Jennifer has an MS in Chemistry and a BS in Biological Sciences.
In mathematics, most operations have an inverse operation. The inverse operation is the opposite of the original operation. So, addition is the opposite of subtraction and division is the inverse of multiplication. Squared is the inverse of the square root.
Trigonometric functions are no different. They all have inverse operations.
- The inverse of sine is arcsine.
- The inverse of cosine is arccosine.
- The inverse of tangent is arctangent.
- The inverse of secant is arcsecant.
- The inverse of cosecant is arccosecant.
- The inverse of cotangent is arccotangent.
Inverse trigonometric functions have practical applications in physics, engineering, and navigation.
The sine (sin) of an angle in a right triangle is a trigonometric function that can be defined as the ratio of the side opposite the angle to the hypotenuse.
The sine function can be used to solve problems involving a right triangle. For example:
A railroad track is inclined at 1°. If the train travels 3000 meters up the hill, what is the vertical height that the train will have traveled?
sin'x' = opposite/hypotenuse
sin1° = opposite/3000 m
opposite = (sine1°)(3000 m)
opposite = 52.36 m
So, the train rises 52.36 meters.
The arcsine is the inverse function of the sine function. It is most useful when trying to find the angle measure when two sides of a triangle are known.
To solve problems using the arcsine, you need a scientific calculator. It should have a button, usually above the sin button.
The arcsine will allow you to find the measure of an angle when you know the ratio of the side opposite the angle to the hypotenuse.
1.) Find the angle measure Θ in the following triangle.
sin'Θ' = 7/13
sin'Θ' = 0.538
x = 57.4°
2.) Let's look at the railroad example from above. If you knew that the train rose 3 vertical miles on a track that was 5 miles long, you could determine the angle of incline of the track using the arcsine.
sin'x' = 3 miles/5 miles (opposite/hypotenuse)
sin 'x' = 0.6
Now, to find the measure of that angle, simply find the arcsine of 0.6
x = 36.9°
As with any other inverse operations, you can always check your answer by doing the opposite.
Check this answer by calculating sin(36.9°)
sin 36.9 = 0.6
This confirms that our calculation was correct.
The arcsine is the inverse function of the sine function. This means that they are opposite functions, and one will cancel out the other. The arcsine is mainly used to determine the measure of an angle when two sides of a right triangle are known. In order to use this function, you must know the hypotenuse and the side opposite the angle you are trying to determine. It has applications in navigation, engineering, and other sciences.
When this lesson ends, you'll be able to:
- Define inverse operations
- List the inverse operations pertaining to trigonometric functions
- Describe how to solve the inverse operation for sine
- Solve trigonometric examples using arcsine and check answers
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