Trigonometric Functions of an Acute Angle
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1 Trigonometry Module T06 Trigonometric Functions of an Acute Angle Copyright This publication The Northern Alberta Institute of Technology All Rights Reserved. LAST REVISED December, 2008
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3 Trigonometric Functions of an Acute Angle Statement of Prerequisite Skills Complete all previous TLM modules before completing this module. Required Supporting Materials Access to the World Wide Web. Internet Explorer 5.5 or greater. Macromedia Flash Player. Rationale Why is it important for you to learn this material? Trigonometry is simply the study of triangles. It has been used since the time of the Greeks. The trigonometric ratios introduced in this module are used in a wide variety of applications today including surveying, navigation, engineering, and construction. Learning Outcome When you complete this module you will be able to Solve problems involving the six trigonometric functions. Learning Objectives. Define and list the six trigonometric functions in terms of the sides of a right triangle. 2. Determine the function values of angles between 0 and 90 degrees inclusively.. Determine the angle from a given function value. 4. Calculate the function value of the acute angles in any right triangle, given two sides of the triangle. 5. Determine all function values of an acute angle given one function value of the angle. Connection Activity Consider the following Triangle: A c = 5 b B a = 4 C Your work with right triangles allows you to solve for side b. What is the ratio of side a to side c? What if a = 8 and c = 0? What is the ratio of side a to side c now? Notice the ratio has not changed. The proportionate increase in the sides a and c means that angle B did not change and the ratio of the sides remained the same. This is a hint at how the ratio of two sides of a triangle can help you determine the angle between the sides.
4 OBJECTIVE ONE When you complete this objective you will be able to Define and list the six trigonometric functions in terms of the sides of a right triangle. Exploration Activity Define and list the six trigonometric functions in terms of the sides of a right triangle. There are six possible trigonometric functions. Following is a list of these functions and their abbreviations. The basic functions: sine - sin cosine - cos tangent - tan The reciprocals of the basic functions: cosecant - csc secant - sec cotangent - cot The above trigonometric functions are meaningless in themselves, you must relate them to angles of a triangle. These side-angle relationships are illustrated in the following right angle triangle that identifies the sides relative to θ. These relationships hold true only for triangles with a 90º angle. These relationships must be memorized: opposite sin θ = hypotenuse adjacent cos θ = hypotenuse opposite tan θ = adjacent Hypotenuse Opposite θ csc θ = sec θ = hypotenuse = opposite hypotenuse = adjacent adjacent cot θ = opposite = sin θ cosθ tan θ θ opposite Adjacent to θ 2
5 NOTE:. The sin and csc functions are related in that the csc is the reciprocal of the sin. 2. The cos and sec functions are reciprocals of each other.. The tan and cot functions are also reciprocals of each other. We can name our right triangle ABC, and identify the functions in terms of the lettered sides. sin A = c a cos A = c b c B a tan A = b a A b C csc A = a c sec A = b c cot A = a b NOTE: When labeling a triangle the same letter is used for the angle and the side opposite it. Use capitals for angles and lowercase for sides. Observe the reciprocal relationships in these fractions, i.e. sin A a = = csc A = c c csc A a cos A b = = sec A = c c sec A b tan A a = = cot A = b b cot A a
