Sec. 7.4: The Laws of Cosine and Sine Objectives: Use trig-values to determine the measure of a missing side or angle of a non-right triangle. Determine the area of a triangle given side–ange–side information.

Concept: Parts of a Triangle The Law of Cosines is all about the relationship of the given information within a non-right triangle. Triangles are named by their vertices (angles) which use all upper-case letters. B Sides are named by using the same letter of the opposite angle, but in lower case. c a A C b

Concept: Law of Consines The Law of Cosines is used when we have the lengths of all 3 sides (SSS) and need an angle, or When we have an angle and the two sides that make it (SAS) and need the third side.

Concept: Law of Consines cont… The Law of Cosines can be written: a2 = b2 + c2 – (2bc)CosA or b2 = a2 + c2 – (2ac)CosB c2 = a2 + b2 – (2ab)CosC Remaining 2 sides Opposing side and angle: 1 given, 1 variable

Concept: Using the Law of Consines Using the given measurements, solve for the missing part of the triangle: Q 8 80º q = ______ mLP = ______ mLR = ______ 6 P q R Solving for side q: Q and q are opposing parts and mLQ is given. 1. Write the formula q2 = p2 + r2 – (2pr)CosQ 2. Substitute q2 = 62 + 82 – (2)(8)(6)Cos80º

Concept: Using the Law of Consines Using the given measurements, solve for the missing part of the triangle: Q 8 80º q = ______ mLP = ______ mLR = ______ 6 P q R Solving for side q: Since there are no variables on the right side you can just enter this in your calculator the way it is written. 3. Enter into calculator q2 = 36 + 64 – 96Cos80º

Concept: Using the Law of Consines Using the given measurements, solve for the missing part of the triangle: Q 8 80º q = ______ mLP = ______ mLR = ______ 9.13 6 P q R 9.13 Solving for side q: Simplify the right side q2 = 83.32977494… 5. Take the square root 2nd x2 2nd (–) = √q2 = √83.32977494… q = 9.13

Concept: Using the Law of Consines Using the given measurements, solve for the missing part of the triangle: Q 8 80º q = ______ mLP = ______ mLR = ______ 9.13 6 P q R =9.13 Solving for LP 1. Set up the Equation p2 = q2 + r2 – (2qr)CosP 2. Substitute 62 = 9.132 + 82 – (2)(9.13)(8)CosP 3. Simplify 36 = 83.3569 + 64 – 146.08CosP 4. Combine like terms 36 = 147.3569 – 146.08CosP

Concept: Using the Law of Consines Using the given measurements, solve for the missing part of the triangle: Q 8 80º q = ______ mLP = ______ mLR = ______ 9.13 6 P q R =9.13 Solving for LP 36 = 147.3569 – 146.08CosP Move the constant –147.3.. –147.3569 –111.35.. = –146.08CosP

Concept: Using the Law of Consines Using the given measurements, solve for the missing part of the triangle: Q 8 80º q = ______ mLP = ______ mLR = ______ 9.13 6 P q R =9.13 Solving for LP –111.3569.. = –146.08CosP ________ ___________ 6. Divide BS by Coef. –146.08 –146.08 0.7623…= CosP

Concept: Using the Law of Consines Using the given measurements, solve for the missing part of the triangle: Q 8 80º q = ______ mLP = ______ mLR = ______ 9.13 6 40.3º 40.3º P q R =9.13 Solving for LP 0.7623…= CosP 7. Undo cos 2nd cos 2nd (–) ent 40.3º = mLP

Concept: Using the Law of Consines Using the given measurements, solve for the missing part of the triangle: Q 8 80º q = ______ mLP = ______ mLR = ______ 9.13 6 40.3º 40.3º P q R =9.13 Solving for LR Now that you have the measure of 2 angles you can use the Triangle Sum Postulate to find the measure of the missing angle. 80º + 40.3º + mLR = 180º 120.3º + mLR = 180º mLR = 59.7º

Concept: Law of Sines The relationship between the angles of a triangle and the opposite sides can be written as a proportion. Sine of the angle / length of opposite side/ You MUST have values for two opposing parts of the triangle. You need the measure of an angle and the side opposite that angle.

Concept: Law of Sines cont… B Find the length of side b. 85º 8cm mLA = 40º. The side across from it = 8cm. 40º C A b 1. Set up the proportion. Sin 40º Sine 85 _______ _______ = 8 b

Concept: Law of Sines cont… B Find the length of side b. 85º 8cm 2. Cross multiply. 40º C A b Sin 40º Sine 85 _______ _______ = 8 b 8sin85º = bsin40º

Concept: Law of Sines cont… B Find the length of side b. 85º 8cm Divide both sides by sin40º. 40º C A b 8sin85º = bsin40º _______ _______ sin40º sin40º 8sin85º = b sin40º

Concept: Law of Sines cont… B Find the length of side b. 85º 8cm 4. Put into calculator 40º C A b 8sin85º = b sin40º 8 Sin ( 85 ) ENT ÷ sin ( 40 ) ENT b = 12.40 cm

Concept: Area Area of a non-right triangle: You MUST have SAS B Area of a non-right triangle: You MUST have SAS 85º 6cm Area = bcsinA 2 35º C A 8cm Area = (6)(8)sin35º 2 Area =13.77cm2