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1 Vetors - Unit 3.3A - Problem 3.5A 3 49 A right triangle s hypotenuse is of length. (a) What is the length of the side adjaent to the angle? (b) What is the length of the side opposite to the angle? () What is the ratio of the lengths of the two sides? Give answers in terms of and the appropriate trigonometri funtion of. (d) Write down the formula whih relates all three sides algebraially Write an expression for the osine of the angle in terms of the length of the hypotenuse and the length of the adjaent side Here is the first of some trigonometri definitions and their onsequenes whih should be memorized. *Definition of Cosine os = adjaent side hypotenuse (3.1) Note that adjaent side is short for length of adjaent side, and hypotenuse is short for length of hypotenuse. This means that the unit of measure for the osine is For example, the adjaent side might be 4.0 m long, and the hypotenuse 5.0 m long. When 4.0 m is divided by 5.0 m the m anels out. This means that the unit of measure for the osine is dimensionless. Let b = length of adjaent side, and = length of hypotenuse, as given. Use the definition of the osine above to solve for b. Workbook for Introdutory Mehanis Problem-Solving Copyright by Daniel M. Smith, Jr. Sponsored by FIPSE (U.S. Department of Eduation)
2 3 50 Vetors - Unit 3.3A - Problem 3.5A A right triangle s hypotenuse is of length. (a) What is the length of the side adjaent to the angle? (b) What is the length of the side opposite to the angle? () What is the ratio of the lengths of the two sides? Give answers in terms of and the appropriate trigonometri funtion of. (d) Write down the formula whih relates all three sides algebraially From the osine definition, os = b. (3.2) After multiplying both sides of the equation by, we get b = os, (3.3) whih answers part (a) of the question. Write a statement of equation (3.3) in words without using b,, or. *Adjaent Side Length 3.74 The length of the adjaent side is found by multiplying the length of the hypotenuse by the osine of the angle. Beause this result (with vetor magnitude replaing length) will be used frequently in this and later hapters, it should be memorized Write a formula for the length of the adjaent side in the diagram. b
3 Vetors - Unit 3.3A - Problem 3.5A 3 51 A right triangle s hypotenuse is of length. (a) What is the length of the side adjaent to the angle? (b) What is the length of the side opposite to the angle? () What is the ratio of the lengths of the two sides? Give answers in terms of and the appropriate trigonometri funtion of. (d) Write down the formula whih relates all three sides algebraially Beause the adjaent side is now of length, and the hypotenuse is of length b, the answer is = b os. This shows that the formula (3.3) is not important to remember, but the idea given in frame 3.74 is important. It is used many times throughout physis Write an expression for the sine of the angle in terms of the length of the hypotenuse and the length of the opposite side. *Definition of Sine 3.78 Here is another important trigonometri definition whih should be memorized. sin = opposite side hypotenuse (3.4) Beause the sine is the ratio of two lengths, the unit of measure for the sine is. Let b = length of opposite side, and = length of hypotenuse, as given. Use the definition of the sine above to solve for b.
4 3 52 Vetors - Unit 3.3A - Problem 3.5A A right triangle s hypotenuse is of length. (a) What is the length of the side adjaent to the angle? (b) What is the length of the side opposite to the angle? () What is the ratio of the lengths of the two sides? Give answers in terms of and the appropriate trigonometri funtion of. (d) Write down the formula whih relates all three sides algebraially For the same reason as given in frame (3.3) the unit of measure for the sine is dimensionless. From the sine definition, sin = b. (3.5) After multiplying both sides of the equation by, we get b = sin, (3.6) whih answers part (b) of the question. Notie that equations (3.3) and (3.6) are both expressions for b, with two different meanings. This illustrates the importane of understanding the meaning of a symbol. Restate equation (3.6) in words without using b,, or. *Opposite Side Length 3.80 To find the length of the opposite side, multiply the length of the hypotenuse by the sine of the angle. This result (with vetor magnitude replaing length) will also be used frequently in this and later hapters, so it should also be memorized.
5 Vetors - Unit 3.3A - Problem 3.5A 3 53 A right triangle s hypotenuse is of length. (a) What is the length of the side adjaent to the angle? (b) What is the length of the side opposite to the angle? () What is the ratio of the lengths of the two sides? Give answers in terms of and the appropriate trigonometri funtion of. (d) Write down the formula whih relates all three sides algebraially a φ Write a formula for the length of the opposite side in the diagram. d Beause the opposite side is of length d and the hypotenuse of length a, the answer is d=asin φ Write an expression for the tangent of the angle in terms of the length of the opposite side, and the length of the adjaent side The definition of the tangent is as follows: *Definition of Tangent tan = opposite side adjaent side. (3.7) whih answers part () of the problem. This trigonometri definition should also be memorized. The definitions of frames (3.1) and (3.4) give sin os opposite side adjaent side =, hypotenuse hypotenuse (3.8) sin opposite side hypotenuse opposite side = = = tan. os hypotenuse adjaent side adjaent side (3.9) Later, we will need the result just derived: tan = sin os. (3.10)
6 3 54 Vetors - Unit 3.3A - Problem 3.5A A right triangle s hypotenuse is of length. (a) What is the length of the side adjaent to the angle? (b) What is the length of the side opposite to the angle? () What is the ratio of the lengths of the two sides? Give answers in terms of and the appropriate trigonometri funtion of. (d) Write down the formula whih relates all three sides algebraially Draw a diagram of a right triangle, label its sides a, b,and, then write your answer to part (d) of the problem In later hapters, vetor magnitudes will be determined by using the famous theorem due to Pythagoras: 2 = a 2 + b 2 Other answers are also orret beause labels for the sides are arbitrary. a b
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