Warm UP Find the area of an oblique triangle ABC. Where angle A = 32degrees Side b= 19, and side c = 32.

Essential Question: What is the law of Sines and how can I use this rule to solve for the sides and angles of a triangle?

Standard(s): MM4A6. Students will solve trigonometric equations both graphically and algebraically. c. Apply the law of sines and the law of cosines.

The Law of SINES

The Law of SINES For any triangle (right, acute or obtuse), you may use the following formula to solve for missing sides or angles:

Use Law of SINES when ... AAS - 2 angles and 1 adjacent side you have 3 dimensions of a triangle and you need to find the other 3 dimensions - they cannot be just ANY 3 dimensions though, or you won’t have enough info to solve the Law of Sines equation. Use the Law of Sines if you are given: AAS - 2 angles and 1 adjacent side ASA - 2 angles and their included side

Example 1 You are given a triangle, ABC, with angle A = 70°, angle B = 80° and side a = 12 cm. Find the measures of angle C and sides b and c. * In this section, angles are named with capital letters and the side opposite an angle is named with the same lower case letter .*

Example 1 (con’t) A C B 70° 80° a = 12 c b The angles in a ∆ total 180°, so angle C = 30°. Set up the Law of Sines to find side b:

Example 1 (con’t) A C B 70° 80° a = 12 c b = 12.6 30° Set up the Law of Sines to find side c:

Example 1 (solution) A C B 70° 80° a = 12 c = 6.4 b = 12.6 30° Angle C = 30° Side b = 12.6 cm Side c = 6.4 cm Note: We used the given values of A and a in both calculations. Your answer is more accurate if you do not used rounded values in calculations.

Example 2 You are given a triangle, ABC, with angle C = 115°, angle B = 30° and side a = 30 cm. Find the measures of angle A and sides b and c.

Example 2 (con’t) To solve for the missing sides or angles, we must have an angle and opposite side to set up the first equation. We MUST find angle A first because the only side given is side a. The angles in a ∆ total 180°, so angle A = 35°. A C B 115° 30° a = 30 c b

Example 2 (con’t) A C B 115° a = 30 c b 30° a = 30 c b 35° Set up the Law of Sines to find side b:

Example 2 (con’t) A C B 115° a = 30 c b = 26.2 30° a = 30 c b = 26.2 35° Set up the Law of Sines to find side c:

Example 2 (solution) Angle A = 35° Side b = 26.2 cm Side c = 47.4 cm A 115° 30° a = 30 c = 47.4 b = 26.2 35° Angle A = 35° Side b = 26.2 cm Side c = 47.4 cm Note: Use the Law of Sines whenever you are given 2 angles and one side!

The Law of Sines AAS ASA Use the Law of Sines to find the missing dimensions of a triangle when given any combination of these dimensions.

Let’s do three problems together off of tonights homework 1, 6, and 11