The Law of SINES

When Do I use Law of Sines vs. Law of Cosine ? Two sides One opposite angle given Angle opposite side Two angles One opposite side given side given side Two side One angle given Given three sides given any angle

Helpful Web Site nometry/law-of-sines-and- cosines.php nometry/law-of-sines-and- cosines.php

Objectives: CCSS To find the area of any triangle. To use the Law of Sine; Understand and apply. Derive the formula a=1/2 ab sin for the area of a triangle.

Use Law of SINES when... AAS - 2 angles and 1 adjacent side ASA - 2 angles and their included side SSA (this is an ambiguous case) 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:

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) AC 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) teaching note AC 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) Set up the Law of Sines to find side c: AC B 70° 80° a = 12 c b = °

Example 1 (con’t) teaching note Set up the Law of Sines to find side c: AC B 70° 80° a = 12 c b = °

Example 1 (solution) Finally! Gott’em all Angle C = 30° Side b = 12.6 cm Side c = 6.4 cm AC B 70° 80° a = 12 c = 6.4 b = ° 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) AC B 115° 30° a = 30 c b 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°.

Example 2 (con’t) teaching note AC B 115° 30° a = 30 c b 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°.

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

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

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

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

Example 2 (solution) done! Got all parts Example 2 (solution) done! Got all parts AC B 115° 30° a = 30 c = 47.4 b = ° 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.

Applying Law of Sines Due next class worksheet problems # 1-