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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?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.
Since the lengths of the opposite side and the hypotenuse are given, the angle of descent can be calculated by using the sine ratio.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 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.
The angle between the horizon and the viewer's gaze as they look down is an angle of depression. In the diagram, it is labeled as ∠θ.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∘.
Find the length of the awning they should buy. Round the answer to the first decimal place.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.
Let ℓ represent the length of the awning. In order to find its value, the cotangent ratio can be used.LHS⋅7=RHS⋅7
Rearrange equation
Use a calculator
Round to 1 decimal place(s)