26. [Pythagoras / Trigonometry]

Transcription

1 6. [Pythagoras / Trigonometry] Skill 6. Solving simple quadratic equations. Calculate the square numbers on the right-hand side of the equation. Evaluate and simplify the right-hand side of the equation. Take the square root of both sides of the equation to find the value of the unknown. Estimate which positive number, when multiplied by itself, produces the number under the square root. Check your estimation by multiplying your guess by itself. If the number is a decimal number consider the position of the decimal point. Q. Find the positive solution for a: a 0 6. a 0 6 a a 44 a 44 a a a a a) Find the positive solution for c: c 676 c b) Find the positive solution for b: b 44 b b b c) Find the positive solution for a: a 5 a a a d) Find the positive solution for b: b 600 b b b e) Find the positive solution for c: c 6.5 f) Find the positive solution for a: a 0.6 a a a g) Find the positive solution for c: c page 303 c c h) Find the positive solution for a: a a a a a i) Find the positive solution for b: b 5 0 b b b b Maths Mate 5./6. Skill Builder 6

2 Skill 6. Recognising Pythagoras theorem. Determine which is the longest side of the right-angled triangle (hypotenuse). ints: In a triangle, the vertices are labelled with capital letters. ny side length of a triangle is usually labelled with a lower case letter (the same as the letter at the opposite verte or angle). Identify correct statements of Pythagoras theorem or ones derived from it. Pythagoras Theorem: a + b c For any right-angled triangle, the square of the length of the hypotenuse (c) equals the sum of the squares of the lengths of the two perpendicular sides (a and b). c hypotenuse perpendicular side b perpendicular side B a C R: c a + b For any right-angled triangle, the square of the length of the hypotenuse (c) equals the sum of the squares of the lengths of the two perpendicular sides (a and b). R: a c b and b c a For any right-angled triangle, the square of the length of a perpendicular side equals the difference between the square of the length of the hypotenuse and the square of the length of the other perpendicular side. Q. Which statements of Pythagoras theorem are correct? u ) t + u s B ) u s + t t C ) s u t s. Pythagoras statements are: u s + t or s + t u or s u t or t u s The correct statements are B and C. a) Which letter marks the perpendicular side opposite to angle D in this right-angled triangle? E c D b) Which statements of Pythagoras theorem are correct? ) w u + v B ) u v + w C ) v w u d C e v w u and c) Connect the following Pythagoras relationships to their corresponding diagram: d) Connect the following Pythagoras relationships to their corresponding diagram: f d e d e + y z z f y e f d d e f z y z y e y e d + f d f y + z z page Maths Mate 5./6. Skill Builder 6

3 Skill 6.3 Solving more comple quadratic equations. Calculate the square numbers on both sides of the equation. Isolate the pronumeral on the left-hand side of the equation. Evaluate and simplify the right-hand side of the equation. Take the square root of both sides of the equation to find the value of the unknown. Q. Find the positive solution for b: + b 5. + b b 5 b 5 44 b 8 b 8 b a) Find the positive solution for c: + 6 c c c 400 c 400 a 0 b) Find the positive solution for a: a a + a... a c) Find the positive solution for b: 5 + b b b b d) Find the positive solution for a: a e) Find the positive solution for b: 4 + b 5 f) Find the positive solution for c: 9 + c a b g) Find the positive solution for c: c h) Find the positive solution for b: 40 + b 50 i) Find the positive solution for c: c b page Maths Mate 5./6. Skill Builder 6

4 Skill 6.4 Finding the hypotenuse when the other sides of a right-angled triangle are given. Identify the given side lengths on the diagram. Pythagoras Theorem State Pythagoras theorem. Substitute the values into Pythagoras theorem. a + b c Isolate the unknown quantity on the left-hand side of the equation. Evaluate and simplify the right-hand side of the equation. Take the square root of both sides of the equation to find the value of the unknown. int: The most common triplets of numbers that make Pythagoras theorem true are: (3, 4, 5) (5,, 3) (8, 5, 7) (7, 4, 5). e.g (Pythagorean triads) Q. For this triangle use Pythagoras theorem a + b c. Find the length of the hypotenuse. a? a) For this triangle use Pythagoras theorem a + b c. Find the length of the hypotenuse.? b 6. a and b 6 a + b c + 6 c c + 6 c c 400 c b) For this triangle use Pythagoras theorem a + b c. Find the length of the hypotenuse. b 30 a 90 b 0 a 6? c c c 500 c c c... c) For this triangle use Pythagoras theorem c a + b. Find the length of the hypotenuse. b 48? d) For this triangle use Pythagoras theorem a + b c. Find the length of the hypotenuse.? a 4 a b page Maths Mate 5./6. Skill Builder 6

