Why is tangent sine over cosine

There are a few career paths that lead to constant use of these equations. For example, lets say you're a sound engineer working on the production of a hit artist's new album. You know that sound travels in waves, and engineers can manipulate these waves measured by and applying trigonometry to create different computer-generated sounds. What if you're an architect who needs to know the height of an existing building in a neighborhood you're assigned?

You can use the distance you are from the building and the angle of elevation to detemine the height. You can even use trig to figure out the angles the sun will shine into a building or room.

Construction workers also use sine, cosine, and tangent in this way. They need to measure the sizes of lots, roof angles, heights of walls and widths of flooring, and even more. Crime scene investigators use trigonometry to determine the angles of bullet paths, the cause of an accident, or the direction of a fallen object.

What about on a crime scene? Investigators can use trigonometry to determine angles of bullet paths, the cause of an accident, or the direction of a fallen object. NASA uses sine, cosine, and tangent.

Its physicists and astronauts often use robotic arms to complete assignments in space and use trigonometry to determine where and how to move the arm to complete their task. Thinking about studying marine biology? In this career, sine, cosine, and tangent are sometimes used to determine the size of large sea creatures from a distance, and also to calculate light levels at certain depths to see how they affect photosynthesis.

There are dozens of careers that use trigonometry in their daily tasks. So, you can stop saying things like, "I'll never use trigonometry in the real world. While all of this talk about the angles and sides of right triangles and their correspondence to one another through the beauty and magnificence of trigonometry is indeed lovely, it might leave you wondering a bit about the "Why?

By which I mean: Why exactly is this useful in the real world? What are the sin, cos, and tan buttons on my calculator for? And how do they work? When might I ever actually want to calculate the sine or cosine something?

Those, obviously, are all very important and very reasonable questions to ask. And they're also very important questions to answer. We claim that the value of each ratio depends only on the value of the angle c formed by the adjacent and the hypotenuse. To demonstrate this fact, let's study the three figures in the middle of the page.

In this example, we have an 8 foot ladder that we are going to lean against a wall. The wall is 8 feet high, and we have drawn white lines on the wall and blue lines along the ground at one foot intervals.

The length of the ladder is fixed. If we incline the ladder so that its base is 2 feet from the wall, the ladder forms an angle of nearly The ladder, ground, and wall form a right triangle.

On another page we will show that if the ladder was twice as long 16 feet , and inclined at the same angle The ratio stays the same for any right triangle with a If we measure the spot on the wall where the ladder touches o - opposite , the distance is 7. You can check this distance by using the Pythagorean Theorem that relates the sides of a right triangle:.

The ratio of the opposite to the hypotenuse is. Now suppose we incline the 8 foot ladder so that its base is 4 feet from the wall.

As shown on the figure, the ladder is now inclined at a lower angle than in the first example. Decreasing the angle c increases the cosine of the angle because the hypotenuse is fixed and the adjacent increases as the angle decreases. If we incline the 8 foot ladder so that its base is 6 feet from the wall, the angle decreases to about Move the mouse around to see how different angles in radians or degrees affect sine, cosine and tangent. In this animation the hypotenuse is 1, making the Unit Circle.

Notice that the adjacent side and opposite side can be positive or negative, which makes the sine, cosine and tangent change between positive and negative values also. It will help you to understand these relatively simple functions. You can also see Graphs of Sine, Cosine and Tangent. To complete the picture, there are 3 other functions where we divide one side by another, but they are not so commonly used.