Name: Block: What I can do for this unit:

Transcription

1 Unit 8: Trigonometry Student Tracking Sheet Math 10 Common Name: Block: What I can do for this unit: After Practice After Review How I Did 8-1 I can use and understand triangle similarity and the Pythagorean Theorem. 8-2 I can use and apply the Sine, Cosine, and Tangent ratio to solve for missing sides and angles. 8-3 I can use and understand Trigonometry Theory (such as the unit circle, quadrants, and reference angles). 8-4 I can solve problems involving one or more right triangles using the primary trig ratios and/or the Pythagorean theory. Code Value Description N Not Yet Meeting Expectations I just don t get it. MM Minimally Meeting Expectations Barely got it, I need some prompting to help solve the question. M Meeting Expectations Got it, I understand the concept without help or prompting. E Exceeding Expectations Wow, nailed it! I can use this concept to solve problems I may have not seen in practice. I also get little details that may not be directly related to this target correct.

2 Unit 8: Systems of Equations Day 1 Math 10 Common 8-1: I can use and understand triangle similarity and the Pythagorean Theorem. Trigonometry is the study of triangles. If one triangle is a perfect enlargement of another triangle, perhaps rotated or flipped, then they are known as similar. Each angle in a similar triangle pair will equal an angle on the other triangle. There are three tests for triangle similarity. 1. Angle Test - all three angles are equal. 2. Side Ratio Test Ratio for all sides are equal. 3. Side Angle Side Two side ratios are equal and the angle contained between them is equal. Consider the following two triangles. A C E B D F We see that for this question, both triangles have on angle of 52. We can now check the two sides ratios that contain the angle to see if they are equal, thus satisfying the Side-Angle-Side requirements. =3. Thus this = triangle is similar according to SAS. We can write the similarity statement: ΔABC~ΔDEF (A and D are both opposite the short side, B and E are both 52, and C and F are both opposite the long side). Make the correct statement of similarity for each pair of triangles if similar. State the reason for their similarity. 15 C 1) 2) D A B 14 E A C E 21 B 18 Similarity: Reason: D 22.5 F Similarity: Reason:

3 3) A D B F 21 E 4) B 2 A 4 3 D C E C F Similarity: Reason: Similarity: Reason: Find the missing variable using the Pythagorean formula. 5) 6)! 15! 24! 10 7) 8)! 8! 5

4 9) A screen s size is usually stated in terms of its diagonal length. Assuming a screen has proportions 16:9, find the horizontal length of a 2 metre screen to the nearest cm. 10) From point A, travel 10 km east then travel 3 km south, turn west and travel km. How far are you from point A? 11) If the distance between bases in baseball is 90 ft., how far is it from home plate directly to second base? ) A 20 foot ramp rises to a doorway that is 3 feet off the ground. How far away from the building is the ramp? 13) Find the side length of the largest square that can fit inside a circle of diameter ) From point A, travel km south and 10 km west. Then travel 5 km north. How far are you from point A?

5 15) A rectangle is 40 units long and 15 units wide. Find the length of its diagonal. 16) Find the side of the largest square you can fit inside a circle with radius ) How high up a wall does 16 ft. Ladder reach if the bottom is 6 ft. From the base of the wall? 18) If the hypotenuse is 34 and one other side of a right triangle is 16, find the length of the third side of the triangle. Determine if each of the following triangles is a right angled triangle. 19) 20)

6 21) 22) Find the length of each variable. 23) 24) 8! < 11! < 13 25) 26) < <

7 Unit 8: Systems of Equations Day 2 Math 10 Common 8-2: I can use and apply the Sine, Cosine, and Tangent ratio to solve for missing sides and angles. Review: Make the correct statement of similarity for each pair of triangles if similar. State the reason for their similarity. 1) 11.5 A 9.2 E F 2) C B D Similarity: Reason: Similarity: Reason: Determine if each of the following triangles is a right angled triangle. 3) 4) For much of this unit we will be looking at right angled triangles. Every right angled triangle has a special side, the hypotenuse. The hypotenuse is opposite the right angle and is the longest side of the triangle. The other two sides can be labelled relative to an acute angle. Since there are two acute angles we must be careful about to which one we are referring. A The triangle at the right has three angles A, B, and C. Note that each side c has a letter corresponding to the opposite side (ie. side a is opposite angle A, b is opposite angle B and side c is opposite angle C.). This is a common b naming technique. C a B

