Part Five: Trigonometry Review. Trigonometry Review

Transcription

1 T.5 Trigonometry Review Many of the basic applications of physics, both to mechanical systems and to the properties of the human body, require a thorough knowledge of the basic properties of right triangles, which are triangles that have one angle that is equal to 90 o. Furthermore, many of the problems of rotational motion depend upon the radian measurement of angles. Basic Trigonometric Relationships The following trigonometric functions are those that would be used most frequently in physics applications. Refer to the figure below for identification of the terms in each expression. : angle a: side adjacent the angle o: side opposite the angle h: hypotenuse of the right triangle o sin h a cos h o sin tan a cos a o h Do the Self-Check to see if you need to refresh your knowledge of the use of the trigonometry of the right triangle. The answers to the Self-Check are found at the end of this section. Self- Check Use the right triangle shown below to answer questions 1-5.

2 1. sin 2. cos 3. tan A certain right triangle has two sides of lengths a = 14 cm and b = 20 cm. What is the length of the hypotenuse? Use the right triangle shown below to answer questions 7 and a = 8. b = 9. A surveyor wishes to determine the distance between two points A and B, but cannot make a direct measurement because a river intervenes. The surveyor steps off a line AC at 90 o to AB and 264 meters long. With the transit at point C, the surveyor measures the angle between line AB and the line formed by C and B. Angle BCA is measured to be 62 o. What is the distance from A to B?

3 Trigonometry Review Trigonometry is used extensively in physics. Any time a direction must be indicated for a quantity, there is a very good possibility that trigonometry will be required. In many physics applications, knowledge of the properties of right triangles is sufficient for the description of the situation. Right angle trigonometry is based on certain ratios and relationships that hold for all right triangles. Properties of Right Triangles The basic trigonometric functions of the angles of a right triangle are defined in terms of ratios of the sides of the triangle. Angles are measured either in units of degrees or radians. If two lines intersect, the angle between the lines is defined by drawing a circle around the intersecting lines, with the point of intersection at the center of the circle. The complete circle is defined to have a measure of 360 o ; the angle measure in degrees for the intersecting lines is then the fraction of the 360 o enclosed by the lines. For example, if an angle defined by 2 intersecting lines encloses 1/6 of the circle, the angle measure is (1/6)360 0 or 60 o. The radian measure is defined in a similar manner. The circle is defined to have a measure of 2 radians. An angle defined by two intersecting lines that enclose 1/6 of the circle would have a measure of (1/6)(2 radians) = /3 radians. The relationship between degree measure and radian measure can be seen in the figure below. Since a full circle has a measure of 360 o, or alternately, 2 radians, these measures can be used to construct a conversion factor between the two measures. The conversion factor can then be used to convert any degree measure to radians or vice versa.

4 2 rad rad 1 o 360 rad 1 o 180 o Converting 120 o o rad to radians: 120 o rad 3 2.1rad The length of the arc contained by the two intersecting lines is proportional to the angle between the two lines. If the angle is given in radian measure, this proportionality can be expressed as s r, where r is the radius of the circle. This relationship shows that the angle in radians is the ratio between the arc s length s and the angle :. Since the radian is defined as a ratio between r two lengths, it is dimensionless; that is, it has no units associated with it. The trigonometric functions are also defined in terms of ratios. Shown below are three similar triangles, all of which have the same angles. Use the figure and the given information and complete the calculations necessary to fill in the second table below. = 53 o = 37 o Triangle Side a Side b Side c A 22.5 mm 30 mm 37.5 mm B 30 cm 40 cm 50 cm C 45 m 60 m 75 m

5 + + + right angle a c b c a b Triangle A Triangle B Triangle C Notice that the angles are the same for all the triangles. Does the difference in the lengths of the sides make a difference to the value of the angle? What determines the difference between angle and angle? Look at the sum of angles and for each triangle. What do you notice? From this result, what general statement can be made about the sum of the acute angles in a right triangle? A pair of angles such as these, that add up to 90 o, are known as complimentary angles. What general statement can be made about the sum of all of the angles in a triangle? Take a look at the ratio of b over c. For all three of the triangles, this is the ratio of the side opposite the angle to the hypotenuse. What do you notice about these ratios? The ratio of the side opposite an angle in a triangle to the hypotenuse of that triangle is defined to be the sine of the angle, indicated by sin. Using your calculator, find the sine of the angle. Does this verify your result from the table? Look at the figure and determine the value of sin. Verify your result using your calculator. Take a look at the ratio of a over c. For all three of the triangles, this is the ratio of the side adjacent the angle to the hypotenuse. What do you notice about these ratios? The ratio of the side adjacent an angle in a triangle to the hypotenuse of that triangle is defined to be the cosine of the angle, indicated by cos. Using your calculator, find the cosine of the angle. Does this verify your result from the table? Look at the figure and determine the value of cos. Verify your result using your calculator. Take a look at the ratio of a over b. For all three of the triangles, this is the ratio of the side adjacent the angle to the side opposite the angle. What do you notice about these ratios? The

