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| 10 Theory slides |
| 9 Exercises - Grade E - A |
| Each lesson is meant to take 1-2 classroom sessions |
Here are a few recommended readings before getting started with this lesson.
Try your knowledge on these topics.
A submarine sailed underwater near a whale that was swimming 9 feet deeper than the submarine. A sonar on that submarine measured the angles of depression to the whale's mouth and tail to be 42∘ and 14∘, respectively.
How can the sailors use this information to measure the length of the whale?On the way to work, a man started wondering how tall his office building is. Suppose he is standing 10 meters from the building, looking up toward the rooftop at an approximate angle of 70∘. What is the height of the building?
Draw an angle at which the man is looking at the top of the building. Which trigonometric ratio can be used to find the building's height?
Start by drawing an angle at which the man is looking at the top of the building. Then identify a right triangle.
LHS⋅10=RHS⋅10
Rearrange equation
Use a calculator
Round to 1 decimal place(s)
Draw a right triangle so that the hypotenuse shows the expected path of the plane's descent. Analyze which side length is given to determine which trigonometric ratio should be used.
First, draw a right triangle, whose hypotenuse shows the expected path of descent of the plane and label the given distances on the diagram.
ba=b/2a/2
sin-1(LHS)=sin-1(RHS)
Use a calculator
Round to 1 decimal place(s)
In the previous two examples, the angles mentioned can be called the angle of elevation and angle of depression, respectively. To be able to refer to these angles, they should first be properly defined.
An angle of elevation is defined as an angle between the horizontal plane and oblique line from the observer's eye to some object above. Suppose someone is standing on the ground looking up toward the top of a tree.
An angle of depression is defined as an angle between the horizontal plane and oblique line from the observer's eye to some object below. Suppose someone is standing on a cliff looking down toward a ship in the ocean below.
A snowboarder is sliding down a ski hill that is 870 feet long. Its angle of depression is 12∘. What is the height of the ski hill?
Round the answer to the closest integer.Which trigonometric ratio describes the ratio between the height and the length of the ski hill in relation to the angle of depression?
It is given that the angle of depression of the ski hill is 12∘. That angle can be identified on the diagram as ∠θ. To determine the height of the hill, a line parallel to ℓ can be drawn at the bottom of the hill. By the Alternate Interior Angles Theorem, ∠θ and ∠1 are congruent angles.
It is also known that the length of the ski hill is 870 feet. Let h represent the height of the hill.
m∠1=12∘
LHS⋅870=RHS⋅870
Rearrange equation
Use a calculator
Round to nearest integer
Draw a right triangle formed by the ball and the goal post. Analyze the given side length and angle measures to determine which trigonometric ratio should be used.
Draw a right triangle based on the position of the ball and the goal post. In order to score, the ball should not rise higher than 10 feet. Therefore, to find the farthest position of Mark from the goal post, let one leg of the triangle be 10 feet long.
Calculate power
Add terms
Rearrange equation
LHS=RHS
Round to nearest integer
A family has a 7-foot tall sliding-glass door leading to the backyard. They want to buy an awning for the door that will be long enough to keep the Sun out when the Sun is at its highest point with an angle of elevation of 75∘.
Identify parallel lines and use the corresponding theorem to find the measure of an interior angle of a right triangle. Which trigonometric ratio can be used to find the length of the awning?
First, note that the concrete entrance way is parallel to the awning, so by the Alternate Interior Angles Theorem, ∠1 and the 75-degree angle are congruent angles.
LHS⋅7=RHS⋅7
Rearrange equation
Use a calculator
Round to 1 decimal place(s)
A mountaineer is planning to climb the highest mountain in the US, Denali, located in Alaska. When she reaches the peak, she wonders if she would be able to see the most eastern point of Russia, about 660 miles away from Denali.
Use the Pythagorean Theorem to find the distance from the top of the mountain to the horizon. The angle of depression can be found using one of the trigonometric ratios.
The exercise will be solved in two steps. First, the distance to the horizon will be calculated to find the answer to the given question. Then, the angle of depression will be determined.
Begin by representing the problem with a diagram. The horizon, or the farthest point the person could see, is the point where the line of sight of the mountaineer is tangent to the circle of the Earth. Let point C represent the center of the globe.
