Pre-Calc 12

4.3.2 Trigonometric Ratios and Angles

Big Idea:

Using inverses is the foundation of solving equations and can be extended to relationships between

functions

Curricular Competencies:

Model mathematics in situational contexts

Use proper math vocabulary and language

If r =1 then

sin 𝜃 = cos 𝜃 = tan 𝜃 =

Determining the Sign of a Trigonometric Ratio

Reciprocal trigonometric ratios do not change signs so the reciprocal ratios will follow the same

pattern as their counterparts.

Example 1: Find the value of all 6 ratios for each angle 𝜃 in exact value.

a) 𝜃 =5𝜋

6

sin 𝜃 = csc 𝜃 =

cos 𝜃 = sec 𝜃 =

tan 𝜃 = cot 𝜃 =

Pre-Calc 12

b) 𝜃 =5𝜋

3

sin 𝜃 = csc 𝜃 =

cos 𝜃 = sec 𝜃 =

tan 𝜃 = cot 𝜃 =

Example 2: In which quadrant will 𝜃 terminate if angle 𝜃 is in standard position with the given

conditions?

a. cot 𝜃 > 0 c. sin 𝜃 𝑎𝑛𝑑 sec 𝜃 > 0

b. cos 𝜃 > 0 𝑎𝑛𝑑 cot 𝜃 < 0 d. sec 𝜃 < 0 𝑎𝑛𝑑 𝑡𝑎𝑛𝜃 < 0

Finding Angles Given Their Trigonometric Ratios

1. Determine the quadrant(s) the angle will be in by looking at the sign of the ratio.

2. Determine the reference angle and draw a rough sketch in the appropriate quadrant.

3. Determine the rotation angle(s) using the reference angle and the quadrant(s).

Example 3: Determine the exact measure of all angles that satisfy the given conditions.

a. tan 𝜃 = −1,0° ≤ 𝜃 < 360° c. cos 𝜃 =√3

2, 0° ≤ 𝜃 < 360°

Pre-Calc 12

b. cos 𝜃 = −1

2, 0 ≤ 𝜃 < 2𝜋 d. cot 𝜃 = −1,0 ≤ 𝜃 < 2𝜋

Example 4: Determine the approximate measure of each angle. Give answers to the nearest

hundredth of a unit, where possible.

a. sin 𝜃 = 0.42,0 ≤ 𝜃 < 2𝜋 b. cot 𝜃 = −4.87,0 ≤ 𝜃 < 2𝜋

Calculate Trigonometric Ratios for Points not on the Unit Circle

Example 5: The point A(-4,3) lies on the terminal arm of an angle 𝜃 in standard position. What is

the exact value of each trigonometric ratio for 𝜃?

sin 𝜃 = csc 𝜃 =

cos 𝜃 = sec 𝜃 =

tan 𝜃 = cot 𝜃 =

Assignment: p 202 3, 5-7, 10-12, 16, 19