4.3.2 trigonometric ratios and angleskbriggsmath.weebly.com/uploads/2/4/5/0/24504307/4... · 4.3.2...
TRANSCRIPT
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