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Pre- Calculus 12 Date: _____________ Chapter 4– Trigonometry & the Unit Circle Name: ____________________ Section 4.3B – Trigonometry Ratios (Primary& Secondary) Relating the trigonometric ratios to the coordinates of points on the unit circle Determining exact and approximate values for trigonometric ratios Identifying the measures of angles that generate specific trigonometric ratios Solving problems using trigonometric ratios For the following right-angled triangle, AB is the _____________________ of the right triangle. BC is _____________________ to angle θ. AC is _____________________ to angle θ. The following are called the primary trigonometric ratios: we use SOH CAH TOA. opp adj opp sin cos tan hyp hyp adj The following are called the secondary trigonometric ratios. They are the _____________________ of the primary trigonometric ratios. COSECANT SECANT COTANGENT csc θ = sec θ = cot θ = Example 1: The point (4, -3) is on the terminal arm of an angle θ. a) Draw θ in its standard position. c) Find all six trig ratios in exact values. sin θ = csc θ = cos θ = sec θ = tan θ = cot θ = b) Calculate θ. Each quadrant gives us different signs: ASTC Q II QI Q III Q IV Given the following conditions, where would be? a) cos > 0 & tan < 0 b) sin < 0 & cot > 0 Example 2: Example 3: Find sec θ in exact value if θ is in 5 quadrant II and cot θ = . 12 Ex. 4 If θ is on a terminal arm, find the measure(s) of angle θ if sin θ = 1 . 5 Example 4: Determine each of the trigonometric ratios. Draw a diagram to support your answer. a) cos 260 (nearest hundredth) b) csc ( 70 ) (nearest hundredth) c) sin 135 (exact value) d) cot 5 6 Example 5: Determine the measures of all angles that satisfy the following. Use diagrams to support your explanations. a) cos 0.366, 0 360 c) Sin Ѳ = 0.879 , 0 ≤ Ѳ ≤ 2π b) sec d) 2 , 0 2 . 3 To find any trigonometric ratio given an angle: Step 1 Approximation Exact Value Set calculator to appropriate mode (degree Determine the reference angle R and which or radian) quadrant it lies in. Draw the right triangle to the x-axis. 2 Enter trig ratio into calculator Use the special triangles to determine the side lengths of your triangle (the hypotenuse is always 1 in the unit circle!) Fill in the trig ratio including the sides (from 3 step 2) and the sign (from step 1). To find an angle given any trigonometric ratio: Step 1 2 Approximation Exact Value Determine all of the possible quadrants in Determine all of the possible quadrants in which our angle will lie (use ASTC). Draw which our angle will lie (use ASTC). Draw a a picture. picture. Enter inverse trig ratio (-1 exponent) into Determine the reference angle R using calculator (check mode!) to find the special triangles. reference angle. Do not include the sign of the ratio if it is negative! 3 a. Use your diagram and your reference angle Use your diagram and your reference angle to to find all possible angles. find all possible angles. Sin Ѳ = 0.879 , 0 ≤ Ѳ ≤ 2∏ b. Assignment: Page 201 #1-3, 5, 6, 8-10 (every other letter for all), + 12a, 14