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Unit 9 Day 1 – Solving Trig. Equations We will solve simple trig equations giving both general solutions and solutions over an interval and I will practice solving these problems. March 13, 2017 Trigonometric Equations • Trigonometric identity - trig equation that holds true for any angle • Trigonometric equation - involves one or more trigonometric functions and is only true for certain angles • To solve a trigonometric equation: use trig. identities and algebraic techniques to reduce the given equation to an equivalent but more manageable expression. Examples – Solve the equation 2 cos x 1 0 2cosx = -1 cosx = -½ 2π 4π x , 3 3 where on the unit circle is cosx = - ½? Examples – Solve the equation 3 sec x 2 2 sec x 3 3 cos x 2 5π 7π x , 6 6 Examples – Solve the equation 3 cot x 3 3 cot x 3 2π 5π x , 3 3 Examples – Solve the equation 2cos x 1 0 2 2cos x 1 2 1 cos x 2 2 1 cos x 2 2 cos x 2 3 5 7 x , , , 4 4 4 4 Solve the equation 2 cos x 1 0 2 cos x 1 1 cos x 2 2 cos x 2 7 x , 4 4 Are these the only answers?? Solving Trig Equations General rule: Because trig functions are periodic, if there is one solution, then there are an infinite number of solutions. There are 2 ways to give a solution to a trig equation: • Solutions in an interval: include all the solutions that fall in an interval (usually 0 ≤ θ < 2π). This is what we have been doing thus far. • General Solution: θ +2kπ (for sin, cos, sec, csc) θ + kπ (for tan, cot) where θ is one solution to the equation, and k is an integer Solve the equation. Give a general formula for all solutions. 2sinθ + 3 = 2 2sinθ = -1 sinθ = -½ general solution: 7π θ= + 2kπ 6 11π θ= + 2kπ 6 7 11 , 6 6