MATH 109 TOPIC 3 RIGHT TRIANGLE TRIGONOMETRY. 3a. Right Triangle Definitions of the Trigonometric Functions

Transcription

1 Math 09 Ta-Right Triangle Trigonometry Review Page MTH 09 TOPIC RIGHT TRINGLE TRIGONOMETRY a. Right Triangle Definitions of the Trigonometric Functions a. Practice Problems b and Triangles Review b. Practice Problems c. Determining Lengths of Sides c. Practice Problems d. Reference ngles and Trigonometric Functions d. Practice Problems

2 Math 09 Ta-Right Triangle Trigonometry Review Page a. Right Triangle Definitions of the Trigonometric Functions In this section, we present the definitions of the trigonometric functions based on a right triangle. n alternate but equivalent method of introducing the trigonometric functions based on the unit circle will be discussed later. Consider some acute angle,. Form a right triangle, where C = π =90. (Remember that the sum of the interior angles in any triangle is 80 or π radians.) B C Fig. a. We have the following definitions. sin = opposite hypotenuse = BC C cos = adjacent hypotenuse = C B tan = opposite adjacent = sin cos = BC C cot = adjacent opposite = tan = cos sin = C BC sec = hypotenuse adjacent csc = hypotenuse opposite = cos = B C = sin = B BC

3 Math 09 Ta-Right Triangle Trigonometry Review Page In a similiar fashion, we could start with the other acute angle in the right triangle, angle B, and define sin B = opposite hypotenuse = C B, BC cos B = B, etc. Notice that opposite and adjacent are relative to the angle in question but that the definitions do not change. lso remember that since C is a right angle, ( π ) B =90 = radians. PRCTICE PROBLEMS for Topic a-right Triangle Definitions of the Trigonometric Functions B 5 C a.. Find exact values for the following. a) sin = b) cos = c) tan = d) sin B = e) cos B = f) cot B = g) sec = h) sin + cos = i) csc B = ( π ) j) sin = nswers a.. Find exact values for all remaining trig ratios given that sec =. nswer

4 Math 09 Tb and Triangles Review Page b and Triangles Review The two types of triangles mentioned in the title often appear in calculus. For the acute angles in these triangles, we will find the exact values of the associated trigonometric functions by studying the relationships between the functions, the lengths of the sides, and the angles. We begin with Fig. b.. B π C Given: B = = π rad = 5 C = π rad = 90. Fig. b. Since the sum of the interior angles is π or 80, B also = π radians. This means that the triangle is isosceles, which implies C = BC. By the Pythagorean Theorem, (C) +(BC) =, (BC) =, BC = = = C. Thus we have π Fig. b. B π C This implies that sin π =, cos π =, tan π ==cotπ, etc.

5 Math 09 Tb and Triangles Review Page 5 Next consider a triangle. B 60 0 Fig. b. C Given: =0 = π 6 rad B =60 = π rad C =90 = π rad B = We now form Fig. b., which comes from flipping the triangle in Fig. b. about the line C. B C 60 D Fig. b. Evidently, BD is an equilateral triangle since each vertex angle is 60. This means BD = and BC = CD =, since C is perpendicular to BD. (*See remark below.) So we have 0 Fig. b.5 60 B C

6 Math 09 Tb and Triangles Review Page 6 The Pythagorean Theorem enables us to write C = ( ). Specifically, = +(C), or C =. Using right triangle definitions (Review Topic a), we see that sin 0 = sin π 6 =, cos 0 = cos π 6 =, sin 60 = sin π =, cos 60 = cos π =, tan 0 = =, tan 60 = =, etc. *Remark: There is a popular formula that says the side opposite the 0 angle in a right triangle is one half the hypotenuse. Now we see why this is correct. Based on Figures b. and b.5, we can fill in the following table. θ π 6 π π sin θ cos θ tan θ Table b.6

7 Math 09 Tc Determining Lengths of Sides Review Page 7 PRCTICE PROBLEMS for Topic b and Triangles b.. Study Figs. b. and b.5 and be able to reconstruct them by memory. Try to understand how the various quantities are derived. b. From memory, fill in Table b.6. nswer b. Construct an isosceles right triangle with legs of length. Find the sine, cosine, and tangent of π and compare these values to those found in Fig. b.. What conclusions can be drawn? nswer c. Determining Lengths of Sides Right triangle trigonometry can also give the lengths of the sides of a right triangle, if you know at least one of the acute angles and one side of the triangle. For example, in the figure below, find BC and C. B 0 Fig. c. C Since sin 0 = sin π 6 = (see Review Topic b), we can write BC = sin =, or BC =6. Then C can be obtained by suing the cosine or the tangent function (or the Pythagorean Theorem). For example, = tan = BC C. Thus = 6 C,orC =6. Note that since =0 (and C =90 ), angle B =60. Then we could apply the same reasoning to angle B and obtain the lengths of the respective sides. Let s do another example where you have to use a calculator. Consider

