CAPS Mathematics GRADE Sine, Cosine and Area Rules
Outcomes for this Topic. Calculate the area of a triangle given an angle and the two adjacent sides. Lesson. Apply the Sine Rule for triangles to calculate an unknown side or an unknown angle of a given triangle. Lesson 3. Apply the Cosine Rule for triangles to calculate an unknown side or an unknown angle of a given triangle. Lesson 3 4. Apply the Sine and the Cosine rules to solve problems in -dimensions. Lesson 4
CAPS Mathematics GRADE Lesson The Area Rule
Trigonometric Ratios In a Some basic definitions a reminder right angled triangle, the 3 trigonometric ratios for an angle are defined as follows: hypotenuse opposite adjacent sin opposite hypotenuse tan opposite adjacent cos adjacent hypotenuse
The area formula of a triangle Consider a non-right angled triangle ABC. a, b and c are the sides opposite angles A, B and C respectively. ( This is the conventional way of labelling a triangle ). Draw the perpendicular, h, from C to BA. Area of base height Area b ch In ACN, sin A h b --- () bsin A h Substituting for h in () Area cb sin A A c h C N Area bc sin A a B
A similar argument gives the same formula for the area if B is obtuse i.e. B 90 The formula always uses Different forms of the area formula sides and the angle formed by those sides (Included ) Any angle can be used as such in area formula, so absin C bc sin A Area = = = A c b 90 B ca sin B a C h N
A Three possible approaches to find b the area of a triangle Any angle can be used in the formula, so Area c C Area a absin C B b A bc sin b C A a c C Area a B casin B A c B
The area of a triangle Example Find the area of PQR. Solution: We must use the angle formed by the sides with the given lengths. We know PQ and RQ so use the included angle Q Area of PQR QPQR sin Q 64 8 7 sin 64 cm 5, cm
The area of a triangle Example Find the area of ABC. A useful application of the area formula occurs when we have a triangle formed by radii and a chord of a circle. r A Area CA CB sinc C But CA CB r r B Area r sin
Classwork [Area of Triangles] Find the areas of the triangles shown in the diagrams. Give your answers accurate to decimal digits ) ) 8 cm 40 30 0 cm radius 6 cm AOB 0
8 cm Solution [Area of Triangles] 40 30 0 cm ) Area XYZ XY YZ siny sin z x Y 8 0 sin 0 cm Y 80 40 30 0 37,59 cm
Solution [Area of Triangles] Given: radius 6 cm AOB 0 ) Area AOB r sin O 6 cm sin0 5,59 cm
CAPS Mathematics GRADE Lesson The Sine Rule
The Sine Rule for Triangles One way to find unknown sides and angles in triangles is by using the Sine Rule: Suppose ABC is a scalene triangle h In ACN, sin A b In h bsin A h BCN, sin B a h asin B bsin A asin B or b sin a c non -right angled Drop CN A h N a sin b AB B
The Complete Sine Rule for Triangles ABC can be turned so that BC is the base. We then get C b Now h c sin B bsin C sin B sin C b c a A A So c B sin A sin B sinc a b c B c h a b C
When do we use the Sine Rule? The sine rule can be used in a triangle when: Two angles and a side are given To calculate second side Two sides and the non-included angle are given To calculate second angle
Application of the Sine Rule - Example In ABC, find the size of angles A and C. Solution: Use sin a asin B sin A b 0 sin 6 sin A 47,4 A sin b A A is opposite the shorter of the given sides. B A 6 A must be an acute angle. (Only one possibility as can be seen from sketch) Thus C 80 6 47,4 70,6
Application of the Sine Rule - Example In PQR it is given that: QR 5, PR 4 and Q 48. Determine P. P is opposite the longer of the given sides. P 48 P can be an acute or obtuse angle. ( Two possibilities as can be seen from sketch below) Solution : Use psin Q 5sin 48 sin P q 4 P 68,3 or sin Q q sin P p P 80 68,3,7 P P
Application of the Sine Rule - Example 3 In XYZ, find the length XY. Solution : sin z Z y sin Y 3sin 55 z sin 9 z,0 As the unknown is a side, we use the sine rule in its reciprocal form. The unknown side is then at the top. sin z y Z sin Y
Classwork [Sine Rule]. In ABC, find B. (Correct to two decimal places). In PQR, find QR and the area of PQR
Find B. ( decimal places) B 35 B acute or obtuse sin B 0sin 35 7 Solution Given: sin A sin B sin35 sin B a b 7 0 B 55,0 or B 80 55,0 4,98
Obtained: B 55,0 or B 4,98 Solution Given: B 35 B acute or obtuse B (Acute) B (Obtuse)
Solution. Find QR and the area of PQR. QR 67 sin 64 sin80 Given: 67sin 64 QR 6,5 cm sin80 Area of PQR QPQR sin 36 67 6,5 sin36 04,09 cm R 80
Lesson 3 The Cosine Rule CAPS Mathematics GRADE
The Cosine Rule for Triangles The Cosine Rule for ABC is given by: b a c ac cos B or a b c bc cos A Symmetry also implies that: c a b absin C A C b a c B We use this form to find the third side when two sides and included angle are given.
