Chapter 8: Right Triangle Trigonometry

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Haberman MTH 11 Setion I: The Trigonometri Funtions Chapter 8: Right Triangle Trigonometry As we studied in Part 1 of Chapter 3, if we put the same angle in the enter of two irles of different radii,

1 Haberman MTH 11 Setion I: The Trigonometri Funtions Chapter 8: Right Triangle Trigonometry As we studied in Part 1 of Chapter 3, if we put the same angle in the enter of two irles of different radii, we an onstrut two similar triangles; see Figure 1. Figure 1: The angle in both a unit irle and in a irle of radius r, induing similar right triangles. We an use these similar triangles to obtain the following ratios: os ( ) x sin( ) y and 1 r 1 r Solving these ratios for os( ) and sin( ), respetively, gives us the following: x os( ) and sin( ) r y r

2 Haberman MTH 11 Setion I: Chapter 8 To help us remember these ratios, it s best to imagine yourself standing at angle looking into the triangle. Then the side labeled y is on the opposite side of the triangle while the side labeled x is adjaent to you. We use these desriptions (as well as the fat that the side labeled r is the hypotenuse of the triangle) to refer to the sides of the triangle in Fig.. Figure : We use the terms opposite (or ), adjaent (or ), and hypotenuse (or ) to refer to the sides of a right triangle. DEFINITION: If is the angle given in the right triangles in Figure (above), then y sin( ) and r x os( ). r Consequently, the other trigonometri funtions an be defined as follows: sin( ) os( ) tan( ) os( ) sin( ) ot( ) 1 1 se( ) os( ) 1 1 s( ) sin( ) We an use these ratios along with the Pythagorean Theorem (see below) to learn a great deal about right triangles. THE PYTHAGOREAN THEOREM: If the sides of a right triangle (i.e., a triangle with a 90 angle) are labeled like the one given in Figure 3, then a b. Figure 3

3 Haberman MTH 11 Setion I: Chapter 8 3 EXAMPLE 1: Find the value for all six trigonometri funtions of the angle given in the right triangle in Figure 4. (The triangle might not be drawn to sale.) Figure 4 First we need to use the Pythagorean Theorem to find the length of the hypotenuse. (1) (9) We an use this value to label our triangle: 5 15 Thus, Figure sin( ) os( ) tan( ) ot( ) se( ) s( ) 9 3

4 Haberman MTH 11 Setion I: Chapter 8 4 EXAMPLE : Find the value for all six trigonometri funtions of the angle given in the right triangle in Figure 6. (The triangle might not be drawn to sale.) Figure 6 First we need to use the Pythagorean Theorem to find the length of the side labeled a. a (5) (13) a a 144 a 1 We an use this value to label our triangle: Figure To determine the sine and osine values of angle, imagine standing at angle and looking into the triangle. Then, sin( ) 1 13 os( ) 5 13 tan( ) 1 5 ot( ) 5 1 se( ) 13 5 s( ) 13 1

5 Haberman MTH 11 Setion I: Chapter 8 5 We an use the trigonometri funtions, along with the Pythagorean Theorem to solve a right triangle, i.e., find the missing side-lengths and missing angle-measures for a triangle. EXAMPLE 3: Solve the triangle in Figure 8 by finding,, and. (The triangle might not be drawn to sale.) Figure 8 We an use the Pythagorean Theorem to find. (4) (8) Now we an use the tangent funtion to find. Note that we hoose to use tangent, not sine or osine, to find the sine it allows us to use the given info, rather than info that we've found. (If we made a mistake finding, we don't want to ompound that mistake but using the inorret value to find other values.) tan( ) 8 4 tan( ) 1 tan () Note also that we ould have just as easily found first, instead of. No matter whih angle we find first, we an easily find the last angle by using the fat that the sum of the angles in a triangle is 180 :

6 Haberman MTH 11 Setion I: Chapter Let s summarize our findings: 4 5, 63.43, and 6.5. EXAMPLE 4: Solve the triangle in Figure 9 by finding b,, and. (The triangle might not be drawn to sale.) Figure 9 We an use the Pythagorean Theorem to find b. b b 100 Now we an use the sine funtion to find : b 51 b sin( ) sin Although it would be just as easy to use tangent or osine to find, to we hoose to use sine sine it allows us to use the given info, rather than info that we've found, in order to avoid the possibility of ompounding our mistakes.

7 Haberman MTH 11 Setion I: Chapter 8 Now we an use the fat that the sum of the angles in a triangle is 180 : Let s summarize our findings: b 51, 44.4, and EXAMPLE 5: Solve the right triangle given in Figure 10 by finding A, b, and. (The triangle might not be drawn to sale.) Figure 10 First, notie that angle A must measure 30 sine the sum of the angles in a triangle is 180 and our triangle already has angles measuring 60 and 90. The only thing we know about the sides of the triangle is that the side adjaent to the 60 angle is units long. Notie that the osine of the 60 angle is, and we an use this fat to find : os(60 ) 1 os(60 ) 1 (sine os(60 ) 1 ) 14

8 Haberman MTH 11 Setion I: Chapter 8 8 Now that we know the length of two sides of the triangle, we ould use the Pythagorean Theorem to find the length of the third side, b. Instead, we ll use the fat that, on the triangle, the sine of 60 is b : sin(60 ) sin(60 ) 3 b b b b 14 b 3 3 (sine 14) 3 (sine sin(60 ) ) Let s summarize our findings: A 30, b 3, and 14.

Chapter 9: Right Triangle Trigonometry

Haberman MTH 11 Section I: The Trigonometric Functions Chapter 9: Right Triangle Trigonometry As we studied in Intro to the Trigonometric Functions: Part 1, if we put the same angle in the center of two