6 If we wished to find the 6 functions of angle B in the previous diagram we would obtain. sin B = cos B = tan B = opp b hyp c = csc B = = hyp c opp b adj a hyp c = sec B = = hyp c adj a opp b adj a = cot B = = adj a opp b Observe again the reciprocal relationships in the above functions. Sometimes students may get confused as to which side is the "opposite" side from an angle. Remember, the "opposite" side is the side which is not an arm of the angle. The hypotenuse is always opposite the right angle. The adjacent side is always an arm of the angle. EXAMPLE For the triangle shown, determine the ratios of the six trigonometric functions of angle θ. EXACT DECIMAL 2 sin θ = cos θ = 20 2 tan θ = 6 20 csc θ = θ sec θ = = cot θ = 2 4. Pythagorean Theorem Check 4
7 csc, sec, cot could have also been found by using the reciprocals of the first trig functions as shown in the next examples.. sin θ = cscθ = = =.6667 sinθ cos θ = secθ =.2500 cosθ = =. tan θ = cotθ = = =. tanθ
8 Experiential Activity One. In the figure, choose either angle P or angle R. a) OP = tan? OR P b) PR = sec? OP c) OP = cos? PR O d) OP = sin? PR e) PR = csc? OP 2. The three sides of a right triangle are 5, 2, and. Let A be the acute angle opposite the side 5 and let B be the other acute angle. Find the six functions of A and B. Show Me. In the figure, if x = r, find the six functions of θ. R R z x X θ r Z 4. In the figure in question, x = 2r, find the sine, cosine, and tangent of θ. 6
9 Experiential Activity One Answers. a) R, b) P, c) P, d) R, e) R 2. sin A = sin B = 0.92 cos A = 0.92 cos B = tan A = tan B = csc A = csc B =.08 sec A =.08 sec B = cot A = cot B = sin θ = x ;cos θ = r ;tanθ = ;csc θ = z ;sec θ = z ;cosθ = z z x r 4. 2 sin r r = ; cos ; tan 2 z = z = 7
10 OBJECTIVE TWO When you complete this objective you will be able to Determine the function values of angles between 0 and 90 degrees inclusively. Exploration Activity Now we will use the calculator to find the ratios of the 6 trigonometric functions. Refer to your calculator manual for this objective. To find the SINE, COSINE, and TANGENT of any angle, you enter the desired function and the desired angle. EXAMPLE Take: 2 a) Find the trigonometric ratio of sin 2º Step : press the sin key Step 2: press 2º and then the = key Step : display: b) cos 2 Step : press the cos key Step 2: enter 2º Step : display: c) tan 2 Step : press the tan key Step 2: enter 2º Step : display: Ensure calculator is in degree mode when the angle is given in degrees and you are using any of the trig functions on your calculator!! 8
11 EXAMPLES TRIGONOMETRIC RATIOS OF RECIPROCAL FUNCTIONS Finding the COSECANT, SECANT, and COTANGENT of any angle is explained in the following examples: Find csc 79º, sec 79º and cot 79º to four decimal places: PREFERRED METHOD Example : csc 79º Since csc 79º = sin 79 Enter: sin 79 = Display:.087 Example 2: sec 79º Since sec 79º = cos 79 Enter: cos 79 = Display: Example : cot 79º Since cot 79º = tan 79 Enter: tan 79 = Display: ALTERNATE METHOD Find the sin 79º, then take the reciprocal. This will be equal to the csc 79º. Step : press the sin key. Step 2: press 79 and then the = key. Step : press the x key. Step 4: display:.087 (= csc 79º) Find the cos 79º, then take the reciprocal. This will be equal to the sec 79º. Step : press the cos key. Step 2: press 79 and then the = key. Step : press the x key. Step 4: display: (= sec 79º) Find the tan 79º, then take the reciprocal. This will be equal to the cot 79º. Step : press the tan key. Step 2: press 79 and then the = key. Step : press the x key. Step 4: display: (= cot 79º) 9
12 Experiential Activity Two Exercise Set A Perform the following:. sin cos 75. tan tan 2º 5. cos 5º 6. sin 7 7. cos tan 4 9. cos 6 0. sin 0 Exercise Set B Perform the following:. csc 4 2. cot 4. tan sec 5 5. csc 0 6. cos cot 5 8. sec 4 Show Me 9. csc 4 0. sin 8. sin 0 2. tan 0 Experiential Activity Two Answers Answers for Exercise Set A Answers for Exercise Set B
