Lesson 5-3 (non honors) The Law of Sines Math-3 Lesson 5-3 (non honors) The Law of Sines

The Law of Sines is for Triangles that are NOT right triangles. B a b C

vocabulary Solve a triangle: find the unknown lengths of sides and measures of angles of a triangle. The problem will given you some of the sides or angles.

Sin, cos, and tan ratios are for solving Right triangles! What do we do if the triangle is not a right triangle? A B a C c b In this lesson we learn how to solve triangles that are NOT right triangles.

The standard method of labeling triangles is: The length of the side opposite Angle A is lower case a. A B a C c b

The standard method of labeling triangles is: The length of the side opposite Angle A is lower case a. A B a C c b

The standard method of labeling triangles is: The length of the side opposite Angle C is lower case c. A B a C c b

C a b h A B Solving for ‘h’ then setting the equations equal to each other. Eliminating ‘h’, dividing by ‘a’ and ‘b’. We could repeat this for any combination of sides and angle.

Law of Sines By the Transitive Property this means each of the expressions are equal to each other. We could also write it this way (using sequential property of equality steps):

Law of Sines Which one we use depends upon whether we need to find the measure of an unknown angle or an unknown side. Pick the version that puts the unknown variable in the numerator!

When a problem is given, they either (1) Give you the drawing and provide some of the measured sides or angles. Then ask you to solve for the other sides. A B a C c 15 107º 25º In this example we’ll solve for ‘c’.

Pick the version that puts the unknown variable in the numerator! 15 120º 25º Pick the version that puts the unknown variable in the numerator! Replace letters in the formula with numbers from the triangle. “Solve” for ‘c’ (isolate the variable ‘c’ on one side of the equal sign..

Another way a problem is given, is that they just give you some measurements. C = 120º

Draw the general triangle that has capital letters for angles and lower case letters for the lengths of the sides opposite the angles. A B a C c b

A= 35º a = 10 C = 120º 2. Label the triangle with values given in the problem. A B a C c b A B a C c 120º 10 b 35º

Pick the version that puts the unknown variable in the numerator! 120º 10 b 35º Pick the version that puts the unknown variable in the numerator!

Pick the version that puts the unknown variable in the numerator! 120º 10 b 35º Pick the version that puts the unknown variable in the numerator!

Solve for the measure of angle C, given the following triangle. B a C 10 31º 19 Pick the version that puts the unknown variable in the numerator! Replace letters in the formula with numbers from the triangle.

Solve for the measure of angle C, given the following triangle. B a C 10 31º 19 “Solve” for ‘C’ (isolate the variable ‘C’ on one side of the equal sign). First we will have to isolate sin(C)

“Inverse sine (ratio) = angle Sine (angle) = ratio A B a C 10 31º 19 “Inverse sine (ratio) = angle

What if they given you two angles but not the two that you need? C = 107º c = 15 Find “little” ‘a’ A B a 15 C b 107º 25º c Using the Triangle Sum Theorem (angle is a triangle always add up to 180º): Now solve using the Law of Sines.

Your Turn: Draw and label the triangle. A B a C 2. A = ? 75º b C = 75º B = 20º c = 20 solve the triangle. Draw and label the triangle. A B a C 2. A = ? 75º b 3. a = ? 4. b = ? 20º 20

Your turn: Solve for a: A = 28º B = 64º c = 55 A B a b C 28º 64º c 55