8 If we look at the following triangle in terms of angle, we can easily find the hypotenuse h-(opposite the right angle) and the opposite side o-(opposite ). The side that is left over is referred to as the adjacent (a). o h Label o, a, and h for the following triangles relative to the angle indicated. a 5) 6) 7) In a right angled triangle, the tangent ratio is defined as: Tangent =! "" $! %!&%' $ For the following triangle, tan ( = *++*,-./ = * = 5. Note that tan( depends on which 003/ acute angle is 5 Find tan ( for each of the following triangles. Round your answer to the nearest thousandth. 8) 9) 10) Using our scientific calculator, we can calculate tan38 to three decimal places. First, ensure you are working in the degree mode (usually D of DRG). Depending on the model, we may need to type tan, then 35 or 35, then tan. Ex. tan38 = Calculate each value of tan to the nearest thousandth using your scientific calculator. 11) tan 53 ) tan 7 13) tan 61 14) tan 81

9 If we know the value of tan( we can use it to calculate. Suppose tan ( = To solve this algebraically, we need a way to isolate (that is, eliminate the tan function). The opposite function to tan is called arctan, often denoted tan >?. This does not mean?, it means arctan. Your scientific calculator should have an arctan button (usually Shift-Tan or 2 nd F-Tan). % tan( = tan >? (tan() = tan >? (1.072) ( = Note that if we are given tan( we will need to use the arctan function to eliminate it and solve for the angle. Solve for missing variable in the following equations. Round your answer to the nearest tenth of a degree. 15) tan ( = ) tan B = ) tan C = ) tan ( = 3.02 Algebra Review: Solve for D in each of the following to the nearest thousandth. 19) E F = ) 5.H5 E = 6 21) 5 = H.IJ E 22) 0.73 = E K.J? We can use the tangent ratios to solve missing angles in triangles. Consider the following triangle. We know that tan( = 5 = ( = 6 tan>? Solve for in the following triangles to the nearest tenth ) 24) 25) The tangent ratio can also be used to solve for a missing side in a right angled triangle. tan41 = * 0 =?J E D tan41 = 18 D = 18 tan D 18

10 Solve for the missing variable in the following triangles. Round your answer to the nearest tenth. 26) 27) D 28) D D Solve the following problems using the tangent ratio. Draw a diagram for each and round to the nearest tenth. 29) A ladder leaning against a wall makes an angle of 71 with the ground. The foot of the ladder is1.5 m from the base of the wall. How far up the wall will the ladder reach? 30) A ship sees a lighthouse that is 45 m above sea level. If the boat forms an angle of 8 to the top of the lighthouse, how far is the boat from the base of the lighthouse? 31) A flagpole is supported by a guy wire. The wire reaches m up the pole and is attached to the ground 4 m from the base of the pole. What angle does the guy wire form with the ground? 32) A pilot approaching a runway needs to maintain a constant angle of 4. If a pilot begins an approach while at a height of 3500m, what is the land distance from the airport that the pilot should begin her approach?

11 Unit 8: Systems of Equations Day 3 Math 10 Common 8-2: I can use and apply the Sine, Cosine, and Tangent ratio to solve for missing sides and angles. Review: Find tan for each of the following triangles. 1) 2) 3) Solve for in each of the following. Round your answers to the nearest tenth. 4) =3.85 5) =5.1 6) 0.81=. 7) 1.09=. Draw a diagram and answer each question. 8) A wheelchair ramp has a height of 2 m and is 10 m from the base of a building. What is the angle of inclination to the nearest degree? 9) A ladder reaches 3.5 m up the side of a house and forms an angle of 28 with the house. How far is the ladder from the base of the house to the nearest tenth of a metre? Last lesson we learned about the Tangent ratio. Today we will be learning two other trigonometric ratios known as the Sine ratio and the Cosine ratio.

