Question

daffynition deco
europe:
underground garage:
to decode the two daffynitions above: for the first nine exercises, find the
measure of the angle indicated. for the remaining exercises, find the angle measure needed
to solve the problem. round to the nearest degree. each time the answer appears in the
code, write the letter of the exercise below it.
a driveway is built on an incline so
that it rises 3 m over a distance of
20 m. what is the angle of elevation
of the driveway?
each step of a stairway rises 16 cm
for a tread width of 36 cm. what
angle does the
stairway make
with the
floor?
a roof is constructed as shown in the
diagram. find the
pitch (angle of
elevation) of
the roof.
a train decreases its altitude by 8 m
when traveling along 200 m of track.
find the angle of depression of the
track.
algebra with pizzazz!
objective 5 - d: to use trigonometric ratios
find measures of angles of right triang

Explanation:

Step1: Analyze the problem for \(N\)

We have a right - triangle with opposite side \(y = 3\) and hypotenuse \(r=5\). Use the sine function \(\sin\theta=\frac{y}{r}\).
\(\sin x=\frac{3}{5}\)
Then \(x = \sin^{-1}(\frac{3}{5})\)
Using a calculator, \(x\approx37^{\circ}\)

Step2: Analyze the problem for \(U\)

We have a right - triangle with opposite side \(y = 4\) and hypotenuse \(r = 11\). Use the sine function \(\sin\theta=\frac{y}{r}\).
\(\sin x=\frac{4}{11}\)
Then \(x=\sin^{-1}(\frac{4}{11})\)
Using a calculator, \(x\approx21^{\circ}\)

Step3: Analyze the problem for \(C\)

We have a right - triangle with opposite side \(y = 5\) and hypotenuse \(r = 8\). Use the sine function \(\sin\theta=\frac{y}{r}\).
\(\sin x=\frac{5}{8}\)
Then \(x=\sin^{-1}(\frac{5}{8})\)
Using a calculator, \(x\approx39^{\circ}\)

Step4: Analyze the problem for \(A\)

We have a right - triangle with opposite side \(y = 1\) and adjacent side \(x = 12\). Use the tangent function \(\tan\theta=\frac{y}{x}\).
\(\tan x=\frac{1}{12}\)
Then \(x=\tan^{-1}(\frac{1}{12})\)
Using a calculator, \(x\approx5^{\circ}\)

Step5: Analyze the problem for \(Y\)

We have a right - triangle with opposite side \(y = 13\) and hypotenuse \(r = 15\). Use the sine function \(\sin\theta=\frac{y}{r}\).
\(\sin x=\frac{13}{15}\)
Then \(x=\sin^{-1}(\frac{13}{15})\)
Using a calculator, \(x\approx60^{\circ}\)

Step6: Analyze the problem for \(P\)

We have a right - triangle with opposite side \(y = 20\) and adjacent side \(x = 50\). Use the tangent function \(\tan\theta=\frac{y}{x}\).
\(\tan x=\frac{20}{50}=\frac{2}{5}\)
Then \(x=\tan^{-1}(\frac{2}{5})\)
Using a calculator, \(x\approx22^{\circ}\)

Step7: Analyze the problem for \(W\)

We have a right - triangle with opposite side \(y = 21\) and hypotenuse \(r = 30\). Use the sine function \(\sin\theta=\frac{y}{r}\).
\(\sin x=\frac{21}{30}=\frac{7}{10}\)
Then \(x=\sin^{-1}(\frac{7}{10})\)
Using a calculator, \(x\approx44^{\circ}\)

Step8: Analyze the problem for \(I\)

We have a right - triangle with opposite side \(y = 25\) and adjacent side \(x = 45\). Use the tangent function \(\tan\theta=\frac{y}{x}\).
\(\tan x=\frac{25}{45}=\frac{5}{9}\)
Then \(x=\tan^{-1}(\frac{5}{9})\)
Using a calculator, \(x\approx29^{\circ}\)

Step9: Analyze the problem for \(L\)

We have a right - triangle with opposite side \(y = 3\) and adjacent side \(x = 20\). Use the tangent function \(\tan\theta=\frac{y}{x}\).
\(\tan x=\frac{3}{20}\)
Then \(x=\tan^{-1}(\frac{3}{20})\)
Using a calculator, \(x\approx9^{\circ}\)

Step10: Analyze the problem for \(R\)

We have a right - triangle with opposite side \(y = 16\) and adjacent side \(x = 36\). Use the tangent function \(\tan\theta=\frac{y}{x}\).
\(\tan x=\frac{16}{36}=\frac{4}{9}\)
Then \(x=\tan^{-1}(\frac{4}{9})\)
Using a calculator, \(x\approx24^{\circ}\)

Step11: Analyze the problem for \(B\)

We have a right - triangle with opposite side \(y = 10\) and adjacent side \(x = 16\). Use the tangent function \(\tan\theta=\frac{y}{x}\).
\(\tan x=\frac{10}{16}=\frac{5}{8}\)
Then \(x=\tan^{-1}(\frac{5}{8})\)
Using a calculator, \(x\approx32^{\circ}\)

Step12: Analyze the problem for \(T\)

We have a right - triangle with opposite side \(y = 8\) and hypotenuse \(r = 200\). Use the sine function \(\sin\theta=\frac{y}{r}\).
\(\sin x=\frac{8}{200}=\frac{1}{25}\)
Then \(x=\sin^{-1}(\frac{1}{25})\)
Using a calculator, \(x\approx2^{\circ}\)

Answer:

For \(N\): \(37^{\circ}\), for \(U\): \(21^{\circ}\), for \(C\): \(39^{\circ}\), for \(A\): \(5^{\circ}\), for \(Y\): \(60^{\circ}\), for \(P\): \(22^{\circ}\), for \(W\): \(44^{\circ}\), for \(I\): \(29^{\circ}\), for \(L\): \(9^{\circ}\), for \(R\): \(24^{\circ}\), for \(B\): \(32^{\circ}\), for \(T\): \(2^{\circ}\)