Aim: Law of Cosines Course: Alg. 2 & Trig. Aim: What is the Law of Cosines? Do Now: If the measures of two sides and the included angle of a triangle.

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Presentation transcript:

Aim: Law of Cosines Course: Alg. 2 & Trig. Aim: What is the Law of Cosines? Do Now: If the measures of two sides and the included angle of a triangle are known, can the size (area) and shape of the triangle be determine? Yes

Aim: Law of Cosines Course: Alg. 2 & Trig. SAS  SAS B A C 2 sides and included angle: AB  A’B’, BC  B’C’,  B   B’ 2 sides and included angle: AB  A’B’, BC  B’C’,  B   B’ B’ A’ C’B’ A’ C’ Shortcut for proving congruence in triangles: Measurements showed:  ABC   A’B’C’  SAS

Aim: Law of Cosines Course: Alg. 2 & Trig. Area of Triangle y x (b cos A, b sin A) b c a h AB C The area of a triangle is equal to one-half the product of the measures of two sides and the sine of the angle between them.  A case

Aim: Law of Cosines Course: Alg. 2 & Trig. Area of Triangle & Law of Cosines y x (b cos A, b sin A) b c a AB(c,0) C Use the distance formula to find length of a (0,0) AB = c AC = b BC = a

Aim: Law of Cosines Course: Alg. 2 & Trig. Law of Cosines The square of the measure of one side of a triangle is equal to the sum of the squares of the measures of the other two sides minus twice the product of the measures of these two sides and the cosine of the angle between them. Law of Cosines

Aim: Law of Cosines Course: Alg. 2 & Trig. What does it all mean? If you do know the measures of two sides and the included angle of a triangle, then you can determine the size and shape of the triangle. a = ? b c If I know the measures of  A, b, and c, then I can find the measure of side a. ex. Find a if m  A = 35, b = 16, and c = 7 to nearest tenth

Aim: Law of Cosines Course: Alg. 2 & Trig. Model Problem In ∆ABC, if a = 4, c = 6, and cos B = 1/16, find b. Appropriate version of Law of Cosines Substitute and solve: must be positive B A C a b c

Aim: Law of Cosines Course: Alg. 2 & Trig. Draw: Model Problem In ∆RST, if r = 11, s = 12, and m  T = 120, find t to the nearest integer. Appropriate version of Law of Cosines in terms of r, s &T Substitute and solve: S RT t s = 12 r = º t 2 = 397 t = must be positive t = 20 to nearest integer

Aim: Law of Cosines Course: Alg. 2 & Trig. Finding Angle Measures In ∆ABC, a = 5, b = 7, and c = 10. Find cos B. Appropriate version of Law of Cosines solved for cos B Substitute and solve: Law of Cosines solved for cosine of angles

Aim: Law of Cosines Course: Alg. 2 & Trig. Model Problem In isosceles triangle RED, RE = ED = 5 and RD = 8. Find the measure of the vertex angle,  E, to the nearest degree. E DR t r = 5d = 5 120º e = 8 Draw: Appropriate version of Law of Cosines solved for cos E cos E = -.28 Substitute & solve: m  E ≈ 106

Aim: Law of Cosines Course: Alg. 2 & Trig. Regents Question In triangle DEF, side e = 10, f = 8 and m  D = 110. Find the length of the third side to the nearest tenth. 1) ) ) ) 10.5

Aim: Law of Cosines Course: Alg. 2 & Trig. The Product Rule A surveyor at point P sights two points X and Y that are on opposite sides of a lake. If P is 200 m. from X and 350 m. from Y, and m  XPY = 40, find the distance from X to Y to the nearest meter.

Aim: Law of Cosines Course: Alg. 2 & Trig. Model Problem Some nylon fabric will be cut to cover the kite frame shown below. Diagonal AC is 29 inches. What size should the angles be at A, B, C, and D? B D C A 16 in. 26 in. 16 in. 26 in.