LHS−r2=RHS−r2
(a+b)2=a2+2ab+b2
Subtract term
LHS=RHS
Next, to find the angle of depression from the top of the mountain to the horizon, any of the trigonometric ratios can be used.
Throughout the lesson, different real-life problems have been solved using trigonometric ratios. Through the use of the learned methods, the challenge presented at the beginning can now be solved.
A submarine sailed underwater near a whale that was swimming 9 feet deeper than the submarine. A sonar on that submarine measured the angles of depression to the whale's mouth and tail to be 42∘ and 14∘, respectively.
How can the sailors use this information to measure the length of the whale?Use one of the trigonometric ratios to determine the horizontal distance from the submarine to the whale's nose and tale.
It is given that the angle of depression from the submarine to the whale's front is 42∘. Mark this information on the diagram and form a corresponding right triangle.
Using the tangent ratio, the horizontal distance from the submarine to the whale labeled as d1 can be found.
The wooden box has an inside base area of 40×50 square centimeters and a height of 30 centimeters.
The longest pole we can fit inside the box would fit either between the upper left and lower right corner or the box's upper right and lower-left corner. Each combination of corners the pole touches would be on opposite sides of the box.
Notice that the pole forms the hypotenuse of a right triangle. In this case, the shorter leg is the box's height and the longer leg is the diagonal of the base area of the box.
To calculate the diagonal of the box's base area, we can use the Pythagorean Theorem.
The diagonal of the base area of the box is 10sqrt(41) centimeters.
Now we can calculate the length of the longest pole by using the Pythagorean Theorem once more.
The longest pole that fits inside the box is about 70.7 centimeters.
Let's mark the angle of elevation between the pole and the floor of the box.
Let's calculate θ by using the tangent ratio.
The angle of elevation is about 25.1 ^(∘).
A company uses drones to deliver packages from their warehouse to various locations in a big city. The streets are in a uniform grid with 0.1 miles between each street. A package needs to be delivered to a location 7 blocks north and 15 blocks west of the warehouse.
Let's make a graph of the situation. We will let the warehouse W be at the origin of a coordinate plane. This means the delivery address D, will be at the coordinates (15,7). We will also label the angle with which we will send the drone as θ.
To determine θ, we will use the tangent ratio. tan θ = Opposite/Adjacent From the diagram, we see that the opposite side is 15 and the adjacent side is 7. With this information, we can determine θ.
The drone should be sent off at an angle of about 65^(∘) to the north.
From the exercise, we know that the distance from one street to the next is 0.1 miles. This means the warehouse is 0.7 miles north and 1.5 miles east of the warehouse. Let's add this to the diagram.
Since the route of the drone is the hypotenuse of a right triangle, we can determine the distance the drone must fly by using the Pythagorean Theorem.
The delivery address is about 1.7 miles away.
A semicircular tunnel is measured to be 30 feet high at its highest point. One evening, Jordan is hoping to drive through the tunnel while driving her mobile home. Will she make it through if the vehicle measures to be 24 feet wide and 28 feet high?
Let's add the dimensions of the tunnel and of the mobile home to the diagram. We will assume that Jordan is able to drive perfectly in the middle of the road.
If the vehicle can make it through, the distance from the ground to the ceiling must be greater than 28 feet at the top left and top right corners of the mobile home.
Since Jordan is driving in the middle of the tunnel, the distance from the ground to the ceiling at the critical point of the vehicle's corner, can be viewed as the leg of a right triangle with a hypotenuse equal to the tunnel's radius. The second leg will be half the width of the vehicle.
Using the Pythagorean Theorem, we can calculate a. Once again, if it is less than 28 feet, Jordan will not make it through.
The height of the tunnel where the sides of the mobile house would pass through, is less than 28 feet. Therefore, Jordan can not drive her mobile home through the tunnel. Good thing she did the math before trying!
Dominika is running across the diagonal of a rectangular parking lot.
The diagonal can be seen as the hypotenuse of a right triangle where the parking lot's width is the triangle's shorter leg. We also know that the hypotenuse is 123 % greater than the shorter leg. This means that its length is 2.23 times the length of the leg. Therefore, if we label the shorter leg x, the hypotenuse becomes 2.23x.
With the given information, we can determine ∠ θ by using the sine ratio.