13 OBJECTIVE THREE When you complete this objective you will be able to Determine the angle from a given function value. Exploration Activity opposite The trigonometric ratio of is the ratio of the sides hypotenuse for θ = 6.7º. When we are given sin θ = and are asked to determine θ, we know that the only acute angle which has a sine of is 6.7º. Therefore, we should be able to find the angle. Example: sin θ = θ = arc sin θ = 6.7º or sin θ = θ = sin θ = 6.7º Arc sin and sin are the two names used to express this operation. It is very important you understand that sin - θ means the inverse operation of finding the sine of the angle. sin θ is not the same as sinθ Do not confuse this with the reciprocal x function on your calculators!! EXAMPLE Given: sin θ = Find θ in degrees. SOLUTION: The problem is to find the angle. Therefore by our definition θ = Arc sin or θ = sin This means we want to find the angle whose sine is
14 Using the calculator: Refer to your calculator manual as the procedure varies from calculator to calculator. Step : Press the 2nd F key Step 2: press sin key Step : enter and then press the = key Step 4: display: 6.7º Check: sin 6.7º = NOTE: θ = Arc sin is read as "θ is the angle whose sine is ". θ = Arc cos is read as "θ is the angle whose cosine is θ = Arc tan 4.64 is read as "θ is the angle whose tangent is 4.64"... and so on. It is sometimes desirable to find the angle in radians. Again your calculator will do this operation as long as you change it to radian mode. EXAMPLE 2 Find the value of θ in radians for cos θ = θ = Arc cos SOLUTION: Again refer to your own calculator manual for finding an angle in radians, given a trigonometric function. Using your calculator: Step : Put your calculator in radian mode Step 2: Press the 2nd F key Step : press cos key Step 4: enter and then press the = key Step 5: display: Therefore θ = Check: cos(0.2496) = Using only csc, sec and cot, find the angle in degrees or radians. 2
15 EXAMPLE Given sec θ = 5.24, find angle θ in degrees Since the calculators do not have a secant function, it is necessary first to convert this to the cosine function. Since cos θ = secθ cos θ = 5.24 θ = arc cos 5.24 or θ = cos 5.24 θ = 79.0º EXAMPLE 4 Given csc θ =.7965, find angle θ in degrees SOLUTION: Since sin θ = cscθ sin θ =.7965 θ = arc sin.7965 or θ = sin θ =.8º
16 Experiential Activity Three Exercise Set A Solve for θ in degrees, given:. sin θ = cos θ = tan θ = sin θ = tan θ = cos θ = sin θ = tan θ = cos θ =.0000 Exercise Set B Find θ in Radians, given:. sin θ = cos θ = tan θ = sin θ = tan θ = cos θ = sin θ = cos θ = Exercise Set C (a) Find the angle in degrees, given the following:. csc θ = sec θ = cot θ = csc θ = sec θ = cot θ = sec θ = cot θ = csc θ = Show Me 0. sec θ = 6.54 (b) For the exercise above, find the angles in radians. 4
17 Experiential Activity Three Answers Answers for Exercise Set A. 66.0º º. 22.5º º º 6. none this is because cos θ and sin θ are limited to values between and º º 9. 80º Answers for Exercise Set B Answers for Exercise Set C (a.) (b.). 0.5º º º º º º º º º º
18 OBJECTIVE FOUR When you complete this objective you will be able to Calculate the function value of the acute angles in any right triangle, given two sides of the triangle. Exploration Activity If we are given any two sides of a right triangle then according to Pythagoras Theorem we can calculate the third side. This theorem states that in a right triangle the square of the hypotenuse equals the sum of the squares on the other two sides. In the adjoining triangle we can find side b by using Pythagoras' formula: c 2 = a 2 + b = b 2 b 2 = 25 6 b = 9 so: b = c = 5 A b B a = 4 C Once we know the three sides, we can easily find the trigonometric ratios of both angle A and angle B Trigonometric Ratios of A Trigonometric Ratios of B hypotenuse c = 5 A b = Adjacent to A hypotenuse c = 5 A b = Opposite B B a = 4 Opposite A C B a = 4 Adjacent to B C opp 4 sin A = = = hyp 5 opp sin B = = = hyp 5 cos A = = cos B = = tan A = =. tan B = = csc A = = csc B = = sec A = = sec B = = cot A = = cot B = =. 6
