Solving Linear Trig Equations (For Angles NOT on the Unit Circle)

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Solving Linear Trig Equations (for angles NOT on the unit circle) When solving a trig equation for solutions outside the first quadrant it is important to review how to find the “matching” angle in the other 3 quadrants. Remember that the angle in the first quadrant is often referred to as the “reference” angle. Sometimes the reference angle is one of your solutions, however, other times you simply use it as a “reference” to find the solutions that satisfy the equations. How to use a reference angle to find a matching angle: Quadrant II Quadrant I In degrees: 180 – reference angle Angles in the first quadrant are sometimes one of In radians: π – reference angle the solutions, but sometimes used as a reference to find the solutions. You must decide depending upon the value (positive or negative) of the trig relationship you are working with. Quadrant III Quadrant IV In degrees: 180 + reference angle In degrees: 360 – reference angle In radians: π + reference angle In radians: 2π – reference angle Examples: 1) Solve for 0 < x < 360: tan(x) = ¼ In Degrees: First, I notice that ¼ is not a value on my unit circle, so this equation must be solved using the inverse tangent feature on my calculator. Tan-1(1/4) ≈ 14.04 Since Tangent values are positive in the first and 3rd quadrants this reference angle is also one of my solutions. To find the other solution in the 3rd quadrant, I must add 180o 180 + 14.04 = 194.04 Final answer: x ≈14.04o and x ≈ 194.04o 2) Solve for all possible solution: cos (x) = In Radians: Whenever solving a trig equation set equal to a negative value, we first find our reference angle by setting the equation equal to the same value, only positive. Then use the inverse function to solve for your reference angle: Cos-1 ( ) ≈ 1.14 Now we must reconsider the original problem, and ask “in what quadrants will cosine values be negative?” Answer: 2nd and 3rd. To find the solution in the 2nd quadrant we subtract our reference angle from pi, to find our solution in the 3rd quadrant we add our reference angle to pi. x = 3.14 – 1.14 = 2.00 radians x = 3.14 + 1.14 = 3.28 radians Final answer: x ≈ 2.00, 3.28, 2.00 + 2πn, and 3.28 + 2πn (Since the directions said to list all possible solutions we must put “+ 2πn” on each of our answer to represent that there will be another solution every 2π radians after our original solutions, because cosine y-values repeat every 2π radians.) .

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