5 Skill 6.5 Finding a perpendicular side when the other perpendicular side and the hypotenuse of a right-angled triangle are given. Identify the given side lengths on the diagram. Pythagoras Theorem State Pythagoras theorem. Substitute the values into Pythagoras theorem. a + b c Isolate the unknown quantity on the left-hand side of the equation. Evaluate and simplify the right-hand side of the equation. Take the square root of both sides of the equation to find the value of the unknown. int: The most common triplets of numbers that make Pythagoras theorem true are: (3, 4, 5) (5,, 3) (8, 5, 7) (7, 4, 5). e.g (Pythagorean triads) Q. For this triangle use Pythagoras theorem a + b c. Find the length of the side labelled b. a.5 b? 6.5 a) For this triangle use Pythagoras theorem a + b c. Find the length of the side labelled b.. a.5 and 6.5 a + b c.5 + b 6.5 b b b 36 b 36 b 6 b) For this triangle use Pythagoras theorem a + b c. Find the length of the side labelled a. a 5 b? a? b 5... b b b 400 b... c) For this triangle use Pythagoras theorem a + b c. Find the length of the side labelled a. b a? b 4... a... a... a a... d) For this triangle use Pythagoras theorem a + b c. Find the length of the side labelled b. a 9 4 b? a a b b... page Maths Mate 5./6. Skill Builder 6

6 Skill 6.6 pplying Pythagoras theorem (). Locate or draw a right-angled triangle in the diagram. Identify the given side lengths in this right-angled triangle. Identify the required side length in this right-angled triangle and label it with a variable. Use Pythagoras theorem to find the required side length. (see skills 6.4, page 306 and 6.5, page 307) Q. Find the missing length in this diagram showing a T-square cm cm cm cm cm 5 cm 3 missing length: hypotenuse cm Pythagoras theorem 5 cm a) Would clipping shears, 5 cm long, fit inside this rubish bin with its lid on? [bjects not drawn to scale.] b) Would a 6 cm long paint brush fit inside this tin with its lid on? [bjects not drawn to scale.] Find the diagonal of the bin 0 cm 0 cm 5 cm 50 cm Pythagoras theorem clipper 5 cm 5 cm < 30 cm clipper fits inside the bin... yes... page Maths Mate 5./6. Skill Builder 6

7 Skill 6.6 pplying Pythagoras theorem (). c) ow far down this mountain slope is the sleigh descending? d) What is the distance marked on this diagram showing the semaphore which signals letter I? cm 50 m m 30 cm 600 m m 50 cm cm e) ow long is the ramp on which the model boat descends? f).5 m long ladder is leaning against a wall and its end is.5 m from the base of the wall. ow high up the wall is the ladder reaching? 50 cm cm? m.5 m 00 cm.5 m cm m page Maths Mate 5./6. Skill Builder 6

8 Skill 6.7 pplying Pythagoras theorem to find the perimeter of -dimensional shapes. ighlight a right-angled triangle in the diagram. Identify the given side lengths in this right-angled triangle. Identify the missing side length in this right-angled triangle and label it with a variable. Use Pythagoras theorem to find the missing side length. (see skills 6.4, page 306 and 6.5, page 307) Calculate the perimeter of the -dimensional shape. (see skill 3., page 59) Q. Find the perimeter of this rhombus by first calculating the missing side length. 50 m a) Find the perimeter of this rectangle by first calculating the missing side length m m 5 P cm 39 cm cm cm c) Find the perimeter of this isosceles triangle by first calculating the missing side length.. 50 m 5 m 0 m m 60 m P 4 65 m 60 m b) Find the perimeter of this triangle by first calculating the missing side length Pythagoras theorem P... diagonals are perpendicular and bisect each other 5 mm 4 mm mm d) Find the perimeter of this triangle by first calculating the missing side length. mm cm cm 9 cm 9 cm m 8 m 0 m 5 m 6 m P... cm P... m page 30 Maths Mate 5./6. Skill Builder 6

