Lesson #64 First Degree Trigonometric Equations

Transcription

1 Lesson #64 First Degree Trigonometric Equations A2.A.68 Solve trigonometric equations for all values of the variable from 0 to 360 How is the acronym ASTC used in trigonometry? If I wanted to put the reference angle, 75 into the 2 nd, 3 rd, and 4 th quadrants, how would I do so? Find sin30. Find sin390. Find sin(-330 ). Find sin50. Find sin50. Find sin(-20 ). Why do all of these angles have the same sine value? Solve the equation, sin( x ). Is your initial answer the only solution? 2 Solve the equation, cos( x ). Find all solutions between 0 and Solve the equation, sin( x ). Find all solutions between 0 and Solve the equation, cos( x ). Find all solutions between 0 and ~ ~

2 Even though there are really an infinite number of solutions to most trigonometric equations, we will only consider the solutions between 0 and 360 for this course. Here is the method for doing so.. Isolate the trigonometric part of the equation using SADMEP. 2. Determine what quadrants your answer will be in based upon the sign of the trig. value (ASTC). TIP: Write ASTC next to the trig. value. Circle the quadrants where the answers will be. 3. Use the inverse trig function to solve for the angle. 4. If necessary, find the reference angle. 5. Put the reference angle in the quadrants you chose. a. QII: subtract reference angle from 80. b. QIII: add reference angle to 80. c. QIV: subtract reference angle from 360. Ex) Solve for x. Round to the nearest degree. 3cos x 6 8 Example: Solve for in the interval 0 8cos 2 5 5cos 360 to the nearest degree. ) Solve for x on the interval, 0 x 360. sin x 2 sin x 2) Solve for x on the interval, 0 x 360 nearest degree. 2tan x 0 to the ~ 2 ~

3 3) Solve for x on the interval, 0 x 360 to the nearest degree. 3(sin x 5) 4 4) Solve for x on the interval, 0 x 360 to the nearest degree. tan x 2 2 In this unit we will be working with a couple of formulas that are used with triangles. Since we are working with triangles we will only have to consider angles between 0 and 80. The Law of Sines a b sin A sin B The Law of Cosines a b c 2bccos A Solve the following equations for x on the interval, 0 <x<80. Round to the nearest degree. 5 9 sin 30 sin x (4)(7)cos x (6) 2(3.2)(6)cos x Notice, with the law of sines you can get 2 answers, but with the law of cosines you can get only one answer for your angle. Why is this true? ~ 3 ~

4 Lesson #52 - Inverse Trigonometric Functions A2.A.63 Restrict the domain of the sine, cosine, and tangent functions to ensure the existence of an inverse function A2.A.64 Use inverse functions to find the measure of an angle, given its sine, cosine, or tangent A2.A.65 Sketch the graph of the inverses of the sine, cosine, and tangent functions At this point, many students ask the following questions:. If there are two answers between 0 and 360, why does my calculator only give me one of them? 2. Why do I sometimes get a negative answer for my angle? These questions have loaded answers. We will have to use a lot of our knowledge about one-to-one functions, inverses, trig. graphs, and domains to answer them. Graph y=sin(x) on your calculator, in radian mode, with a zoom trig window. You should see the following graph. Let s consider the equation, sin(x)=0. Where is the y- value of the sine curve equal to 0? y sin ( x ) Convert these values to degrees. Are these the only places where the sine curve is equal to 0? How many answers are there? Your calculator cannot give you an infinite number of answers. It works with functions, which only give one output for each input, and expects you to find any other answers you want using your knowledge of reference angles and ASTC. Since functions are predictable, your calculator is predictable in what answer it will give you. Circle a portion of the sine curve that is one-to-one (passes the Horizontal Line Test) and is also closest to the origin. The domain of this piece of the graph is:. Converted to degrees this would be:. ~ 4 ~

