 # Uses of Trigonometry

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More Lessons for High School Geometry
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A series of free, online High School Geometry Video Lessons.
Videos, worksheets, and activities to help Geometry students.

In this lesson, we will learn

• how to define and calculate the angle of elevation and depression
• how to use the sine function to calculate the area of a triangle
• how to use the Law of Sines
• how to use the Law of Cosines

### Angle of Elevation and Depression

In real world situations we often discuss the angle of elevation and depression. The angle of elevation and depression is used often in word problems, especially those involving a persons line of sight as they look up at an object. These angles can be used to solve problems involving trigonometric functions such as sine, cosine, and tangent, and the inverse trigonometric functions.
How to define an angle of elevation or an angle of depression?
The following video describes the angle of elevation and depression. It also gives some examples on how to calculate the angles of elevation and depression.

### Using Sine to Calculate the Area of a Triangle

Using the standard formula for the area of a triangle, we can derive a formula for using sine to calculate the area of a triangle. Using sine to calculate the area of a triangle means that we can find the area knowing only the measures of two sides and an angle of the triangle. There is no need to know the height of the triangle, only how to calculate using the sine function
How to calculate the area of any triangle using two sides and an included angle?
This video explains how to determine the area of a triangle using the sine function when given SAS

This video provides an example of how to determine the area of a triangle using the sine function.

### The Law of Sines

One method for solving for a missing length or angle of a triangle is by using the law of sines. The law of sines, unlike the law of cosines, uses proportions to solve for missing lengths. The ratio of the sine of an angle to the side opposite it is equal for all three angles of a triangle. The law of sines works for any triangle, not just right triangles.
This video derives the law of sines and solve triangles using the law of sines. The Law of Sines: The Ambiguous Case
This video explains how to solve a triangle using the law of sines when given SSA. How to solve word problems using the law of sines?
Example:
Standing on one bank of the canyon, a surveyor notices a tree at a bearing of 115°. Then the surveyor walks 300 meters. The bearing of the tree is then 85°. What is the distance across the canyon?

How to solve applications using the law of sines?
Example:
A group of forest rangers are hiking through Denali National Part to wards Mt. McKinley, the tallest montain in Nort America. From their campsite, they can see Mt. McKinley, and the angle of elevation from their campsite to the summit is 21°. They know that the slope of mountain forms a 127° angle with ground and the verical height of Mt. McKinley is 20,320 feet. How far away is their campsite from the base of the mountain? If they can hike 2.9 miles in an hour, how long will it take to take them to get to the base?

### The Law of Cosines

One method for solving for a missing length or angle of a triangle is by using the law of cosines. The law of cosines, unlike the law of sines, is similar to the Pythagorean theorem, but it works for all triangles, not just right triangles. In order to understand how to use the law of cosines effectively, one must understand the cosine function and inverse trigonometric functions.
This video introduces the law of cosines and solve triangles using the law of cosines. How to solve word problems using using the law of cosines?
Examples:
There is a transmission tower near I-10. The range of the service from the tower form a 47° angle and range of service is 28 miles to one section of I-10 and 31 miles to another point on I-10. If a driver is traveling at 50 mph, how long will she have service?

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