Lesson 6 More Than Right Develop Understanding

Ready

Find the area of each triangle.

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Set

Solve the following application problems using right triangle trigonometry.

6.

Students in a high school mathematics class were given the following problem.

While traveling across a flat stretch of desert, Joey and Holly make note of the top of a butte* in the distance that seems to be directly in front of them. They estimate the angle of elevation to the top as $5 °$ . After traveling $6$ miles towards the butte, the angle of elevation is $25 °$ . Approximate the height of the butte in miles and in feet. $5,280 ft = 1 mile$ . Use the given values to solve for $x$ in miles and $d$ in both miles and feet. $\tan 5 \approx 0.1$ and $\tan 25 \approx 0.5$

* Buttes are tall, flat-topped, steep-sided towers of rock.

7.

Rework problem 6. This time use $\tan 5 \approx 0.0875, \tan 25 \approx 0.4663$ . Or, use the values in your calculator without rounding them.

8.

The situation in problem 6 included approximating angles, so rounding didn’t matter very much. Consider a situation where Joey and Holly were serious rock climbers and were planning on scaling the face of the butte, and they had used an inclinometer* to measure the angles of elevation accurately. Describe the difference between rounding to one decimal place and rounding to four or more decimals.

* An inclinometer is an instrument used for measuring angles of slope, elevation, or depression of an object with respect to a plane.

9.

The Star Point Ranger Station and the Twin Pines Ranger Station are $30 miles$ apart along a straight, mountain road. Each station gets word of a cabin fire in a remote area known as Ben’s Hideout. A straight path from Star Point to the fire makes an angle of $34 °$ with the road, while a straight path from Twin Pines makes an angle of $14 °$ with the road. Find the distance, $d$ , of the fire from the road.

10.

In problem 9 we had two expressions that were both equal to $d$ . Use both of your expressions and the value you found for $d$ in problem 9 to check your answers.

Explain why they were not exactly equal. Does it matter if this application were real life? Why or why not?

Go

Solve for the missing sides and angles in the right triangles. Write answers in radical form. Do NOT use a calculator.

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Write a rule for finding the sides of an isosceles right triangle when you know the hypotenuse and the measure of the hypotenuse does NOT show a $\sqrt{2}$ .

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Write a rule for finding the missing sides in a $30 ° - 60 ° - 90 °$ triangle when you know the side opposite the $60 °$ angle, but the measurement doesn’t show a $\sqrt{3}$ .

Fill in the missing measurements.

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18.

Fill in the ratios for the given functions. Do not use a calculator. Answers should be in radical form.

19.

$\sin 45 ° =$
$\cos 45 ° =$
$\tan 45 ° =$

20.

$\sin 30 ° =$
$\cos 30 ° =$
$\tan 30 ° =$

21.

$\sin 60 ° =$
$\cos 60 ° =$
$\tan 60 ° =$