Proof of the Cosine Rule In CAD : x cos A and b x h b In BCD : a h c x h c cx x b x cbcos A b h a x c x a c x b c bc cos A Proofs for symmetrical results are similar.
A second form of the Cosine Rule Know: a b c bc cos A bc cos A b c a b c a cos A bc We use this form to find any angle of a triangle when we know all 3 sides.
Applications of the Cosine Rule - Example Find p in the PQR Q Apply the Cosine Rule 6 P 0 p 7 p q r qr cos P R p 7 6 76cos0 7 p, 3 decimal accuracy
Applications of the Cosine Rule - Example Find X in the XYZ Solution: Use the Cosine Rule cos X cos X y z x yz 8 6 4 (8)(6) X 6 8 Y 4 Z X 9,0 ( dec )
Applications of the Cosine Rule Example 3 Find side c and B in the given ABC. Cosine rule: c b a ba cosc b 5 A c Sine rule: sin B b C 30 c 5 9 (5)(9) cos30 c 9,6 ( decimal places ) a 9 sin C 5 sin 30 sin B c 9,6 B 5,3 ( dec. ) B
Classwork. Given ABC with AB 6 cm; BC 4 cm and ABC 60. Find AC correct to decimal digits.. Find all the angles in XYZ, giving your answers to one decimal place accuracy.
Find AC Solution ( dec accuracy): Given: AC BC AB BC AB cos ABC 4 6 4 6 cos60 8 AC 8 5,9 cm
Solution Determine all angle measures of XYZ. cos X y z x yz Given: Hence cos X Now 4 9 7 4 9 X 48, sin X siny y sin X 4sin 48. siny x y x 7 Y 5, Then Z 80 X Y 06,6
Lesson 4 CAPS Mathematics GRADE Basic Applications: Problems in -D
Problems in dimensions: Example. Points A and B are in the same horizontal plane as C, the foot of a vertical tower PC. B 4 ; PAC 65 and AB 5 m. Calculate PC. Sine rule: BPA 65 4 3 AP 5 sin 4 sin 3 5sin 4 AP 4,8 m sin 3 4 3 65 B A PC 5 m sin 65 AP PC APsin65 4,8sin65 38,8 m P C
. In the figure Given: QP Problems in dimensions: Example QR represents a proposed tunnel. Q and R are visible from a point P. The three points are in the same plane. 00 m; PR 60 m and QPR 0 Q Calculate the length of tunnel. 00 60 0 QR 00 60 00 60 cos0 P R QR 33 m
Activity. From the ends of a bridge AB, 0 metres long, the angles of depression of a point P on the ground directly under the bridge is 4, and 70,. Find the height, h, of the bridge under this point. A 0 m 4, 70, h B P
Solution A 0 m 4, 70, h B Question: Find h P APB 80 4, 70, 67, 7 AP 0 AP sin 70, sin 67,7 0 msin 70, sin 67,7 AP 0,65 m But sin 4, h 0,65 h 0,65 sin 4, 68,95 m
Activity. ABCD is a wall of a room, AD being the line of the ceiling. EF is a picture rail, with E being directly below A. AE = metres; ACB x and ECB y cos x (a) Prove that EC sin( x y) (b) Find the length and height m y x of the wall if x 33 and y 0.
(a) Prove that cos x EC sin( x y) Solution m y x x Now ACE x y and CAD x Hence, CAE 90 x From AEC : EC sin 90 sin x EC sin 90 x x sin x y cos x sin x y y
Solution b Know: EC cos x sin x y x 33 and y 0 m y x EC cos33 sin3 7,46 m BC cos y BC EC cos y EC Length of room BC 7, 46 m cos 0 7,0 m