8 Math 09 Tc Determining Lengths of Sides Review Page 8 Fig. c., where angle B =50 and C =. Find the lengths of B and BC. 50 B Fig. c. C Using a calculator (set in degree mode!) sin 50 =.766. Thus, Next,.766 = sin 50 = sin B = B, or B = =5. (approximately)..766 tan B = tan 50 =.9 = BC, or BC =.9 =.6. In calculus, sometimes the following problem is encountered. If sin θ = x, and θ is in the first quadrant, find cos θ and tan θ. To solve this problem, we first note that x = x. This allows us to create the following picture, since sin θ = opposite hypotenuse. θ x Fig. c. By the Pythagorean Theorem, we can solve for the missing side (call it a), at least algebraically. That is, c = a + b, or = a + x, or a = x.

9 Math 09 Tc Determining Lengths of Sides Review Page 9 So we now have the following. θ x x Fig. c. This means cos θ = x, and tan θ = x x. PRCTICE PROBLEMS for Topic c-determining Lengths of Sides c.. B Fig. c.5 C Suppose =5 = π radians. If BC =, find C and B. nswer c.. B C Fig. c.6 Suppose B = 8 and angle = radian. Find C and BC. nswer c.. If cos θ = x and θ is in the first quadrant, find sin θ and tan θ. nswer

10 Math 09 Td-Reference ngles and the Trigonometric Functions Reviews Page 0 d. Reference ngles and the Trigonometric Functions The definitions of the trigonometric functions in Review Topic a were based on having an acute angle (less than 90 ). What happens if > 90 = π radians? For example, Let be the angle pictured in Fig. d.. Now, what is meant by sin, cos, etc.? Fig. d. To answer this question, we consider the reference angle of, which is the acute angle between the terminal side of and the x axis. In this case, the reference angle is (π ) as in Fig. d.. (For the rest of our discussion, the reference angle of will also be denoted by ref.) ref ref Fig. d. Fig. d. We next form a right triangle between the terminal side of and the x axis, as indicated in Fig. d.. Then we define sin = sin(ref angle). Note that since (ref ) is an acute angle with its own right triangle, we can again use definitions involving an opposite side, adjacent side, etc., just as we did for Fig. a. in Review Topic a.

11 Math 09 Td Reference ngles and the Trigonometric Functions Review Page For example, consider Fig. d. below. 5 ref Fig. d. Then we define sin = sin(reference angle)= 5,, cos = 5, tan = =, etc. Observe that the adjacent side of the reference angle is ( ), since x is negative. lso note that since we have a right triangle, the numbers,, and 5 satisfy the Pythagorean Theorem. The situation is similar if the terminal side of lies in the third or fourth quadrant, as in Fig. d.5. ref 5 Fig. d.5 Fig. d.6 We form a right triangle with the reference angle as indicated in Fig. d.6, and we suppose the lengths of the sides are as shown. Then sin = sin(reference angle) = 5 5, cos =, tan =, etc.

12 Math 09 Td-Reference ngles and the Trigonometric Functions Reviews Page PRCTICE PROBLEMS for Topic d-reference ngles and the Trigonometric Functions d.. Consider angle as in Fig. d.7, and suppose its reference angle forms a right triangle with the dimensions shown in Fig. d.8. Find sin, cos, tan. ref 5 Fig. d.7 Fig. d.8 nswer Evidently, based on Figures.d, d.6, and d.8, we can conclude that sin θ > 0 for θ in the first and second quadrants and sin θ < 0 for θ in the third and fourth quadrants. Similarly, cos θ is positive in the first and fourth quadrants and negative in the second and third quadrants, while tan θ is positive in the first and third quadrants and negative in the second and fourth quadrants. The table below summarizes these observations. st Q. nd Q. rd Q. th Q. sin θ + + cos θ + + tan θ + + Table d.9 - Q means quadrant.

13 Math 09 Td Reference ngles and the Trigonometric Functions Review Page PRCTICE PROBLEMS for Topic d-continued d.. Determine whether the following expressions are positive or negative. a) tan 7π 6, b) sec π, c) sin π 6, d) sin 5π 6, e) cos 5π, f) tan θ, if π <θ<π. nswers d.. Name the quadrant(s) where θ must terminate, given the following information. a) sin θ<0 b) sin θ>0 and tan θ<0. nswers Finally, we can use the idea of reference angles to extend what we know about and triangles from the first quadrant to the other quadrants. For example, let = π. Let B, C, and D be the 6 positive angles in the second, third, and fourth quadrants respectively that have reference angle π 6. moment s reflection yields that B = 5π 6,C= 7π 6, and D = π 6. = π 6 ref B B = 5π 6 ref C C = 7π 6 D = π 6 ref D Fig. d.0. ll reference angles above = π 6.

