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

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

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

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

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

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

Activity 4-2: Trig Ratios of Any Angles Part 3: Exact Values of the Trigonometric and Reciprocal Trigonometric Ratios for Special Angle You have completed this presentation, go back to the activity page and complete the rest of the lesson.