Review Triangles 11th

TRIANGLES AND TRIGONOMETRIC RATIOS

REVIEW 11TH

BRAINSTORMING

1. What is triangle?2. How can classify the triangles? 3. What is the sum of the internal angles in a triangle?

Types of Triangles

AREA IN A TRIANGLE

The area is half of the base times height.

"b" is the distance along the base"h" is the height (measured at right angles to the base) Area = ½ × b × h

AREA IN A TRIANGLE

The base can be any side, Just be sure the "height" is measured at right angles to the "base":

ASSESMENT QUESTIONS

ASSESMENT QUESTIONS

ASSESMENT QUESTIONS

ASSESMENT QUESTIONS

What are congruent figures?

When are two triangles similar?

TRIGONOMETRY

Trigonometry ... is all about triangles.

Trigonometry helps us find angles and distances, and is used a lot in science, engineering, video games, and more!

Right-Angled Triangle

The triangle of most interest is the right-angled triangle

Another angle is often labeled θ, and the three sides are then called: Adjacent: adjacent (next to) the angle θ Opposite: opposite the angle θ and the longest side is the Hypotenuse

Sine, cosine and tangent

The triangle of most interest is the right-angled triangle

They are simply one side of a right-angled triangle divided by another. For any angle "θ":

EXERCISE EXAMPLE

PROBLEM EXAMPLE

Cotangent, Secant, Cosecant)

Similar to Sine, Cosine and Tangent, there are three other trigonometric functions which are made by dividing one side by another:

Cosecant Function: csc(θ) = Hypotenuse / Opposite Secant Function: sec(θ) = Hypotenuse / Adjacent Cotangent Function: cot(θ) = Adjacent / Opposite

EXERCISE

Jacob is measuring the height of a Sitka spruce tree in North Carolina. He stands 45 feet from the base of the tree. He measures the angle of elevation from a point on the ground to the top of the tree to be 59°. How can he estimate the height of the tree ?

EXERCISE

If angle X is an acute angle with sin x = 3/4 , what is the value of cot x?

EXERCISE

Solve the right triangle shown below, given that cot N = 4/5 . Find the exact side lengths and approximate the angles to the nearest degree.

The law of sines

Examples

Finding sides

Examples

Finding angles

The law of cosines

Examples

Finding sides

Examples

Finding angles

When to use sine or cosine rules?

Sine Rule

Cosine Rule

The Law of Cosines is useful for finding: -the third side of a triangle when we know two sides and the angle between them. -the angles of a triangle when we know all three sides

This law is useful for finding a missing angle when given an angle and two sides, or for finding a missing side when given two angles and one side.

These examples illustrate the decision-making process for a variety of triangles:

Now, you:

Let's GO