Download 8.5 angles of elevation and depression 2016 ink.notebook

8.5 angles of elevation and depression 2016 ink.notebook
March 02, 2017
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8.5 Angles of
Elevation and
Depression
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8.5 Sine and
Cosine as
Complementary
Angles
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8.5 angles of elevation and depression 2016 ink.notebook
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Lesson Objectives
March 02, 2017
Standards
Lesson Notes
8.5 Angles of Elevation and Depression Press the tabs to view details.
Lesson Objectives
Standards
Lesson Notes
After this lesson, you should be able to successfully solve problems using angles of elevation and depression.
Lesson Objectives
Standards
Lesson Notes
G.SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
G.MG.3 Apply geometric methods to solve design problems.
G.CO.10 Prove theorems about triangles.
Press the tabs to view details.
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8.5 angles of elevation and depression 2016 ink.notebook
When an observer is looking up, the angle of _____________ is the angle between a horizontal line from the observer and the line of sight to an object that is above the horizontal line. In the diagram, AB is the _____________ line. x is the _________________________ from the observer at A to the object at C.
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When an observer is looking down, the angle of _____________ is the angle between a horizontal line from the observer and the line of sight to an object that is below the horizontal line. In the diagram, PQ is the horizontal line. y is the angle of _____________ from the observer at P to the object at R.
In the diagram, ∠1 is the angle of elevation from the
tower T to plane P.
∠2 is the angle of depression from the plane to tower.
m∠1 = m∠2
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8.5 angles of elevation and depression 2016 ink.notebook
vertical height
lin
e o
f s
i
gh
t
Angles of depression and elevation are equal.
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ight
of s
line
angle of elevation
angle of depression
line
of sig
ht
1. Classify each as an angle of elevation or an angle of depression.
horizontal length
Use SOHCAHTOA
Angles are alternate interior angles.
The angle of depression is not usually inside the triangle, that's why you put the angle measure on the angle of elevation inside the triangle at the ground level.
∠1 ____________________ ∠3___________________
∠2 _____________________∠4 __________________
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8.5 angles of elevation and depression 2016 ink.notebook
2. The angle of elevation from point A to the top of a cliff is 34°. If point A is 1000 feet from the base of the cliff, how high is the cliff?
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3. An airplane over the Pacific sights an ship at a 20° angle of depression. If the plane is 475 m above water, how many meters is it from a point 475 m above the ship?
Let x = the height of the cliff
Where does the 34° go?
The height of the cliff is about _______ feet.
4. HILL TOP The angle of elevation from point A to the top of a hill is 49°. If point A is 400 feet from the base of the hill, how high is the hill?
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8.5 angles of elevation and depression 2016 ink.notebook
5. SUN Find the angle of elevation of the Sun when a 12.5­meter­tall telephone pole casts an 18­meter­long shadow. March 02, 2017
6. AIR TRAFFIC From the top of a 120­foot­high tower, an air traffic controller observes an airplane on the runway at an angle of depression of 19°. How far from the base of the tower is the airplane?
Name the angle of depression or angle of elevation in each figure.
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8.
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8.5 angles of elevation and depression 2016 ink.notebook
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9. MOUNTAIN BIKING On a mountain bike trip along the Gemini Bridges Trail in Moab, Utah, Nabuko stopped on the canyon floor to get a good view of the twin sandstone bridges. Nabuko is standing about 60 meters from the base of the canyon cliff, and the natural arch bridges are about 100 meters up the canyon wall. If her line of sight is 5 meters above the ground, what is the angle of elevation to the top of the bridges? Round to the nearest tenth degree.
10. SHADOWS Suppose the sun casts a shadow off a 35­foot building. If the angle of elevation to the sun is 60°, how long is the shadow to the nearest tenth of a foot?
11. INDIRECT MEASUREMENT Kyle is at the end of a pier 30 feet above the ocean. His eye level is 3 feet above the pier. He is using binoculars to watch a whale surface. If the angle of depression of the whale is 20°, how far is the whale from Kyle's binoculars? Round to the nearest tenth foot.
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8.5 angles of elevation and depression 2016 ink.notebook
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In general, the sine of an acute angle is equal to the cosine of the complement of that angle. sin A = cos B and cos A = sin B
The trigonometric function of the complement of an angle is called a cofunction. The sin and cosines are cofunctions of each other.
Example 1: Find the sine and cosine of the acute angles in the right triangle shown. Start with the sine and cosine of ÚA.
Then, find the sine and cosine of ÚB.
In general, the sine of an acute angle is equal to the cosine of the complement of that angle. sin A = cos B and cos A = sin B
The trigonometric function of the complement of an angle is called a cofunction. The sin and cosines are cofunctions of each other.
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The acute angles of a right triangle are complementary angles. If the measure of one of the two acute angles is given, the measure of the second acute angle can be found by subtracting the given measure from 90¡.
