Pre-Calculus 11 Notes

By: Gen L.

In partnership with Hyperion University, 2023




Lesson 5: Trigonometric Ratios






Note:

  • All angles are in standard position.
  • (diagonal Distance)


Finding Exact Distance

Using Trig Ratios, we can find the exact distance a point is from the origin and the exact ratios for the angle.

Suppose we have an angle with a point somewhere on the terminal arm. Each coordinate represents the horizontal () and vertical () distance, combining to result in a diagonal () distance.

Determining

We can use Pythagoras to solve for the value of as long as is on the terminal arm. ( is terminal arm.)

  • Our distance:

Determining Exact Ratios

Assume we had a point that is on the terminal arm. What are the exact Ratios?

  1. Find

Determining Exact Ratios, cont.

  1. Determine Ratios
    • Sine
    • Cosine
    • Tangent

Positivity Rule: CAST

The CAST Law is a rule which states which ratios are positive in each quadrant

C: Q(IV) A: Q(I) S: Q(II) T: Q(III)
sin(x) - + + -
cos(x) + + - -
tan(x) - + - +

Example 1: Exact Values

Determine the exact value of

  1. Define Variables
    • let
      • why look at ? It's our key to finding the answer.
    • let
  2. Determine value

Example 2: Completing the Set

Assume is an angle with a terminal arm () and . Determine exact values of .

  1. Define variables:
    • let
    • Since
    • Using Pythagoras, we can state

Example 2, cont.

  1. Complete the Set
    • Cosine
    • Tangent

Example 3: Range Condition

Solve

  1. Cut the boundary.
    • Since .
    • This means that the answer must be in .
  2. Determine
    • Since there is a

Example 3, cont.

  1. Determine
    • Use Reference Formulae

Finish