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How long a ladder is needed to reach a windowsill 50 feet above the ground if the ladder rests against the building making an angle of $\text{\hspace{0.17em}}\frac{5\pi}{12}\text{\hspace{0.17em}}$ with the ground? Round to the nearest foot.
About 52 ft
Access these online resources for additional instruction and practice with right triangle trigonometry.
Trigonometric Functions | $$\begin{array}{cc}\text{Sine}\hfill & \phantom{\rule{1em}{0ex}}\text{sin}t=\frac{\text{opposite}}{\text{hypotenuse}}\hfill \\ \text{Cosine}\hfill & \phantom{\rule{1em}{0ex}}\text{cos}t=\frac{\text{adjacent}}{\text{hypotenuse}}\hfill \\ \text{Tangent}\hfill & \phantom{\rule{1em}{0ex}}\text{tan}t=\frac{\text{opposite}}{\text{adjacent}}\hfill \\ \text{Secant}\hfill & \phantom{\rule{1em}{0ex}}\text{sec}t=\frac{\text{hypotenuse}}{\text{adjacent}}\hfill \\ \text{Cosecant}\hfill & \phantom{\rule{1em}{0ex}}\text{csc}t=\frac{\text{hypotenuse}}{\text{opposite}}\hfill \\ \text{Cotangent}\hfill & \phantom{\rule{1em}{0ex}}\text{cot}t=\frac{\text{adjacent}}{\text{opposite}}\hfill \end{array}$$ |
Reciprocal Trigonometric Functions | $\begin{array}{cc}\text{sin}t=\frac{1}{\text{csc}t}\hfill & \phantom{\rule{1em}{0ex}}\text{csc}t=\frac{1}{\text{sin}t}\hfill \\ \text{cos}t=\frac{1}{\text{sec}t}\hfill & \phantom{\rule{1em}{0ex}}\text{sec}t=\frac{1}{\text{cos}t}\hfill \\ \text{tan}t=\frac{1}{\text{cot}t}\hfill & \phantom{\rule{1em}{0ex}}\text{cot}t=\frac{1}{\text{tan}t}\hfill \end{array}$ |
Cofunction Identities | $\begin{array}{c}\text{cos}t=\mathrm{sin}\left(\frac{\pi}{2}-t\right)\hfill \\ \text{sin}t=\mathrm{cos}\left(\frac{\pi}{2}-t\right)\hfill \\ \text{tan}t=\mathrm{cot}\left(\frac{\pi}{2}-t\right)\hfill \\ \text{cot}t=\mathrm{tan}\left(\frac{\pi}{2}-t\right)\hfill \\ \text{sec}t=\mathrm{csc}\left(\frac{\pi}{2}-t\right)\hfill \end{array}$ |
For the given right triangle, label the adjacent side, opposite side, and hypotenuse for the indicated angle.
When a right triangle with a hypotenuse of 1 is placed in a circle of radius 1, which sides of the triangle correspond to the x - and y -coordinates?
The tangent of an angle compares which sides of the right triangle?
The tangent of an angle is the ratio of the opposite side to the adjacent side.
What is the relationship between the two acute angles in a right triangle?
Explain the cofunction identity.
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.
For the following exercises, use cofunctions of complementary angles.
$\mathrm{cos}\left(\mathrm{34\xb0}\right)=\mathrm{sin}\left(\_\_\_\xb0\right)$
$\mathrm{cos}\left(\frac{\pi}{3}\right)=\mathrm{sin}\left(\_\_\_\right)$
$\frac{\pi}{6}$
$\mathrm{csc}\left(\mathrm{21\xb0}\right)=\mathrm{sec}\left(\_\_\_\xb0\right)$
$\mathrm{tan}\left(\frac{\pi}{4}\right)=\mathrm{cot}\left(\_\_\_\right)$
$\frac{\pi}{4}$
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.
$\mathrm{cos}\text{\hspace{0.17em}}B=\frac{4}{5},a=10$
$\mathrm{sin}\text{\hspace{0.17em}}B=\frac{1}{2},a=20$
$b=\frac{20\sqrt{3}}{3},c=\frac{40\sqrt{3}}{3}$
$\mathrm{tan}\text{\hspace{0.17em}}A=\frac{5}{12},b=6$
$\mathrm{tan}\text{\hspace{0.17em}}A=100,b=100$
$a=\mathrm{10,000},c=\mathrm{10,00.5}$
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