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Main Trigonometric Functions
There are six main trigonometric functions:
- Sine (sin)
- Cosine (cos)
- Tangent (tan)
- Secant (sec)
- Cosecant (csc)
- Cotangent (cot)
These functions are used to relate the angles of a triangle with the sides of that triangle. Trigonometric functions are important when studying triangles and modeling periodic phenomena such as waves, sound, and light.
To define these functions for the angle theta, begin with a right triangle. Each function relates the angle to two sides of a right triangle. First, let's define the sides of the triangle.
- The hypotenuse is the side opposite the right angle. The hypotenuse is always the longest side of a right triangle.
- The opposite side is the side opposite to the angle we are interested in, theta.
- The adjacent side is the side having both the angles of interest (angle theta and the right angle).
The relationship between the trigonometric functions and the sides of the triangle are as follows:
- sine(theta) = opposite / hypotenuse
- cosecant(theta) = hypotenuse / opposite
- cosine(theta) = adjacent / hypotenuse
- secant(theta) = hypotenuse / adjacent
- tangent(theta) = opposite / adjacent
- cotangent(theta) = adjacent / opposite
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These trigonometric functions have practical applications in surveying, building, engineering, and even medicine. Here's one practical way to use these functions to solve a problem:
The angle of elevation of an airplane is 23 degrees, and its altitude is 2500 meters. How far away is it?
We are trying to solve this right triangle for the hypotenuse x. Since the side length we know is opposite the angle we know, we can use the sine function.
Sin(23) = 2500m / x
x = 6398.3 meters
How to Solve Problems
You can use these ratios to solve for any side or angle of a right triangle. The information you are given will help you determine which function to use.
Solve for b if you know that c is 2.5 km and B is 15.7 degrees.
Since we know the measurements of the angle opposite the side we are trying to find and the hypotenuse, we can use either the sine or cosecant functions. Most often when solving these problems, the sine, cosine, and tangent functions are used because they are easier to calculate with a calculator. So, we will use the sine function for this problem.
Sin(15.7) = b / 2.5 km
0.271 = b / 2.5 km
b = 0.6765 km
Try this one:
Solve triangle ABC given that A is 35 degrees and c is 15 feet.
We don't know much about this triangle, but because it is a right triangle and we know at least two other sides or angles, we can use trigonometric functions to solve for the rest. The easiest place to start is to find the angle B. Since all triangles have angle measures that add up to 180 degrees, to solve for B, just subtract.
180 - 90 - 35 = B
B = 55 degrees
Then we can use sine and cosine to solve for sides a and b. Using angle A, and the hypotenuse, the equation to solve for side a is:
Sin(35 degrees) = a / 15 feet
a = 8.6 feet
The equation to solve for side b is:
Cos(35 degrees) = b / 15 feet
b = 12.3 feet
The six main trigonometric functions are sine, cosine, tangent, secant, cosecant, and cotangent. They are useful for finding heights and distances, and have practical applications in many fields including architecture, surveying, and engineering.
As soon as you've reviewed the lesson, apply your knowledge in order to:
- State the six trigonometric functions
- Name the sides of a triangle
- Recognize the relationships between triangular sides and trigonometric functions
- Use trigonometric functions to solve problems
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Trigonometric Functions: Definition & Examples
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