9 Skill 6.8 pplying Pythagoras theorem in a variety of -dimensional shapes. ighlight a right-angled triangle in the diagram. Identify the given lengths in this right-angled triangle. Identify the missing length in this right-angled triangle. Use Pythagoras theorem to find the missing length. (see skills 6.4, page 306 and 6.5, page 307) Q. Find the missing length in this trapezium Pythagoras theorem a) Find the missing length in this rectangle Pythagoras theorem b) Find the missing length in this triangle c) Find the missing length in this triangle d) Find the missing length in this trapezium page 3 Maths Mate 5./6. Skill Builder 6

10 Skill 6.9 Finding a side length in isosceles right-angled triangles (). Use Pythagoras theorem in the isosceles right-angled triangle to find an unknown side length. (see skills 6.4, page 306 and 6.5, page 307) Q. ow much wire was used for the outside square of this clothes line? [Leave your answer in surd form.]. 80 cm cm cm 60 cm 50 cm 40 cm 30 cm 80 cm aerial view 60 cm 50 cm 40 cm 30 cm Perimeter wire 4 Pythagoras theorem cm (appro. 000 cm) a) ow long is each blade of this windmill, if they are all the same length and the distance between the tips of two consecutive blades is 6 m? [Leave your answer in surd form.] b) What is the distance between the flags when this semaphore is signalling letter N as shown in the diagram? [Leave your answer in surd form.] 6 m m 90 cm 90 cm m cm Pythagoras + 6 theorem m cm page 3 Maths Mate 5./6. Skill Builder 6

11 Skill 6.9 Finding a side length in isosceles right-angled triangles (). c) ow long are this ladder s legs, if they are m apart? [Leave your answer in surd form.] d) ow long is the inner wire on this clothes line? [Leave your answer in surd form.] m m m 60 cm 50 cm 40 cm 30 cm cm +... m aerial view cm e) ow long is each of these helicopter blades, if they are all the same length and the distance between the tips of two consecutive blades is 5 m? [Leave your answer in surd form.] f) Find the missing length in this diagram showing a ceiling fan. [Leave your answer in surd form.] m m 6 m 80 cm 80 cm cm... m cm page 33 Maths Mate 5./6. Skill Builder 6

12 } Skill 6.0 pplying Pythagoras theorem to find the distance between two points located on a Cartesian plane (). Draw a horizontal line through the first point. Draw a vertical line through the second point. Mark the point at the intersection of these lines. Join the three points (the two given points and the point at the intersection) to form a triangle. Count the units along the horizontal and vertical sides of the triangle. Use Pythagoras theorem in this right-angled triangle to find the hypotenuse. (see skill 6.4, page 306) Q. Find the distance G in this Cartesian plane. [Leave your answer in surd form.] G 3 Y X. horizontal line G 3 Y 0 G GP + P G G G 45 G 45 5 vertical line X } 3 P intersection point P Pythagoras theorem missing length: hypotenuse a) Find the distance CD in this Cartesian plane. [Leave your answer in surd form.] 3 Y 0 C CD CP + PD CD CD 80 CD D X P 4 b) Find the distance MN in this Cartesian plane. [Leave your answer in surd form.] Y M N 3 4 MN MP + PN... P X page 34 Maths Mate 5./6. Skill Builder 6

13 Skill 6.0 pplying Pythagoras theorem to find the distance between two points located on a Cartesian plane (). c) Find the distance B in this Cartesian plane. [Leave your answer in surd form.] 4 3 Y X d) Find the distance PQ in this Cartesian plane. [Leave your answer in surd form.] P 4 3 Y 0 Q 3 4 X B B... PQ e) Find the distance EF in this Cartesian plane. [Leave your answer in surd form.] E 4 3 Y f) Find the distance ʹBʹ in this Cartesian plane. [Leave your answer in surd form.] Bʹ 4 3 Y X X F ʹ EF ʹB ʹ page 35 Maths Mate 5./6. Skill Builder 6