5 This is called a restricted domain. When you use the the inverse sine function, sin ( x ), the calculator will always give you an answer between -90 and 90, inclusive. In radians this would be: Restricted Domain for Sine so that the inverse will be a function. Graph y sin ( ) x, and sketch it on the graph provided. Note: You must be in radian mode when graphing inverse trig. functions. Compare this graph with the piece of y=sin(x) you circled on the previous page. Graph y=cos(x) on your calculator, in radian mode, with a zoom trig window. You should see the following graph. Let s consider the equation, cos(x)=0. Where is the y-value of the cosine curve equal to 0? y cos ( x ) Convert these values to degrees. Are these the only places where the cosine curve is equal to 0? How many answers are there? Your calculator cannot give you an infinite number of answers. It works with functions, which only give one output for each input, and expects you to find any other answers you want using your knowledge of reference angles and ASTC. Since functions are predictable, your calculator is predictable in what answer it will give you. ~ 5 ~

6 Circle a portion of the cosine curve that is one-to-one (passes the Horizontal Line Test) and is also closest to the origin. The domain of this piece of the graph is:. Converted to degrees this would be:. This is the restricted domain for cosine. When you use the inverse cosine function, cos ( x ), the calculator will give you an answer between 0 and 80, inclusive. In radians this would be: Restricted Domain for Cosine so that the inverse will be a function. Graph y cos ( ) x, and sketch it on the graph provided. Compare this graph with the piece of y=cos(x) you circled on the previous page. y tan ( x ) Graph y=tan(x) on your calculator, in radian mode, with a zoom trig window. You should see the following graph. Let s consider the equation, tan(x)=0. Where is the y-value of the tangent curve equal to 0? Convert these values to degrees. Are these the only places where the tangent curve is equal to 0? How many answers are there? Your calculator cannot give you an infinite number of answers. It works with functions, which only give one output for each input, and expects you to find any other answers you want using your knowledge of reference angles and ASTC. Since functions are predictable, your calculator is predictable in what answer it will give you. ~ 6 ~

7 Circle a portion of the tangent curve that is one-to-one (passes the Horizontal Line Test) and is also closest to the origin. The domain of this piece of the graph is:. Converted to degrees this would be:. This is the restricted domain for tangent. When you use the inverse cosine function, cos ( x ), the calculator will give you an answer between -90 and 90, exclusive. In radians this would be: Restricted Domain for Tangent so that the inverse will be a function. Graph y tan ( ) x, and sketch it on the graph provided. Compare this graph with the piece of y=tan(x) you circled on the previous page. This whole explanation is important for your math understanding which ultimately leads to better retention and better grades, but the information you will be directly tested on is in the thickly outlined textboxes. Other important information about trigonometric inverses. The trigonometric functions can have alternate names, Arc. a. y sin x(also known as y Arc sin x ) b. c. y cos x(also known as rccosx y A ) y tan x(also known as tan y Arc x ) 2. The value for the angle that your calculator gives you is called the principal value. 3. Unless the problem says to solve for x between 0 and 360, you can assume that you are looking for the principle value. 4. For the following problems we will not be graphing. Just as in Unit #6, if you are asked to find the answer in radians, complete the problem in degree mode and convert at the end. ~ 7 ~

8 . What is the principal value of? ) 3) 9 What is the principal value of, in degrees and radians. 2) 4) 2 The value of is ) 0 3) 2) 4) 0 What is the smallest positive value of x, in radians, that satisfies? 3 The value of is ) 20 3) 90 2) 05 4) 75 4 If, then x is equal to ) 3) 2) 4) 5 If and, the measure of angle x is ) 45º 3) 225º 2) 35º 4) 35º 6 What is the value of x in the equation? ) 3) 2) 4) Find the value of, in degrees. 2 If, what is the value of angle A to the nearest minute? ) 3) 2) 4) 3 If, find the value of positive acute angle A to the nearest minute. 4 If, find the value of positive acute angle x to the nearest minute. 5 If, find the value of positive acute angle to the nearest minute. 7 If, what is the measure of angle, in degrees? 6 If, find the measure of positive acute angle to the nearest minute. 8 What is the principal value of? ) 3) 2) 4) ~ 8 ~

