ft

    Real-World Applications

    52) 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?

    53) 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?

    Answer

    \(22.6506\) ft

    54) 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.

    55) 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.

    Answer

    \(368.7633\) ft

    56) 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?

    Contributor