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Trigonometry Introducing Mathematics (Fx350ex/Fx82ex)

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Trigonometry Introducing Mathematics (Fx350ex/Fx82ex) Barry Kissane Module 4 Trigonometry A calculator is a useful tool for many aspects of trigonometry, both for solving problems involving measurement and understanding relationships among angles. To start with check that your calculator is set to use degrees (by looking for a small D symbol on the display). Use T to change it if necessary. Start this module in Calculate mode, by tapping p1. Trigonometry and right triangles Definitions of trigonometric functions can be based on right triangles, such as the one shown below, for which C is a right angle. A C B AC BC AC For this triangle, sine B = , cosine B = and tangent B = . AB AB BC In addition, because the triangle is right angled, the Theorem of Pythagoras allows us to see the relationship between the lengths of the two sides (AC and BC) and the hypotenuse (AB): AC 2 BC 2 AB2 Together, these relationships allow us to determine all the sides and the angles of a right triangle, even when only some of the information is known. For example, if we know that angle B is 38o and that AB = 4.2 m, we can find the length of AC using the definition of sine above: AC AC sin B = = , so AC = 4.2 sin B. AB 4.2 Computations like this are easily performed on the calculator. It is a good practice to include the bracket after the angle size, even though the calculator will calculate correctly without it. Notice that it is not necessary to use a multiplication sign in this case to find that AC ≈ 2.59 m. The screen above shows that the calculator gives the result to many places of decimals, but care is needed in deciding the level of accuracy of calculations like this. Since the original measurement of AB, which is used in the calculation, is given to only one place of decimals, it would be consistent with this to give AC to only one place of decimals too: AC ≈ 2.6 m. To find the length of the other side, BC, a similar process could be employed, using the cosine of B. Alternatively, the Pythagorean Theorem can be used to see that © 2016 CASIO COMPUTER CO., LTD. CASIO Worldwide Education website http://edu.casio.com/ 4-1 Module 4: Trigonometry BC 2 AB2 AC 2 and so BC AB2 AC 2 This can be readily determined from the above calculator result (which shows AC) as follows: So BC ≈ 3.3 m. Notice that it has not been necessary to record any intermediate results here, and nor has it been necessary to use approximations (such as AC ≈ 2.6), which is likely to introduce small errors. In general, it is better to not round results until the final step in any calculations on calculators or computers. In this case, use of an approximation for AC results in a (slightly) different and slightly less accurate result for BC: The calculator can also be used to determine angles in a right triangle, using inverse trigonometric relationships. For example the inverse tangent of an angle can be accessed with ql?. If you know a perpendicular height and a distance, you can use this to find an angle of elevation. In the diagram below. QR represents the height of the Eiffel Tower in Paris. The tower is 324 metres high. P is a point one kilometre away from the base, measured on level ground. What is the angle of elevation from P to the top of the tower? R 324 m 1 km P Q RQ Since tan P = , PQ RQ Angle P = tan1 . PQ The calculator gives the result using decimal degrees, as shown below on the left. To represent this angle using degrees, minutes and seconds, tap the x key, to get the result shown at right above. Once again, care is needed to express results to a defensible accuracy. In this case, one kilometre from the Eiffel Tower, the angle of elevation from the ground is about 18o. Introducing Mathematics with ClassWiz 4-2 Barry Kissane When using the calculator for trigonometry, it is sometimes necessary to enter angles in degrees, minutes and seconds. However, the calculator works with decimal degrees. To see the relationship between these two ways of using sexagesimal measures, enter an angle in degrees, minutes (and possibly seconds also) and tap = to see the decimal result. To enter the angle, tap the x key after each of the degrees, minutes and seconds are entered; although a degree symbol is shown each time, the calculator interprets the angle correctly, as the first screen shows after tapping =. Tap the x key to see the angle in decimal degrees. Tap the x key again to see the angle in degrees, minutes and seconds. Tables of values It is instructive to examine tables of values for trigonometric functions, in order to see how the functions depend on the angle. To do this use p3 to select Table mode and enter the functions concerned, using x as the variable. In the screens below, we consider the sine and cosine functions. Tap = after entering each function. As only 30 values are permitted, a good choice for the table is to start at x = 0o and end at x = 180o, with a step of 10o. (Degree symbols are not used, however.) Tap = after each value is entered. Use the cursor keys to scroll the table, showing the values of the sine and cosine of the angle in the final two columns. Notice how the values of sine increase up to a maximum of 1 from 0o up to 90o, and then decrease to 0 again at 180o. The cosine values also show a pattern, but it is a different one. These tables will allow you to quickly sketch graphs of the two functions, which will also help to illustrate the relationships involved. Check also that angles the same distance from 0o and 180o have the same sine value, as illustrated above for 20o and 160o.This illustrates the general relationship that sin x = sin (180o – x). Again, the relationship is different for cosines, with cos x = - cos (180o – x). You can also compute values for sin x and cos x for 180o ≤ x ≤ 360o and observe other relationships. Remember that the calculator is restricted to 30 rows in a table, however, so a different choice of table step (such as 15o) is needed. © 2016 CASIO COMPUTER CO., LTD. 4-3 Module 4: Trigonometry While the sine and cosine functions have a special relationship with each other, the tangent function follows a different pattern. To explore this, firstly construct a table of values for f(x) = tan x from 0o to 180o in steps of 10o. (Merely tap = instead of defining a second function, to construct a table of just one function.) Parts of this table are shown below. As there is no value defined for tan 90o, the calculator shows an error at that point. To examine more closely the behaviour of the function near 90o, tabulate values of the function that are close to 90o. There are two ways of doing this. In the screens below, the step has been changed to 0.01, and the table range changed to ensure no more than 30 values are obtained. An alternative method is to change individual x values by highlighting them and entering a new value, followed by =. The tabulated values are changed accordingly. The screens below show some examples of this, starting with the original table. Notice from the last two screens that values exceeding the table’s display width can both be entered and displayed. You can see from these tables that there is a discontinuity at x = 90o, around which the tangent function changes sign, from a large positive value to a large negative value. These tables will also help you to visualise the shape of a graph of the tangent function from 0o to 180o. Exact values Although practical measurement always involves approximations (as no measurement is ever exact), it is also interesting to study the exact values of some trigonometric relationships. You may have noticed that the calculator provides some of these, three of which are shown below. It is often possible to see the origins of these values, by drawing suitable triangles. For example, an isosceles right triangle includes angles of 45o, while an equilateral triangle has angles of 60o. In addition, you can use these and other exact values to find the exact values of the sine, cosine and tangent of other angles, using various formulae for combinations of angles. To illustrate, consider the formula for the tangent of a sum of two angles: tan A tan B tan( A B) 1 tan Atan B Introducing Mathematics with ClassWiz 4-4 Barry Kissane You can use the exact values of tan 45o = 1 and tan 30o = 1/√3 to calculate for yourself tan 75o. You should be able to obtain the same exact result as the calculator, shown below. Experiment in this way with other addition and subtraction trigonometric formulae for sine, cosine and tangent. The calculator will also help you to experiment to see how trigonometric values repeat. For example, the screens below show how values for the sine function repeat every 360o. Radian measure Radian measure provides a different way of measuring angles from the use of degrees, by measuring along a circle. An angle has a measure of 1 radian if it cuts an arc of a circle that is one radius long. As there are 2π radius measures in the circumference of a circle, the measure of a full rotation in a circle is 2π (radians), the same angle as 360o in degrees (or sexagesimal measure.) If you are using radian measures often, it is best to use SET UP to convert the angle unit to radians.
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