9 Lesson #65- Trigonometric Application Formulas A2.A.73 Solve for an unknown side or angle, using the Law of Sines or the Law of Cosines For all problems in this lesson, round to the nearest tenth. Triangle Review Sum of the Degrees in a Triangle: Labeling a Triangle: lowercase letters for sides. UPPERCASE letters for angles. The same letter for a side and the opposite angle. The smallest angle is across from the smallest side,. The largest angle is across from the largest side,. B a c b A m BAC = 36 m ABC = 93 C m BCA = 5 In what types of triangles can you use the Pythagorean Theorem and SOH-CAH-TOA? Finding the sides and angles in triangles that are not right triangles requires the use of the trig laws. You are given these formulas on the A2&T reference sheet. Therefore the main focus of this unit is learning how and when to use them to solve different types of problems. ALWAYS DRAW A PICTURE!!!! The Law of Sines: a b c sin A sin B sin C. Given: a=2, m A 25, m C 58. Find side c. Connection to Proofs: Use when given: ASA or AAS Trick CIRCLE The PAIRS You must have Angle/Side Pair where you know the values. 2. Example: Solve for x. B C 70 x A ~ 9 ~

10 Remember, with the Law of Sines you always need a known side angle pair and one other piece of information. 3. Example: Solve for x x Law of Cosines: a b c 2bc cos A Connection to Proofs: Use when given: SSS or SAS 4. Given: c=2, b=5, m A 84. Find a. Law of Cosines WORKING WITH 3 SIDES An Angle/Side pair must start and finish the equation. One of them will be unknown since it is what you are finding. 5. Given: a=0, b=5, and c=20, find m A. The letters are less important in the formula than the actual placement of the sides. ~ 0 ~

11 The gist of the Law of Cosines is: ( st side)ü= (2 nd side)ü+ (3 rd side)ü 2(2 nd side)(3 rd side)cos(angle opposite the st side) You can choose which side you want for the st side based upon what you want to find. 6. Solve for x x 7. Find the measure of angle B. C 5 A 2 B ~ ~

12 Directions: Round sides to the nearest tenth of a unit. Round angles to the nearest degree. ) In triangle ABC, a=2, b=5, and m C 60. Find c. 2) If m C 82, m A 55, and a=8, find c. 3) In triangle ABC, a=20, b=6, and c=32. Find m B. 4) In triangle ABC, if m B 0, m A 30, and a=5, find b. ~ 2 ~

13 5) In triangle ABC, a=20, c=25, and m B 98. Find b. 6) In triangle ABC, if m A 6, b=92, and m B 20 find c. 7) In triangle ABC, b=20, c=23, and a=30, find m A. 8) If m A 75, m B 55, and c=5, and find a. 9) In triangle ABC, a=9, b=4, and c=2. Find m C. ~ 3 ~

14 Lesson #66 The Ambiguous Case & the Donkey Theorem (SSA) A2.A.75 Determine the solution(s) from the SSA situation (ambiguous case) Solve for x in each equation on the interval 0 <x<80 because these are the only angles that could be in a triangle. Round your answers to the nearest degree.. sin x cos x cos x.5678 When working with triangles, what is the only trigonometric function that can give us two answers for the angle? Abiguous/Ambiguity (from Webster dictionary) a : doubtful or uncertain especially from obscurity or indistinctness <eyes of an ambiguous color> **2 : capable of being understood in two or more possible senses or ways <an ambiguous smile> <an ambiguous term> <a deliberately ambiguous reply> In lesson #72, we looked at solving triangles when given AAS, ASA, SAS, and SSS. You will remember from last year that these are all ways to prove triangles congruent. When two triangles are congruent it means we could find all of their sides and angles, so we know that they are EXACTLY THE SAME. What about the donkey theorem, SSA? This is not one of our ways to prove triangles congruent, which means that given this pattern, we do not really know what the remaining parts of the triangle will be. It is AMBIGUOUS. You will want to look for this SSA pattern, but the fact that this situation is unclear arises naturally when we use the law of sines. Why: When given SSA we have a known SIDE-ANGLE pair, so we would use the law of sines to find the other Angle. There are two possible answers for the angle, one between 0 and 90 degrees as well as an answer between 90 and 80 degrees. You just have to figure out if one, both, or none of the angles will fit in your triangle with the angle you are given. ~ 4 ~