12 Remember that in any right angled triangle you can label the sides relative to an acute angle as opposite, adjacent, and hypotenuse. The sine ratio is defined as follows: Sine =,--,./01. (We abbreviate this to sin but we still pronounce it sine). 23-, Thus for our triangle shown, sin = 6 7 = =0.6. Note that since the hypotenuse of a right angled triangle is always the longest side, sin can never be greater than 1. Determine the sine ratio for the indicated variable. 10) 11) ) Calculate each value of sin to the nearest thousandth using your scientific calculator. 13) sin32 14) sin5 15) sin85 16) sin52 If we know the value of sin we can use it to calculate. Suppose sin = To solve this algebraically, we need a way to isolate (that is, eliminate the sin function). The opposite function to sin is called arcsin, often denoted sin :. This does not mean, it means arcsin. Your scientific calculator should have an arcsin button (usually Shift-Sin or 2 nd F-Sin)../4 sin =0.782 sin : (sin)=sin : (0.782) = Note that if we are given sin we will need to use the arcsin function to eliminate it and solve for the angle. Also, because sin cannot exceed 1, an equation such sin =1.59 is unsolvable. Solve for missing variable in the following equations. Round your answer to the nearest tenth of a degree. 17) sin = ) sin> = ) sin? = ) sin = 2.09

13 Cosine is defined as: Cosine (We abbreviate this to cos). 23-, Thus for our triangle shown, cos = D 7 =E =0.8. Again, since the hypotenuse of a right angled triangle is always the longest side, cos can never be greater than 1. Thus sin and cos cannot exceed 1 but tan can. A handy mnemonic to memorize all three of these ratios is SohCahToa (sin = 6 7, cos = D 7, tan = 6 D ). Determine the cosine ratio for the indicated variable. 21) 22) 23) Calculate each value of cosine to the nearest thousandth using your scientific calculator. 24) cos15 25) cos37 26) cos71 27) cos84 The function cosine also has an inverse function, arcosine, denoted by cos :, just like sine and tangent. Solve for missing variable in the following equations. Round your answer to the nearest tenth of a degree. 28) cos = ) cos> = ) cos? = ) cos =1.34 Determine if you should use the sin, cos, or tan ratio to find the missing side, then solve. Round your answers to the nearest tenth. 32) 33) 34) F

14 35) 36) 37) 40 F Determine if you should use the sin, cos, or tan ratio to find the missing angle, then solve. Round your answers to the nearest tenth. 38) 39) 40) ) 42) 43)

15 Unit 8: Systems of Equations Day 4 Math 10 Common 8-2: I can use and apply the Sine, Cosine, and Tangent ratio to solve for missing sides and angles. To solve a triangle means to find the measure of all unknown angles and find the length of all unknown sides. Solve the following triangles using SohCahToa. Round all answers to one decimal place. 1) 2) 3) A B ) 5) 6)

16 Sketch a diagram and answer each of the following questions. Round answers to the nearest tenth. 7) A surveyor measures the angle of inclination to the top of a building to be 51 at a distance of 152 m from the building. What is the height of the building? 8) A tree casts a shadow that is m long. If the angleof elevation to the sun is 52, find the height of the tree. 9) Find the area of the shaded region below. 10) A kite is flying on 150 feet of string. The wind pulls the kite at an angle of 42. How high is the kite off the ground? cm

17 11) Find the areaof the shaded region. ) A 6.5 m ladder is leaning against a house. If the ladder makes an angle of 21 with the house, how high up the house does the ladder reach? 24 cm 58 Use a protractor to measure each indicated angle to the nearest degree, then determine the length of to the nearest integer. 13) 14) 15.4 in 8.3 m 15) 16) 18 cm 6.5 m

18 Unit 8: Systems of Equations Day 5 Math 10 Common 8-3: I can use Trigonometry Theory (such as the unit circle, quadrants, and reference angles) to solve various problems. Solve the following triangles using SohCahToa. Round all answers to one decimal place. 1) 2) 3) B A B So far in this unit, we have only dealt with sine, cosine, and tangent with angles less than 90. However, angles can vary from 0 to 360 (and beyond, but we won t look at that today!) To better understand sine, cosine, and tangent we look at the unit circle. Mathematicians typically define the unit circle as a circle of radius one centered at the origin. The terminal arm is the line formed from the origin to a point on the unit circle. The value (horizontal) of the point on the circle is equal to cos. The value (vertical) is equal to sin. The slope of the terminal arm is tan. Thus, looking at the following figure we see that for a given angle, cos and sin can never be greater than 1 or less than 1 because the circle has radius 1. We can also better understand why tan can exceed these values (because it is the slope of the line). Consider the Cartesian plane we looked at earlier. We notice that if 0 90 we are in Quadrant I. Quadrant II is , Quadrant III is , and Quadrant IV is

