U Can: Physics I For Dummies - Steven Holzner

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one angle and one side of the triangle, you can find all the other sides. Here are some other forms of the trig relationships – they’ll probably become distressingly familiar before you finish any physics course, but you don’t need to memorize them. If you know the preceding sine, cosine, and tangent equations, you can derive the following ones as needed:

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      To find the angle

, you can go backward with the inverse sine, cosine, and tangent, which are written as sin– 1, cos– 1, and tan– 1. Basically, if you input the sine of an angle into the sin– 1 equation, you end up with the measure of the angle itself. Here are the inverses for the triangle in Figure 2-1:

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      And here’s one more equation, the Pythagorean theorem. It gives you the length of the hypotenuse when you plug in the other two sides:

      If you need a more in-depth refresher, check out Trigonometry For Dummies, by Mary Jane Sterling (Wiley).

       Example

      Q. If the side labeled x in Figure 2-1 is 30 centimeters, and the angle θ is 30 degrees, what is the length of the hypotenuse, rounded to the nearest centimeter?

      A. The correct answer is 35 centimeters.

      1. Solve the equation

for h to get

      2. Plug in the given values:

       Practice Questions

      1. Given the hypotenuse h and the angle θ, what is the length x equal to?

      2. If x = 3 m and y = 4 m, what is the length of the hypotenuse?

       Practice Answers

      1. x = hcosθ. Your answer comes from the definition of cosine.

      2. 5 m. Start with the Pythagorean theorem:

      Plug in the numbers, and work out the answer:

      Interpreting Equations as Real-World Ideas

      After teaching physics to college students for many years, we’re very familiar with one of the biggest problems they face – getting lost in, and being intimidated by, the math.

       Tip: Always keep in mind that the real world comes first and the math comes later. When you face a physics problem, make sure you don’t get lost in the math; keep a global perspective about what’s going on in the problem, because doing so helps you stay in control.

      In physics, the ideas and observations of the physical world are the things that are important. Math operations are really only a simplified language for accurately describing what is going on. For example, here’s a simple equation for speed:

      In this equation, v is the speed, s is the distance, and t is the time. You can examine this equation’s terms to see how this equation embodies simple common-sense notions of speed. Say that you travel a larger distance in the same amount of time. In that case, the right side of the equation must be larger, which means that your speed, on the left, is also greater. If you travel the same distance but it takes you more time, then the right side of this equation becomes smaller, which means that your speed is lower. The relationship between all the different components makes sense.

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