Pre-Calculus 11 Notes

By: Gen L.

Lesson 4: Angles & Special Triangles

4.1: Angles in Standard Position

Angles in Standard Position

When dealing with angles, special cases emerge. To investigate, we need to use a Cartesian Plane ( graph).

Terminology:

• Initial Arm
  • Start position; always lays along the -axis
• Terminal Arm
  • Final position; location after rotation around the origin.
• Angle in Standard Position
  • Initial arm must be positive
  • Origin is vertex.
  • Measured between initial & terminal arms
  • Always positive when drawn anti-clockwise

Angular Range

Since any angle in standard position must always be positive, we know the range of possible answers per quadrant.

Reference Angle:

• An acute angle ()
• Vertex is origin
• Formed between terminal arm & -axis
• Identified by

Reference Angle Formulae

Based on the quadrant, the formula changes. It's a challenge when you have a reference angle (i.e. ), you can actually have 4 different angles that satisfy the reference angle: ,

4.2: Special Triangles

Special Right Triangles

Certan angles we encounter allow us to determine the exact value of Trigonometric ratios.
These triangles have angles:
Let's look at specific cases.

45-45-90 Triangle, defined

If there is a right triangle with an angle of , the other (non-right) angle is also .
If the side lengths of the legs equal unit, the hypotenuse will be equal to .

45-45-90 Triangle, part 2

Ratios: 45-45-90

Using this special triangle, we can determine exact trigonometric ratios for each case:

• Sine
• Cosine
• Tangent

Application 1: Exact Values

• Find exact values for
• Since
  • We can define
• We have a proportion:
  • By cross multiplying & dividing:

30-60-90 Triangle, defined

If we draw a triangle with an angle of , the other (non-right) angle must be . This triangle is half of an equilateral triangle with side length .
If we use Pythagoras to find the height, we get an exact value of , which is always opposite the angle.

30-60-90 Triangle, part 2

Ratios: 30-60-90

• Sine
• Cosine
• Tangent

Applications 2: Exact Values

• Find exact values for
• /
  • Each can be rewritten as proportions:
• /
  • Cross multiplying & dividing gives

Next Lesson