6 ratio of the side adjacent an angle in a triangle to the side opposite the angle of that triangle is defined to be the tangent of the angle, indicated by tan. Using your calculator, find the tangent of the angle. Does this verify your result from the table? Look at the figure and determine the value of tan. Verify your result using your calculator. Take the ratio of sin to cos. Compare this to your answer for tan. What did you find? In the diagram shown, angles and are complimentary, and angles and are complimentary. Using what you have learned about the basic trig functions, use the diagram to fill in the table below. Angle Sin (Angle) Cos (Angle) Look at the results in your table. Can you express a general statement about complimentary angles and their trigonometric functions? If you are having trouble with this look, for example, at the cos and the sin. What do you notice? For each triangle shown in the previous figure, square the two legs (not the hypotenuse) of the triangle and add them together. For Triangle D: For Triangle E: Now square the hypotenuse of each triangle and compare that result to the sum for the corresponding triangle. For Triangle D: For Triangle E: What relationship is there between the square of the hypotenuse of a right triangle and the sum of the squares of its two legs? This

7 relationship is known as the Pythagorean theorem, and it is valid for all right triangles. Inverse Trigonometric Functions The trigonometric functions can also be used to find unknown angles if the lengths of the sides of the triangle are known. Consider again triangles D and E. From the calculations already done, we know that the sin = 4/5 = 0.8. But how do we find? What we are really asking here is What angle has a sine which is 0.8? This question is answered by the inverse trigonometric functions. The inverse sine, written sin -1, gives a result that is the angle which has the sine given by the argument of the sin -1. For example, sin will give as a result the angle that has a sine of 0.8. Likewise, cos would give as a result the angle that has a cosine of 0.8, and the tan would give the angle whose tangent is 0.8. For most calculators, the inverse trigonometric functions are obtained by first pushing an inverse or 2 nd button, followed by the desired trigonometric function. If you are unsure of how to obtain the inverse trig functions on your particular calculator, ask your instructor. Use your calculator and your results for Triangles D and E to find the unknown angles. = = = = Practice Problems in Trigonometry Convert the following angles to radian measures, and find their sine, cosine and tangent values using your calculator o o

8 3. 37 o 4. Use the figure shown to fill in the following table. Triangle A b A B 5. One acute angle of a right triangle is 20 o. The length of the hypotenuse is 6 cm. Find the lengths of the other two sides of the triangle. 6. A car is traveling northeast (45 o N of E). If the car traveled a total distance of km, how far north did it travel? How far east? 7. Melissa left her house and jogged a distance of 12 blocks due east, then 15 blocks due north before stopping at her favorite snack bar for a fruit smoothie. How far, in a straight line, was she from her house when she stopped? What angle does that straight line distance make from due east? 8. To find the height of a tall building, a physics student steps 75 paces (each 1 meter) from the base of the building. Using a ruler at arm's length (1 meter), the student finds that at this distance, the building appears to be 50 centimeters high as compared to the ruler. Determine the approximate height of the building.

9 Answers to Self- Check sin cos tan = 37 o 5. = 53 o cm 24.4 cm 7. a = 34.5 cm 8. b = 28.9 cm 9. AB = tan 62 o = m Answers to Practice Problems 1. = 1.05 rad; sin = 0.866; cos = 0.5; tan = = rad; sin = 0.799; cos = 0.602; tan = = rad; sin = 0.602; cos = 0.799; tan = Triangle A: a = 1.91, b = 0.58, = 17 o Triangle B: a = 4.53, b = 3.94, = 41 o 5. a = 6 cos 20 o = 5.64 cm; o = 6 sin 20 o = 2.05 cm 6. N = sin 45 o = 30 m; E = cos 45 o = 30 m 7. d = 19.2 blocks; = 51.3 o 8. h = 37.5 m

10