Trig Ratios of Angles in Standard Position

DAY 3

Suppose θ is any angle in standard position, and P(x, y) is any point on its terminal arm, at a distance r from the

origin.

• Remember to use the Pythagorean Theorem to find r.

Use a reference triangle to determine the three primary trigonometric ratios in terms of x, y and r.

Primary Trig Functions

Question

• Would the value of x, y and r remain the same in all four quadrants? Why or why not?

Patterns in the Trig Ratios

• All families of angles (those with the same reference angle), use the same congruent triangle to calculate trig ratios

• Due to the position of the triangle, the x and/or y-values will change from positive to negative as you change quadrants.

Families of Angles

• Examples:

The CAST rule shows where each trig ratio is positive.

• Quadrant I – all trig ratios are positive.

• Quadrant II – only Sine is positive.

• Quadrant III – only Tangent is positive.

• Quadrant IV – only cosine is positive.

Review Example 1: Given P (3, -4); sketch the triangle and use it to calculate the trig ratios.

Solution:

• Sketch, Find Length of Terminal Arm, and determine the 3 Primary Trig Ratios

• sin θ =

• cos θ =

• tan θ =

Example 2: The following points are on the terminal arm of angle θ. Find the 6 trig ratios.

a. (5, 12)

Solution:

1. Draw (5, 12)

2. Solve using Pythagorean Theorem.

3. Determine the 3 trig ratios.

sin θ = cos θ = tan θ =

Solve for an Angle Given its Exact Trig Ratio Value

• Ex. Solve for θ:

a. sin θ = 1/2, 0˚≤ θ < 360˚

Solution:

1. Check the value, is it positive or negative?

• What possible quadrants could this value be in? (I, II)

2. What is the angle that is given by sin θ = 1/2?

(30˚, use it to find other angle)

3. Find the two answers using the reference angles formulas.

Possible answers are in QI and II so, θ = 30˚ or 180˚-30˚ = 150˚

NOTE:

• To do this type of solving, we must be familiar with the CAST rule and the reference angles formulas.

Example 2: Solve for the missing angles.

1.

2.

3.

1cos

2

Example 3: Solve for the missing angles.

1tan

3

Solve These Equations using Exact Values

Solve These Equations using Exact Values

Assignment

• P. 84 #10a, 11, 23a,b

• P. 96 #2(a,b), 3(b,c), 4, 5(b,c), 6, 8(a,b), 9, 12