15 Before we start solving these problems, the following three pictures show how there could be 0,, or 2 different possible triangles when we are given the SSA pattern. In each triangle the lengths of sides a, b, as well as B are given. No triangles One triangle Two triangles a) Determine the number of possible triangles. b) Find the measures of the three angles of each possible triangle. Express approximate values to the nearest degree. Steps: a=4, b=6, and m A=30. Once you recognize the SSA pattern, draw 2 triangles. 2. Set up proportions to perform law of sines to find a missing angle. 3. If sin(x), find the missing angle. (If sin(x) >, you know there are triangles) 4. Since sine is positive in QI and Q II, find the 2 possibilities for the angle. 5. Put each answer into one of the triangles you drew. See if neither (0), one (), or both (2) of them fit with your given angle. 6. For each possible triangle, find the remaining angle measure. ~ 5 ~

16 Practice: c) Determine the number of possible triangles. d) Find the measures of the three angles of each possible triangle. Express approximate values to the nearest degree.. a=7, b=6, and m B=50 2. a=6, b=4, and m A=50 Note: Some of these problems will make intuitive sense. Look at your answers to # and #2. How could you figure out those answers without using the law of sines? 3. a=6, b=8, and m A=40 4. In triangle ABC, if A=30, a=6, and b=8, the number of distinct (different) triangles that can be constructed is: a. b. 2 c. 3 d In triangle ABC, if A=30, a=5, and b=0, the number of distinct (different) triangles that can be constructed is: a. b. 2 c. 3 d. 0 ~ 6 ~

17 Lesson #67 Area of Triangles and Parallelograms A2.A.74 Determine the area of a triangle or a parallelogram, given the measure of two sides and the included angle Degrees-Minutes-Seconds We typically use decimals and the base ten system to express parts of a number. There is another way to represent a part of an angle. A full rotation is split into 360. We can also split a degree up into smaller measurements based on multiples of 60 using words that will be familiar to you. One minute of a degree is /60 th of a degree. One second of a degree is /60 th of a minute (/360 th of a degree). You can easily convert between decimal degrees and degrees-minutes-seconds on your calculator. Follow the directions below. Convert 57 45' 7'' to decimal degrees: In either Radian or Degree Mode: Type 57 45' 7'' and hit Enter. is under Angle (above APPS) # ' is under Angle (above APPS) #2 '' use ALPHA (green) key with the quote symbol above the + sign. Answer: Convert to degrees, minutes, seconds: Type DMS Answer: 48 33' 8'' The DMS is #4 on the Angle menu (2 nd APPS). This function works even if Mode is set to Radian. A. Convert the following measures to decimal degrees. Round to the nearest hundredth. a) b) 8 23 c) B. Convert the following measures to degrees-minutes-seconds. Then round them to the nearest minute. d) e) f) For this unit, we will always work in decimal degrees. Therefore, if you are given an angle in DMS, convert it to decimal degrees. If you are asked to find an angle in DMS, convert it at the end. ~ 7 ~

