Triangles: Trigonometry Right Triangles Trigonometric Ratios Rules

Right Triangle Basics Pythagorean Theorem Pythagorean Triples: Trios of Natural numbers that are related by the Pythagorean equality Multiples: Any multiple of a Pythagorean Triple is also a Pythagorean Triple Trigonometric Ratios Inverse Trigonometry

Problem Solving Note: for applied triangle problems such as these, we will use DEGREES to measure angles Minutes and seconds are not in the curriculum Later in the unit, we will discuss RADIANS

Activity Exercise 9.2: 2, 3, 4 Vocabulary: Angles of Depression and Elevation

Triangles: Trigonometry Sine Rule Cosine Rule

New Material Today Trigonometry for triangles that are NOT right triangles Sine Rule (Law of Sines) Cosine Rule (Law of Cosines)

The Sine Rule Useful when an angle and the side opposite that angle are known The numerator/ denominator distinction is not important as long as you are consistent

Sine Rule: Ambiguous Case AC is 17 cm BC is 9 cm Angle A is 29 degrees Solve for all missing sides and angles

Sine Rule: Ambiguous Case AC is 17 cm BC is 9 cm Angle A is 29 degrees Given: angle, opposite and adjacent sides (NOT the sides that form the angle) The two values for the unknown angle are supplementary

Cosine Rule Similar to Pythagorean theorem, with an adjustment for the fact that the triangles are not always RIGHT Note: cos(90) is 0, so the Pythagorean Theorem is a special case

Cosine Rule The cosine rule allows us to find an angle if all sides are known. You need only to solve for angle A, though a formula is given in your formula packet.

Activity 9.5.1: 1-20, multiples of : 1-20, multiples of : 1-20, multiples of 5 Exercise 9.5.6: 2-8, even Skills: Use the Sine Rule and Cosine rules to solve triangles. Identify and solve the ambiguous case. Use trigonometry in context

Triangles: Trigonometry Review of Sine and Cosine Rules Area 3-D Geometry and Trigonometry

Problem(s) of the Day

Today Formula Sheet Reminder, HW Check Review of: Right Angle Trigonometry Sine Rule and Cosine Rule Area of Triangles 3-D Geometry and Trigonometry (Quiz on Tuesday)

Area of a Triangle Given: two sides and the angle they form Use the triangle on the board (and trigonometric ratios) to determine a general formula for the area of a triangle Hint: in the end, you will need to use the “one half base times height” definition

Area of a Triangle If the “height” or altitude is not given Given: two sides and the angle they form

Practice with Area Exercise: 9.4 5, 6, 7 Hint on Parallelograms: Diagonals bisect each other

3-D Trigonometry Look for steps that will allow you to analyze two-dimensional parts of the three-dimensional figure Example 9.3, #1

Practice with 3-D Geometry and Trigonometry Example: 1 Exercise 9.3: 2, 3, 4, 5

Homework 9.3: 2, 3, 4, 5 9.4: 4, 9

Triangles: Trigonometry Review of Mensuration

Problem of the Day From a point A, 150m due south of a tower, the angle of elevation of the top of the tower is 30 degrees. From a point B, due east of the tower, the angle of elevation of the top of the tower is 40 degrees. Draw a diagram (or two) to display this How far apart are points A and B

More 2-D  3-D Geometry Remember your two-dimensional distance and midpoint formulae:

More 3-Dimensional Geometry Here are the distance and midpoint formulae for three dimensions:

Practice with 3-D Coordinate Geometry Find the length and the midpoint of the segment defined by the points C and D.

Quiz Next Class (Tuesday) Print out the formula sheet!!! Pythagorean Theorem, Angle Sums Right Triangle Trigonometry Sine Rule Recognize the ambiguous case Cosine Rule Area of Triangles Basic 3-D Problems

Homework , 14, 18 Challenge: 16 There is a quiz next class. Additional problems may be selected.