14 Skill 6. pplying Pythagoras theorem to find the area of -dimensional shapes. ighlight a right-angled triangle in the diagram. Identify the given side lengths in this right-angled triangle. Identify the missing side length in this right-angled triangle and label it with a pronumeral. Use Pythagoras theorem to find the missing side length. (see skills 6.4, page 306 and 6.5, page 307) Calculate the area of the -dimensional shape. (see skills 3.5 to 3.7, pages 65 to 67) Q. Find the area of this triangle.. Let perpendicular height m m rea of triangle bh a) Find the area of the parallelogram by first calculating the missing side length base parallelogram parall. bh 4... c) Find the area of the square by first calculating the missing side length. + square... m 3 m m 5 m m 5 m m 480 m b) Find the area of the right-angled triangle by first calculating the missing side length triangle m cm... d) Find the area of the rectangle by first calculating the missing side length. rectangle m m... perpendicular height bisects the base 40 cm 39 m 34 4 cm 6 3 m cm 36 m page 36 Maths Mate 5./6. Skill Builder 6

15 Skill 6. Recognising trigonometric functions (sine, cosine, tangent). Identify the hypotenuse of the triangle, and the opposite and adjacent sides corresponding to the marked angle (α - alpha, β - beta, θ - theta, etc). Label each side of the triangle with, and. Decide which two sides of the triangle are given R which side and angle of the triangle are given. Use one of the S - C - T relations to decide which trigonometric ratio can be used to find the unknown angle R the unknown side. int: Use the S - C - T rules to remember the trigonometric ratios. Trigonometric ratio (function) sine oppos ite sin θ sin θ Sineppositeypotenuse (S) hypotenus e Trigonometric ratio (function) cosine adjac ent cos θ cos θ Cosinedjacentypotenuse (C) hypotenus e Trigonometric ratio (function) tangent oppos ite tan θ tan θ Tangentppositedjacent (T) adjac ent θ hypotenuse adjacent opposite Q. Which trigonometric ratio would be used to find the unknown side? ) sine B ) cosine 3 C ) tangent. 3 The answer is C. (opposite) and (adjacent) the tangent ratio T a) Which trigonometric ratio would be used to find angle θ? ) sine 7 θ B ) cosine 8 C ) tangent cosine C b) Which trigonometric ratio would be used to find the unknown side? ) sine 6 B ) cosine 4 C ) tangent c) Which trigonometric ratio would be used to find angle α? ) sine 44 α B ) cosine C ) tangent 55 d) Which trigonometric ratio would be used to find the unknown side y? ) sine y 8 B ) cosine C ) tangent 9 e) Which trigonometric ratio would be used to find angle β? ) sine β B ) cosine 0 5 C ) tangent f) Which trigonometric ratio would be used to find the unknown side? ) sine B ) cosine 56 C ) tangent 5 page 37 Maths Mate 5./6. Skill Builder 6

16 Skill 6.3 Calculating the value of basic trigonometric ratios in right-angled triangles. Mark the angle whose trigonometric ratio is required. Label each side of the triangle with, and. Use one of the S - C - T relations to calculate the required trigonometric ratio. Trigonometric ratio (function) sine oppos ite sin θ sin θ Sineppositeypotenuse (S) hypotenus e Trigonometric ratio (function) cosine adjac ent cos θ cos θ Cosinedjacentypotenuse (C) hypotenus e Trigonometric ratio (function) tangent oppos ite tan θ tan θ Tangentppositedjacent (T) adjac ent θ hypotenuse adjacent opposite Q. Calculate the value of cos D in this triangle.. F 5 60 D 65 E F 5 60 D 65 adjacent cos D hypotenuse cosine C E a) Calculate the value of sin Bʹ in this triangle. sine S 39 5 C 36 B sin Bʹ c) Calculate the value of cos E in this triangle. 70 D E b) Calculate the value of tan M in this triangle. P M 00 tan M... d) Calculate the value of sin ʹ in this triangle. 3 B C N F cos E... sin ʹ... page 38 Maths Mate 5./6. Skill Builder 6

