1.4 Right triangle trigonometry  (Page 6/12)

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Algebraic

For the following exercises, use cofunctions of complementary angles.

$\mathrm{cos}\left(\text{34°}\right)=\mathrm{sin}\left(\text{__°}\right)$

$\mathrm{cos}\left(\frac{\pi }{3}\right)=\mathrm{sin}\text{(___)}$

$\frac{\pi }{6}$

$\mathrm{csc}\left(\text{21°}\right)=\mathrm{sec}\left(\text{___°}\right)$

$\mathrm{tan}\left(\frac{\pi }{4}\right)=\mathrm{cot}\left(\text{__}\right)$

$\frac{\pi }{4}$

For the following exercises, find the lengths of the missing sides if side $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ is opposite angle $\text{\hspace{0.17em}}A,$ side $\text{\hspace{0.17em}}b\text{\hspace{0.17em}}$ is opposite angle $\text{\hspace{0.17em}}B,$ and side $\text{\hspace{0.17em}}c\text{\hspace{0.17em}}$ is the hypotenuse.

$\mathrm{cos}\text{\hspace{0.17em}}B=\frac{4}{5},a=10$

$b=\frac{20\sqrt{3}}{3},c=\frac{40\sqrt{3}}{3}$

$\mathrm{tan}\text{\hspace{0.17em}}A=\frac{5}{12},b=6$

$\mathrm{tan}\text{\hspace{0.17em}}A=100,b=100$

$a=10,000,c=10,000.5$

$a=5,\text{\hspace{0.17em}}\measuredangle \text{\hspace{0.17em}}A={60}^{\circ }$

$b=\frac{5\sqrt{3}}{3},c=\frac{10\sqrt{3}}{3}$

$c=12,\text{\hspace{0.17em}}\measuredangle \text{\hspace{0.17em}}A={45}^{\circ }$

Graphical

For the following exercises, use [link] to evaluate each trigonometric function of angle $\text{\hspace{0.17em}}A.$

$\mathrm{sin}\text{\hspace{0.17em}}A$

$\frac{5\sqrt{29}}{29}$

$\mathrm{cos}\text{\hspace{0.17em}}A$

$\mathrm{tan}\text{\hspace{0.17em}}A$

$\frac{5}{2}$

$\mathrm{csc}\text{\hspace{0.17em}}A$

$\mathrm{sec}\text{\hspace{0.17em}}A$

$\frac{\sqrt{29}}{2}$

$\mathrm{cot}\text{\hspace{0.17em}}A$

For the following exercises, use [link] to evaluate each trigonometric function of angle $\text{\hspace{0.17em}}A.$

$\mathrm{sin}\text{\hspace{0.17em}}A$

$\frac{5\sqrt{41}}{41}$

$\mathrm{cos}\text{\hspace{0.17em}}A$

$\mathrm{tan}\text{\hspace{0.17em}}A$

$\frac{5}{4}$

$\mathrm{csc}\text{\hspace{0.17em}}A$

$\mathrm{sec}\text{\hspace{0.17em}}A$

$\frac{\sqrt{41}}{4}$

$\mathrm{cot}\text{\hspace{0.17em}}A$

For the following exercises, solve for the unknown sides of the given triangle.

Technology

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

$b=15,\text{\hspace{0.17em}}\measuredangle B={15}^{\circ }$

$a=55.9808,c=57.9555$

$c=200,\text{\hspace{0.17em}}\measuredangle B={5}^{\circ }$

$c=50,\text{\hspace{0.17em}}\measuredangle B={21}^{\circ }$

$a=46.6790,b=17.9184$

$a=30,\text{\hspace{0.17em}}\measuredangle A={27}^{\circ }$

$b=3.5,\text{\hspace{0.17em}}\measuredangle A={78}^{\circ }$

$a=16.4662,c=16.8341$

Extensions

Find $\text{\hspace{0.17em}}x.$

Find $\text{\hspace{0.17em}}x.$

188.3159

Find $\text{\hspace{0.17em}}x.$

Find $\text{\hspace{0.17em}}x.$

200.6737

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 $\text{\hspace{0.17em}}36°,$ and that the angle of depression to the bottom of the tower is $\text{\hspace{0.17em}}23°.\text{\hspace{0.17em}}$ How tall is the tower?

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 $\text{\hspace{0.17em}}43°,$ and that the angle of depression to the bottom of the tower is $\text{\hspace{0.17em}}31°.\text{\hspace{0.17em}}$ How tall is the tower?

498.3471 ft

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 $\text{\hspace{0.17em}}15°,$ and that the angle of depression to the bottom of the tower is $\text{\hspace{0.17em}}2°.\text{\hspace{0.17em}}$ How far is the person from the monument?

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 $\text{\hspace{0.17em}}18°,$ and that the angle of depression to the bottom of the tower is $\text{\hspace{0.17em}}3°.\text{\hspace{0.17em}}$ How far is the person from the monument?

1060.09 ft

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 $\text{\hspace{0.17em}}40°.\text{\hspace{0.17em}}$ From the same location, the angle of elevation to the top of the antenna is measured to be $\text{\hspace{0.17em}}43°.\text{\hspace{0.17em}}$ Find the height of the antenna.

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 $\text{\hspace{0.17em}}36°.\text{\hspace{0.17em}}$ From the same location, the angle of elevation to the top of the lightning rod is measured to be $\text{\hspace{0.17em}}38°.\text{\hspace{0.17em}}$ Find the height of the lightning rod.

27.372 ft

Real-world applications

A 33-ft ladder leans against a building so that the angle between the ground and the ladder is $\text{\hspace{0.17em}}80°.\text{\hspace{0.17em}}$ How high does the ladder reach up the side of the building?

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