18 Area of a Triangle There is one more trigonometric formula that you will be given to you on the regents. You already know that the area of a triangle can be calculated with the formula. This formula is limited because you must know the base and the height of the triangle. If you know one side of the triangle, you can make that side the base, but you do not always know the height. We can use trigonometry to substitute known information for the height. Observe below: A similar proof can be used to show that this formula works an obtuse triangle like the second triangle ABC above. To summarize, the area of a triangle, K, is given by the following formula: K absin C 2 You will notice that the formula looks slightly different than the one in the proof. You should be comfortable with the fact that the letters do not matter; it is their relative position on the triangle. Therefore, the information we need is SAS or 2 sides and the included angle.. Find the area of a triangle where: a=8, b=2, and m C 00 square unit.. Round to the nearest Area of a Triangle You need a known corner (SAS). 2. Find the area of a triangle where: b=20, c=30, and m A 34 square unit.. Round to the nearest ~ 8 ~

19 3. Find the area of the triangle below. Round to the nearest tenth Challenge: A triangular plot of land has sides that measure 5 meters, 7 meters, and 0 meters. What is the area of this plot of land, to the nearest tenth of a square meter? Area of Parallelograms A parallelogram can be divided into two equal triangles. Therefore, what formula could we use to find the area of the parallelogram below? 5. To the nearest tenth, find the area of a parallelogram with sides of 6 and 8 and an angle of Find the area of the parallelogram below to the nearest unit ~ 9 ~

20 Lesson #68 Using Trig. Apps. in Word Problems A2.A.74 Determine the area of a triangle or a parallelogram, given the measure of two sides and the included angle A2.A.73 Solve for an unknown side or angle, using the Law of Sines or the Law of Cosines This lesson we will be looking at different situations where we can use the trig. laws and the area of a triangle formula. Below are some common shapes that arise in these problems and their most important properties. Fill in everything else that you know on each shape. Isosceles Triangles 2 sides congruent (called the legs) Base angles congruent A Parallelograms Opposite Sides Parallel Opposite Sides Congruent Opposite Angles Equal Adjacent Angles Supplementary Can be cut into two congruent triangles 0 2 B 69 C 7 67 Rhombuses A parallelogram with congruent sides 5 Isosceles Trapezoids Legs Congruent Base Angles Congruent Diagonals Congruent 46 B 3 D 5 72 A 5 C ) If the area of an isosceles triangle is 25 square feet, and the leg length is 9 feet, find the measure of the angles of the triangle to the nearest minute (assume all angles are acute). ~ 20 ~

21 2) Find, to the nearest tenth, the area of a triangle with side lengths, 22, 34, and 50. 3) In, m<a = 50º, m<b = 35º, and a = 2. Find the missing sides and angle. (nearest tenth, nearest degree) 4) 5) The lengths of the adjacent sides of a parallelogram are 2 cm and 4 cm. The smaller angle measures 58. What is the length of the longer diagonal? Round your answer to the nearest centimeter. 6) If the area of a triangle is 4 square feet, one side is 5 units, and another side is 6 units, find the sine of the included angle. ~ 2 ~

22 7) The side length of a rhombus is 5 feet and the longer diagonal is 23 feet. Find the angles of the rhombus in degrees-minutes-seconds. 8) A ship at sea heads directly toward a cliff on the shoreline. The accompanying diagram shows the top of the cliff, D, sighted from two locations, A and B, separated by distance S. If m DAC 27, m DBC 50, and S = 25 feet, what is the height of the cliff, to the nearest foot? 9) An angle of a parallelogram has a measure of 45. If the sides of the parallelogram measure 9 and 3 centimeters, what is the area of the parallelogram to the nearest tenth? 0) A cross-country trail is laid out in the shape of a triangle. The lengths of the three paths that make up the trail are 2000 m, 200 m, and 800 m. Find to the nearest degree the measure of the smallest angle formed by the legs of the trail. ~ 22 ~