14 Math 09 Td-Reference ngles and the Trigonometric Functions Reviews Page s shown in Review Topic b, sin π 6 =. This leads to Fig. d.. θ = π 6 θ Fig. d. Figs. d.0 and d. immediately yield the following information. θ = 5π 6 θ = 7π 6 θ = π 6 Fig. d.. Fig. d. enables us to create the table below. θ sin θ cos θ 5π 6 7π 6 π 6 tan θ Table d.

15 Math 09 Td Reference ngles and the Trigonometric Functions Review Page 5 We remark that similar reasoning applies for negative angles. For example, if θ = π, it has the same reference angle as the drawing on the right in 6 ( Fig. d. when θ = π ) (. Thus, sin π ) = ( 6 6, cos π ) = 6, etc. s an additional example, θ = π would have the same reference angle as θ = 5π. ( nother observation is that sin π ) ( π ) ( = sin and cos π ) = ( π ) cos. We will see in Review Topic that this observation is true in 6 general. That is, for any x, sin( x) = sin x (sin x is an odd function), cos( x) = cos x (cos x is an even function), and tan( x) = tan x (tan x is an odd function). The concept of even and odd functions is explained in Review Topic. PRCTICE PROBLEMS for Topic d-continued d.. d.5. d.6. What are the positive angles in the second, third, and fourth quadrants that have π as their reference angle? nswer What are the positive angles in the second, third, and fourth quadrants that have π as their reference angle? nswer rguing as we did in Fig. d.0, d., and d., complete the tables below. θ π π 5π sin θ cos θ tan θ

16 Math 09 Td-Reference ngles and the Trigonometric Functions Reviews Page 6 Table d. d.6. Continued θ sin θ π 5π 7π cos θ tan θ Table d.5 d.7. Find the exact value for each expression. ( a) cos π ) ( b) tan π ) c) sin ( π ) nswers nswers Beginning of Topic 09 Study Topics 09 Skills ssessment

17 Math 09 Ta-Right Triangle Definitions of the Trigonometric Functions nswers Page 7 NSWERS to PRCTICE PROBLEMS (Topic a-right Triangle Definitions of the Trigonometric Functions) a.. a) sin = 5 ; b) cos = 5 ; c) tan = ; d) sin B = 5 ; e) cos B = 5 ; f) cot B = ; g) sec = 5 ; h) sin + cos =; i) csc B = 5 ; j) sin ( π ) = 5 ; a.. sec = implies we have the following picture. The Pythagorean Theorem implies that the missing side must equal 7 6 9= 7. Thus, sin =, cos =, 7 tan =, cot =, and csc =. 7 7

18 Math 09 Tb and Triangles nswers Page 8 NSWERS to PRCTICE PROBLEMS (Topic b and Triangles nswers) b. θ π 6 π π sin θ cos θ tan θ nswer b. b. The values of the trigonometric functions in Fig. b. will be the same for any isoceles right triangle.

19 Math 09 Tc Determining Lengths of Sides nswers Page 9 NSWERS to PRCTICE PROBLEMS (Topic c-determining Lengths of Sides) c.. tan π =. lso, tan π = BC BC. Thus, = C =. Note that C C since the triangle is isosceles, elementary geometry also implies that C =. Next sin π = = BC B = B. Thus B =. (Reminder: We can also find B by using the Pythagorean Theorem.) c.. Using a calculator set in radian mode, we find that sin() =.8. This means BC =.8, or BC =6.7. Next, cos() =.5. Since 8 cos = C B = C, we have C =8(.5) =.. 8 c.. Given cos θ = x we can draw the following picture. θ x Fig. c.7 The Pythagorean Theorem implies the missing side has length x. Then, sin θ = x, and tan θ = x x.

20 Math 09 Td-Reference ngles and the Trigonometric Functions nswers Page 0 NSWERS to PRCTICE PROBLEMS (Topic d-reference ngles and the Trigonometric Functions) d.. sin = 5 5, cos =, tan = = 5. d.. a) positive b) negative, since sec θ = cos θ. c) negative d) positive e) negative f) negative d.. a) rd quadrant or th quadrant b) nd quadrant d.. π, π, 5π d.5. π, 5π, 7π

21 Math 09 Td-Reference ngles and the Trigonometric Functions nswers Page NSWERS to PRCTICE PROBLEMS (Topic d-reference ngles and the Trigonometric Functions) d.6. θ π π 5π sin θ cos θ tan θ Table d.5 θ π 5π 7π sin θ cos θ tan θ Table d.5

22 Math 09 Td-Reference ngles and the Trigonometric Functions nswers Page NSWERS to PRCTICE PROBLEMS (Topic d-reference ngles and the Trigonometric Functions) d.7. a), b) c). NOTE: The solution for part a) can be derived in two ways. First, if we use the concept of reference angle, then θ = π has reference angle π. From Table b.6, cos π =. Since π is in the fourth quadrant where the x coordinate is positive, we have ( cos π ) =. s a second approach, since cos x is an even function, (i.e., cos( x) = ( cos x, we can simply say cos π ) ( π ) = cos =. Similar comments apply for answers b) and c).