19 Focusing on the vertical component, we see that the vertical component () is positive in quadrants I and II. Thus if 0 180, sin>0. If it is in quadrants III or IV, is negative. Thus if , sin0. We also see that if =0 or =180 that =0 so sin=0 if =0 or =180. Since cos is the horizontal component, we note that is positive in quadrants I and IV and negative in quadrants II and III. Thus cos>0 if 0 90 or and cos0 if We remember that tan is the slope of the terminal arm. Since the slope is positive in quadrants I and III, tan>0 if 0 90 or We also see that tan is undefined at =90 and =270 because the line is vertical at those values. For each quadrant, state the angle of the quadrant and whether each of the following is greater than zero or less than zero (ie.quadrant I: 0 90, &'(>0, etc.). 4) Quadrant I 5) Quadrant II 6) Quadrant III 7) Quadrant IV Using =30, we see that 30, 150, 210, and 330 have and coordinates that are very similar. For example, we notice that at 30 and 150 the value is the same (sin30 =sin150 ). We also see that at 210 and 330 will be the negative value that it is at 30. Thus sin210 =sin330 =sin30. Similarly, if we look at the horizontal components () we see that cos30 =cos330 and cos150 = cos210 =cos30. Looking at the slope, tan30 =tan210 and tan150 =tan330 =tan30.

20 Since 30, 150, 210, and 330 are so closely related to each other, we say that 30 is the reference angle for those other angles. The reference angle is always between 0 and 90 and is the angle formed from the terminal arm to the axis. Find the reference angle for each of the following angles. 8) 2 9) 49 10) 3 11) 215 ) ) 18 14) ) 298 Determine which quadrant is in if 16) sin>0 and tan0. 17) cos0 and tan>0. 18) tan>0 and sin0. 19) sin>0 and cos0. 20) sin>0 and tan>0. 21) sin0 and tan0. 22) cos0 and tan0. 23) sin>0 and cos>0. Determine the exact value of &'(,0&, and ( for each point given on the terminal side. 24) (2,1) 25) (4,3) 26) (3,5) 27) (2,3)

21 Answer each of the following. 28) Given sin= 6 and , find 7 the exact value of cos and tan. 29) Given tan= 7 and , find => the exact value of sin and cos. 30) Given tan= = and , find > sin and cos exactly. 31) Given sin= 6 7 and what quadrant must be in? 32) If sin=0.848 and tan0, find to the nearest tenth of a degree. 33) If tan=1.96 and cos0, find to the nearest tenth of a degree. 34) If cos=0.933 and sin>0, find to the nearest tenth of a degree. 35) If tan=0.317, find all possible values of to the nearest tenth of a degree. 36) If sin=0.617, find all possible values of to the nearest tenth of a degree. 37) If cos=0.253, find all possible values of to the nearest tenth of a degree.

22 Unit 8: Systems of Equations Day 6 Math 10 Common Review: Make the correct statement of similarity for each pair of triangles if similar. State the reason for their similarity. 1) B D 2) A 28 C E A C E 30 B 35 D Similarity: Reason: 35 Similarity: Reason: F 3) Find the area of the largest square that can fit inside a circle of diameter 30. 4) An 18 metre ramp rises to a doorway that is 5 metre off the ground. How far away from the building is the ramp to the nearest centimetre? For each triangle, label each side opposite, adjacent, or hypotenuse relative to, then determine to the nearest degree. 5) 6) 4 in cm 6 in 8 cm

23 7) 8) Solve the following triangles using SohCahToa. Round all answers to one decimal place. 9) 10) 11) 6 B A /. 42 1

24 ) 13) 14) + B ) Find the area of the shaded region below. 16) A woman stands at the top of a building that is 25 m tall. She measures the angle of inclination to the top of a taller 45 building to be 32. If the buildings are 20 m apart, how high is the taller building? 18 cm