17 Skill 6.4 Finding an unknown side of a right-angled triangle when a trigonometric ratio of an angle and another side of the triangle are given (). Label each side of the triangle with, and. Use the S or C or T relation corresponding to the given trigonometric ratio value. Substitute the numeric values in the relation. Solve the equation for the unknown side length. MM5. MM Trigonometric ratio (function) sine oppos ite sin θ sin θ Sineppositeypotenuse (S) hypotenus e Trigonometric ratio (function) cosine adjac ent cos θ cos θ Cosinedjacentypotenuse (C) hypotenus e Trigonometric ratio (function) tangent oppos ite tan θ tan θ Tangentppositedjacent (T) adjac ent θ hypotenuse adjacent opposite Q. 9 m support wire is attached to a flagpole and makes an angle of 65 with the ground. If cos , find the approimate distance from the end of the wire to the base of the flagpole. 9 m 65 m adjacent. cos 65 hypotenuse 0.4 cosine C 9 4 Cross multiply m 9 m 65 m a) 3 m ladder is leaning against a wall and makes an angle of 6 with the wall. If cos , how high up the wall is the top of the ladder? b) kite s string makes an angle of 40 with the horizon. The string length is 000 cm and sin If the boy s height is 60 cm, how high above the ground is the kite flying? 3 m 6 m cm 000 cm 40 cos adjacent hypotenuse Cross multiply m sin 40 height cm page 39 Maths Mate 5./6. Skill Builder 6

18 Skill 6.4 Finding an unknown side of a right-angled triangle when a MM5. MM trigonometric ratio of an angle and another side of the triangle are given (). c) road sign casts a shadow which is 4. m long when the sun is at an angle of 40 in the sky. If tan , find the height of the road sign. [Give your answer correct to decimal places.] d) In this profile view the vacuum cleaner makes an angle of 50 with the ground. If sin , how high above the ground is the handle of the vacuum cleaner? m m tan 40 cm cm sin 50 m cm e) If cos , what is the height of this tent above the ground? m 65.5 m f) You are observing a hot air balloon which is 5 km away from you and makes an angle of 8 with your eye level. If sin , how high above eye level is the balloon? m 5 km km 8 height m km page 30 Maths Mate 5./6. Skill Builder 6

19 Skill 6.5 Calculating the value of trigonometric ratios in right-angled triangles by first applying Pythagoras theorem (). Label each side of the triangle with, and. pply Pythagoras theorem to calculate the unknown side length of the triangle. (see skills 6.4, page 306 and 6.5, page 307) Pythagoras Theorem a + b c Use one of the S - C - T relations to calculate the required trigonometric ratio. sin θ (S) cos θ (C) tan θ (T) Q. Calculate the value of sin P in this triangle. [int: Pythagoras theorem will help.] P 5 R 0 Q P sine S 5 R 0 opposite. sin P hypotenuse Q PQ PR + RQ PQ PQ PQ 65 PQ 5 RQ 0 sin P PQ RQ PQ Pythagoras unknown PQ a) Calculate the value of tan U in this triangle. [int: Pythagoras theorem will help.] W 50 opposite VW tan U adjacent UV VW UW UV VW VW 48 VW 48 tan U UV 4... c) Calculate the value of cos Bʹ in this triangle. [int: Pythagoras theorem will help.] cos Bʹ 5 B C V 4 U BʹCʹ BʹCʹ BʹCʹ 0 cos Bʹ... tangent T b) Calculate the value of sin B in this triangle. [int: Pythagoras theorem will help.] 60 C sin B B B B sin B... d) Calculate the value of cos N in this triangle. [int: Pythagoras theorem will help.] P N cos N 80 0 NP NP NP cos N... M B page 3 Maths Mate 5./6. Skill Builder 6

20 Skill 6.5 Calculating the value of trigonometric ratios in right-angled triangles by first applying Pythagoras theorem (). e) Calculate the value of cos J in this triangle. [int: Pythagoras theorem will help.] 0 J I 5 cos J J J J cos J... f) Calculate the value of cos L in this triangle. [int: Pythagoras theorem will help.] K M cos L 5 7 KL KL KL cos L... L g) Calculate the value of tan D in this triangle. [int: Pythagoras theorem will help.] B 6 tan D 4 C D h) Calculate the value of tan in this triangle. [int: Pythagoras theorem will help.] G I tan i) Calculate the value of sin Q in this triangle. [int: Pythagoras theorem will help.] Q sin Q 45 5 S R j) Calculate the value of sin M in this triangle. [int: Pythagoras theorem will help.] M 75 sin M N page 3 Maths Mate 5./6. Skill Builder 6