Example 3: Write cos 36¡ in terms of the sine.
Example 4: Write sin 28¡ in terms of its cofunction. Example 5: Write cos 51¡ in terms of its cofunction.
Example 6: Find two angles that satisfy the equation: sin (2x – 4)¡ = cos (3x + 9)¡ If sin (2x – 4)¡ = cos (3x + 9)¡, then sin (2x – 4)¡ = cos (3x + 9)¡ are the measures of complementary angles. The sum of the measures must be 90¡.
(2x – 4) + (3x + 9) = 90 x = ______
The measurements of the two angles are _____ and _____.
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8.5 angles of elevation and depression 2016 ink.notebook
Find two angles that satisfy the equation.
12. sin (3x + 2)¡ = cos (x + 44)¡
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Find two angles that satisfy the equation.
13. sin (2x + 20)¡ = cos (3x + 30)¡
On the Worksheet
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8.5 angles of elevation and depression 2016 ink.notebook
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Name the angle of depression or angle of elevation in each figure.
1.
HOMEWORK
8.5 Practice WS on Angles of Elevation and Depression
2.
3. What is the angle between the ground
and the top of the tree?
5. A building near Atlanta, Georgia, is 181 feet tall. On a particular day at noon it casts a 204­foot shadow. What is the sun's angle of elevation at that time?
4. Find the value of x.
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8.5 angles of elevation and depression 2016 ink.notebook
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6. SKIING A ski run is 1000 yards long with a vertical drop of 208 yards. Find the angle of depression from the top of the ski run to the bottom.
7. A car drives up a 100 yard incline at a steepness of 5°. What vertical height has the car ascended?
8. To find the height of a flagpole, a surveyor moves 150 feet away from the base of the pole and then, with a tripod 3 feet tall, measures the angle of elevation to the top of the pole to be 26°. What is the height of the pole?
9. CONSTRUCTION A roofer props a ladder against a wall so that the top of the ladder reaches a 30­foot roof that needs repair. If the angle of elevation from the bottom of the ladder to the roof is 55°, how far is the ladder from the base of the wall? Round your answer to the nearest foot.
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10. WATER TOWERS A student can see a water tower from the closest point of the soccer field at San Lobos High School. The edge of the soccer field is about 110 feet from the water tower and the water tower stands at a height of 32.5 feet. What is the angle of elevation if the eye level of the student viewing the tower from the edge of the soccer field is 6 feet above the ground? Round to the nearest tenth.
11. TOWN ORDINANCES The town of Belmont restricts the height of flagpoles to 25 feet on any property. Lindsay wants to determine whether her school is in compliance with the regulation. Her eye level is 5.5 feet from the ground and she stands 36 feet from the flagpole. If the angle of elevation is about 25°, what is the height of the flagpole to the nearest tenth?
12. INDIRECT MEASUREMENT Mr. Dominguez is standing on a 40­foot ocean bluff near his home. He can see his two dogs on the beach below. If his line of sight is 6 feet above the ground and the angles of depression to his dogs are 34° and 48°, how far apart are the dogs to the nearest foot?
13. LIGHTHOUSES Sailors on a ship at sea spot the light from a lighthouse. The angle of elevation to the light is 25°. The light of the lighthouse is 30 meters above sea level. How far from the shore is the ship? Round your answer to the nearest meter.
30m
25
?
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8.5 angles of elevation and depression 2016 ink.notebook
14. RESCUE A hiker dropped his backpack over one side of a canyon onto a ledge below. Because of the shape of the cliff, he could not see exactly where it landed.
From the other side, the park ranger reports that the angle of depression to the backpack is 32°. If the width of the canyon is 115 feet, how far down did the backpack fall? Round your answer to the nearest foot.
Find the cosine and sine of the acute angles in the triangles shown.
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17.
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15. AIRPLANES The angle of elevation to an airplane viewed from the control tower at an airport is 7°. The tower is 200 feet high and the pilot reports that the altitude is 5200 feet. How far away from the control tower is the airplane? Round your answer to the nearest foot.
Write each trigonometric function in terms of its cofunction.
18. sin 64¡
19. cos 84¡
20. cos 38¡
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8.5 angles of elevation and depression 2016 ink.notebook
Write each trigonometric function in terms of its cofunction.
21. sin 24¡
22. cos 72¡
Find two angles that satisfy the equation.
25. cos (5x + 49)¡ = sin (3x + 57)¡
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Find two angles that satisfy the equation.
24. sin (2x + 30)¡ = cos (3x + 5)¡
23. sin 45¡
Find two angles that satisfy the equation.
26. sin (5x – 12)¡ = cos (x + 54)¡
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8.5 angles of elevation and depression 2016 ink.notebook
Find two angles that satisfy the equation.
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Answers:
27. cos (3x – 10)¡ = sin (3x – 20)¡
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