25 Use a protractor to measure each indicated angle to the nearest degree, then determine the length of 7 to the nearest integer. 17) 18) 26 cm m Find the reference angle for each of the following angles. 19) ) ) ) 61 Determine which quadrant is in if 23) tan8>0 and sin8<0. 24) sin8<0 and cos8>0. 25) sin8>0 and cos8<0. 26) sin8<0 and tan8<0. Determine the exact value of ;<=8,->;8, and?+=8 for each point given on the terminal side. 5, 1) 3,2)

26 Answer each of the following. 29) Given sin8= E and tan8<0, find the FG exact value of cos8 and tan8. 30) Given tan8= I and 180 <8<360, find G the exact value of sin8 and cos8. Answer each of the following. 31) What does sin8 represent in the unit circle? 32) What does cos8 represent in the unit circle? 33) What does tan8 represent in the unit circle? 34) Why can sin8 and cos8 never be greater than 1 or less than 1? Solve graphically for ;<=8, ->;8, and?+=8 for each of the following angles. 35) 8=0 36) 8=90 37) 8=180 38) 8=270

27 Unit 8: Trigonometry Key Math 10 Common Day 1 1) ~, Angle 2) ~, Ratio 3) ~, SAS 4) ~, Ratio 5) ) ) ) 13 9) cm 10) 3.6 km 11) 7.3 ft ) 19.8 ft 13) ).2 km 15) ) ) 14.8 ft 18) 30 19) no 20) yes 21) no 22) no 23) ) ) 16 27) 27.4 Day 2 1) ~, Ratio 2) ~, SAS 3) yes 4) no 8) 9) ) 11) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) 4.4 m 30) m 31) ) m Day 3 1) ) ) ) 1.3 5) ) ) 3.5 8) 11 9) 1.9 m 10) ) 48.6 ) ) ) ) ) ) ) ) ) undefined 21) 22) 23) 24) ) ) ) ) ) ) ) undefined 32) ) ) ) ) ) ) ) ) ) ) ) 33.5 Day 4 1) =20,=9.2, =9.8 2) =17.1, =55, =20.9 3) =57.1, =32.9,"=13.6 4) =39,# =4.3, =15.9 5) =61.2, =29.8,# =23.7 6) =62,% =6.5, =13.9 7) m 8) 15.4 m 9) cm 10) ft 11) 269 cm ) 6.1 m 13) 13.8 in 14) 9.2 m 15).7 m 16) 5.3 m Day 5 1) =13.0,=66.9, =23.1 2) =23.0, =52, =29.2 3) =28,% =13.1,"=24.7 4) 0 <) <90,*+,)>0,.*)>0,/#,)>0 5) 90 <) <180,*+,)<0,.*)>0,/#,)<0 6) 180 <) <270,*+,)<0,.*)<0,/#,)>0 7) 270 <) <360,*+,)<0,.*)>0,/#,) <0 8) 58 9) 49 10) 48 11) 35 ) 10 13) 18 14) 23 15) 62 16) II 17) III 18) III 19) II 20) I 21) IV 22) II 23) I 24) *+,)=,.*)=,/#,)= 25) *+,)=,.*)=1,/#,) = 26) *+,)=,.*)=,/#,) = ) *+,)=,.*)=,/#,)= 28) tan) =,cos) 1 = 1 29) sin) =,cos) = 30) sin) =,cos) = 31) III 32) ) ) ) 162.4, ) 38.1, ) 104.7, 255.3

28 Day 6 1) ~, AAA 2) No 3) 450 4) 17.3 m 5) 42 6) 34 7) 39 8) 41 9) # =2.8, =6.6,=35 10) # =11.4,=8.9, =48 11) =23.2, =66.8,8=38.1 ) # =5.9, =9.1, =50 13) =57.6, =32.4,# = ) % =2.5,=3.5, =55 15) ) 37.5 m 17) ) =32,% = ) )=63,% = ) 17 20) 45 21) 65 22) 61 23) III 24) IV 25) II 26) I 27) sin) = 28) sin)=,cos) =,tan) = 30) sin)=,cos) =,cos) =,tan)= 29) cos) =,tan) = 31) vertical component 32) horizontal component 33) slope of line segment formed by the point and the origin 34) hypotenuse of unit circle is 1 and they must always be shorter than the hypotenuse 35) sin)=0,cos) =1,tan)=0 36) sin) =1,cos)=0,tan) =undefined 37) sin)=0,cos) = 1,tan) =0 38) sin) = 1,cos)=0,tan) =undefined