19 EXAMPLE If a = and b = 4 in the right triangle ABC to the right, find sin A and tan B. since c 2 = a 2 + b 2 c 2 = c 2 = 25 c = c = a + b c = 5 A c b = 4 B a = C so: sin A = 5 = and tan B = 4 =. NOTE: The trigonometric ratio is equally acceptable in either fraction form or in decimal form. See example above. However when entering answers into the computer the decimal form must be used. i.e. /5 would be entered as Enter answers with 4 decimal places. 7
20 Experiential Activity Four. Find the indicated ratios from the given triangle where a =, b =, and c = 2. Write your ratios in exact form. a) sin A, sec B, cot A b) csc A, sin B, cot B c) tan A, cos B, sec A d) tan B, csc B, cos A c B a A b C 2. Find the indicated ratios from the given triangle where b =2.89, and c =.4, and side a is unknown. Round your ratios to significant digits. a) sin A, sec B, cot A b) csc A, sin B, cot B c) tan A, cos B, sec A d) tan B, csc B, cos A c B a A b C. Determine the indicated ratios. The listed sides are those shown in the given triangle. Write your ratios in exact form. a) a =, b = 4. Find tan A and cos B b) a = 5, c =. Find cos A and csc B c) b = 9, c = 4. Find cot A and cos B d) a = 8, c = 9. Find sin A and sec B c B a Round your ratios to significant digits. e) a = 2, c = 4. Find sec A and tan B f) b = 4, c = 2. Find csc A and cos B g) a = 2, b = 75. Find sin A and tan B A b C 8
21 4. Given the coordinates as shown, find the values requested: Round trigonometric ratios to significant digits. Round angles to the one decimal place. tan A A = = Y P(.6, 4.2) csc A = Show Me A X 5. Given the coordinates as shown, find the values requested: Round trigonometric ratios to significant digits. Round angles to the one decimal place. B sin B = = Y P(5.8, 2.) sec B = B X 9
22 Experiential Activity Four Answers. a) sin A = 2, sec B = 2, cot A = b) csc A = 2, sin B =, cot B = 2 2 c) tan A =, cos B =, sec A = 2 d) tan B =, csc B = 2, cos A = 2 2. a) sin A = 0.5, sec B =.88, cot A =.60 b) csc A =.88, sin B = 0.848, cot B = c) tan A = 0.626, cos B = 0.5, sec A =.8 d) tan B =.60, csc B =.8, cos A = a) tan A = 4, cos B = 5 2 b) cos A =, csc B = c) cot A =, cos B = d) sin A =, sec B = 9 8 e) sec A =.5, tan B =.7 f) csc A =.26, cos B = 0.79 g) sin A = 0.869, tan B = a) tan A =.7, A = 49.4º, csc A =.2 5. a) B = 2.6º, sin B = 0.69, sec B =.08 20
23 OBJECTIVE FIVE When you complete this objective you will be able to Determine all function values of an acute angle given one function value of the angle. Exploration Activity EXAMPLE Find the values of the trigonometric ratios of A 2 given that sin A =. opposite We know that sin A = hypotenuse Therefore: c B a 2 sin A = A b C Since 2 is the side opposite A and is the hypotenuse we can identify the following sides: B a = 2 and c = Use the Pythagorean theorem to find side b: c 2 = a 2 + b 2 2 = b 2 b = 25 b = 5 Hypotenuse c = A b = 5 Adjacent to A a =2 Opposite A C Trigonometric ratios of A 2 sin A = = cos A = = tan A = = csc A = =.08 sec A = = cot A = =
24 EXAMPLE 2 Given that tan A = 4, find sin A and csc A. opp Since tan A = and from the question tan A =, we can identify the following sides adj 4 on the triangle below: a = and b = 4 B c a A b C Use the Pythagorean Theorem to find side c: c 2 = a 2 + b 2 c 2 = c = 7 opp sin A = = hyp 7 = hyp 7 csc A = = opp =
25 Experiential Activity Five. Find the values of the other five trigonometric functions of A, given in exact values. Write your answers in exact form a) sin A = b) cot A = c) tan A = d) cos A = 2 e) cos A = 7 f) cot A = 2 Experiential Activity Five Answers a) cos A =, tan A =, csc A =, sec A =, cot A = b) sin A =, cos A =, tan A =, csc A =, sec A = c) sin A = d) sin A = e) sin A = f) sin A = , cos A =, csc A =, sec A =, cot A = , tan A =, csc A =, sec A =2, cot A = , tan A =, csc A =, sec A =, cot A = , cos A =, tan A =, csc A =, sec A = Practical Application Activity Complete the Trigonometric Functions of an Acute Angle assignment in TLM. Summary This module introduced the student to the six trigonometric functions. 2
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