1. Activity 4-2: Trig Ratios of Any Angles
Part 3: Exact Values of the Trigonometric and Reciprocal Trigonometric Ratios for Special Angles

2. Activity 4-2: Trig Ratios of Any Angles
Part 3: Exact Values of the Trigonometric and Reciprocal Trigonometric Ratios for Special Angle
Let us review some special cases
CASE 1: 45o or π/4 Acute Angle with a Radius of √2 x y
Primary Ratio
Reciprocal Ratio
(1, 1)
Sine Ratio
sin(π/4) = 1/√2
Cosecant Ratio
csc (π/4) = √2 √2 1
Cosine Ratio
cos(π/4) = 1/√2
Secant Ratio
sec (π/4) = √2
θ= π/4 1
Tangent Ratio
tan(π/4) = 1/1 =1
Cotangent Ratio
cot (π/4) = 1/1 =1

3. Activity 4-2: Trig Ratios of Any Angles
Part 3: Exact Values of the Trigonometric and Reciprocal Trigonometric Ratios for Special Angle
Let us review some special cases
CASE 2: 30o or π/6 Acute Angle with a Radius of 2 x y
Primary Ratio
Reciprocal Ratio
Sine Ratio
sin(π/6) = 1/2
Cosecant Ratio
csc (π/6) = 2
(√3, 1) 2 1 1
Cosine Ratio
cos(π/6) = √3/2
Secant Ratio
sec (π/6) = 2/√3
θ= π/6 √3
Tangent Ratio
tan(π/6) = 1/√3 =1
Cotangent Ratio
cot (π/6) = √3/1 = √3

4. Activity 4-2: Trig Ratios of Any Angles
Part 3: Exact Values of the Trigonometric and Reciprocal Trigonometric Ratios for Special Angle
Let us review some special cases
CASE 3: 60o or π/3 Acute Angle with a Radius of 2 x y
Primary Ratio
Reciprocal Ratio
(1, √3)
Sine Ratio
sin(π/3) = √3/2
Cosecant Ratio
csc (π/3) = 2/√3 2 √3
Cosine Ratio
cos(π/3) = 1/2
Secant Ratio
sec (π/3) = 2/1 = 2
θ= π/3 1
Tangent Ratio
tan(π/3) = √3/1 = √3
Cotangent Ratio
cot (π/3) = 1/√3

5. Activity 4-2: Trig Ratios of Any Angles
Part 3: Exact Values of the Trigonometric and Reciprocal Trigonometric Ratios for Special Angle
Let us review some special cases
Example 1: Find the EXACT value of all of the trigonometric ratios for θ=5π/3 x y
Primary Ratio
Reciprocal Ratio
2π/3
π/3
Sine Ratio
sin(π/3) = -√3/2
Cosecant Ratio
csc (π/3) = -2/√3
Cosine Ratio
cos(π/3) = 1/2
Secant Ratio
sec (π/3) = 2/1 = 2 1
3π/3
-√3
Tangent Ratio
tan(π/3) = -√3/1 = -√3
Cotangent Ratio
cot (π/3) = -1/√3 2
(1, -√3)
4π/3
θ= 5π/3
Related Acute Angle=π/3

6. Activity 4-2: Trig Ratios of Any Angles
Part 3: Exact Values of the Trigonometric and Reciprocal Trigonometric Ratios for Special Angle
Let us review some special cases
Example 2: Find all the possible angles between 0≤θ≤2π for the following: cot θ=-1/√3 x y
Since the ratio is negative, the angle resides in Quadrant 2 and Quadrant 4
cot θ = x/y = -1/√3
.: x = 1 or -1
.: y = √3 or -√3 √3
θ= 2π/3 1
When x = -1 and y = √3, the angle that is formed is 2π/3 -1
5π/3
-√3
When x = 1 and y = -√3, the angle that is formed is 5π/3
The possible angles for cotθ=-1/√3 are:
θ = 2π/3 and θ = 5π/3

7. Activity 4-2: Trig Ratios of Any Angles
Part 3: Exact Values of the Trigonometric and Reciprocal Trigonometric Ratios for Special Angle
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