23 ) If the base angle of an isosceles triangle measures 34 and the base of the triangle is 8 inches, find the length of other sides of the triangle to the nearest tenth. 2) In a triangle, two sides that measure 4 cm and 7cm form an angle of 60. Find the measure of the smallest angle of the triangle to the nearest degree. 3) In ABC, AC 8, BC 0, and cosc. find the area of ABC to the nearest tenth of a 2 square unit. (Hint: Find angle C first). 4) In an isosceles triangle, the vertex angle is 30º and the base measures 2 cm. Find the perimeter of the triangle to the nearest integer. ~ 23 ~

24 5) Points A & B are on one side of a river, 00 feet apart, with C on the opposite side. The angles A and B measure 70º and 60º respectively. What is the distance from point A to point C, to nearest foot? 6) A triangular field has side lengths of 00 feet, 250 feet, and 300 feet. Find the area of the field to the nearest square foot. 7) In the accompanying diagram, angle R is an obtuse angle, not a right angle. Find the length of PQ to the nearest foot. 8) ~ 24 ~

25 Lesson #69 Forces and Vectors A2.A.74 Determine the area of a triangle or a parallelogram, given the measure of two sides and the included angle A2.A.73 Solve for an unknown side or angle, using the Law of Sines or the Law of Cosines Imagine you have an aerial view of a situation. Two people are pushing on a heavy object in different directions represented by the circled x. Each one is exerting a certain amount of force. The first person is pushing with a force of 25 pounds while the second person is pushing with a force of 30 pounds. The angle between the two forces they are exerting is 60. In what direction will the object end up moving if they are both pushing at the same time? What is the result of their combined forces? 25 pounds pounds If we form a parallelogram with the two given forces, the resultant force will be the diagonal from the object to the other opposite corner of the parallelogram. Label everything else you know about the parallelogram. Use that information to find the resulting force to the nearest tenth. 25 pounds pounds Find the angle between the larger original force and the resultant to the nearest degree. The forces picture always looks the same! Think of the parallelogram as two congruent triangles. You often have to work with the supplementary angle, not the one you are given. The resultant is always closer to the larger force. In other words, there is a smaller angle between them. Smaller Force Resultant Larger Force ~ 25 ~

26 Set up a diagram and a method for solving the following problems. ) Two forces of 33 newtons and 80 newtons act on an object with a resultant of 70 newtons. Find to the nearest degree, the angle between two applied forces. 2) If you completely solved the last question, the angle between the two forces is 9. Using the same information, find the angle between the resultant and the larger applied force to the nearest degree. 3) Two forces act on a body so that the resultant is a force of 46 pounds. If the angles between the resultant and the forces are 20 degrees and 46 degrees, find the magnitude of the larger applied force to the nearest pound. 4) Two forces act on an object. The first force has a magnitude of 63 pounds and makes an angle of 35 degrees with the resultant. The magnitude of the resultant is 80 pounds. Find the magnitude of the second applied force to the nearest tenth of a pound. ~ 26 ~

27 Solve the following problems. 5) If forces of 47 pounds and 52 pounds act on object such that the angle between them is 70, what is the resultant force to the nearest pound? 6) Two forces of 42 newtons and 57 newtons act on an object with a resultant of 70 newtons. a. Find to the nearest degree, the angle between two applied forces. b. Next, find the angle between the resultant and the larger force to the nearest degree. 7) Two forces act on a body so that the resultant has a force of 35 newtons. If the angles between the resultant and each of the forces are 72 degrees and 2 degrees, find the magnitude of the larger applied force to the nearest tenth of a newton. ~ 27 ~

28 8) Two forces act on an object. The first force has a magnitude of 75 pounds and makes an angle of 34 degrees with the resultant. The magnitude of the resultant is 0 pounds. c. Find the magnitude of the second applied force to the nearest tenth of a pound. d. To the nearest tenth of a degree, find the angle the second force makes with the resultant. 9) Two forces of 80 pounds and 00 pounds act on object such that the angle between them is 05. e. What is the resultant force to the nearest pound? f. What is the angle between the resultant force and the smaller force to the nearest minute? ~ 28 ~