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+ Using Pythagorean and Quotient Identities in Trig Equations By Briana J. Period 8 Algebra 2/ Trig + Quick Review Before we can get straight into what these identities are and how to use them we need to do a quick review on the following: n Right Triangles and labeling accordingly n Trig Functions n Reciprocal Trig functions + What is a right triangle and how do I label it correctly? A right triangle is simply a triangle that has an angle equaling 90°. Now how do you label it? a Hypotenuse Adjacent Opposite There are names for each side of a right triangle. The longest side is called the hypotenuse. The side next to the angle your trying to find, lets name that angle a, is labeled as adjacent. Lastly, the side opposite angle a is labeled opposite. + Trig functions There are three main trig functions you should know, those are Sine, Cosine, and Tangent Sine: Opposite θ Adjacent Cosine: Adjacent Hypotenuse Adjacent Hypotenuse Tangent: Opposite Adjacent Opposite A good way to remember these trig functions is using the acronym SOHCAHTOA + Reciprocal Trig Functions Just like there are trig functions there are reciprocal trig functions. The three you will need to know are Cosecant, Secant, and Cotangent. They can all be expressed two ways. Cosecant: Hypotenuse Opposite Secant: OR Hypotenuse Adjacent Cotangent: 1 θ 1 OR Adjacent Opposite Sin Cosine 1 OR H A O Tangent I like to write the acronym CSCSECCOT under SOHCAHTOA to remember which function is the opposite of the other. + Pythagorean Identities What is a Pythagorean Identities? There are three Pythagorean Identities we will be using. *Note: There are variations that pop up so make sure you know them. They are: n Sin2 θ+ Cos2θ=1 OR Sin2 θ=1- Cos2 θ OR Cos2 θ= 1- Sin2 θ n 1+ Tan2 θ= Sec2 OR Tan2 θ= Sec2 θ-1 n 1+ Cot2 θ=Csc2 θ OR Cot2 θ= Csc2 θ-1 + Using Pythagorean Identities Pythagorean Identities are used to find missing Trigonometric values and simplifying trigonometric expressions. Take a look at this example: If Csc θ= 5/3 and tan θ= 3/4, Find the remaining trig functions. First lets draw a triangle and label it Lets identify our Trig Functions. Tangent is Opposite over Adjacent. Cosecant is Hypotenuse over opposite, which is the opposite of sine. So the hypotenuse is 5, the Adjacent is 4, and the Opposite is 3. A=4 Now lets label θ Now that we have our Triangle labeled we can solve for the missing trig function which is Cosine. Cos θ= 4/5 Cos2 θ= 1- Sin2 θ Cos2 θ= 1- 16/25 Cos2 θ= 1- (3/5)2 θ Cos2 θ= 16/25 H=5 θ O=3 Since all values are positive this triangle must be in Quadrant 1. + Quotient Identities There are two quotient Identities you need to know. Tan θ= Sin θ Cos θ AND Cot θ= Cos θ Sin θ They are used to find tangent and the opposite of Tangent which is Cotangent + Using Quotient Identities Here is a practice problem: If Cos θ= 5/13 and Sin θ= 12/13, What is the value of Tan θ? First lets write the formula we will be using Tan θ= Sin θ Cos θ Now lets plug in our values Tan θ= Sin θ Cos θ 12 13 5 13 Multiply the Numerator and the Denominator by the reciprocal 12 13 13 X 5 5 X 13 13 5 Tan θ= 12/5 *After simplifying + Using Identities in Trig Equations Now lets practice what we learned. This is a trig equation. We have to solve it using the identities we learned. 2 Cos2x + 3 Sin x- 3 = 0 This equation uses a Pythagorean Identity. Now Factor and Solve It uses the Identity Sin2 x + Cos2 x = 1. Now lets plug it in. 2 Cos2x + 3 Sin x- 3 = 0 2(1- Sin2x)+ 3 sin x- 3=0 Now simplify 2(1- Sin2x)+ 3 sin x- 3=0 2-2 sin2x+ 3 sin x-3=0 -2 sin2x + 3sin x-1=0 Multiply by -1 -1(-2 sin2x + 3sin x-1)=0 2 sin2 x – 3 sin x + 1=0 2 sin2 x – 3 sin x + 1=0 (2 sin x -1)(sin x – 1)=0 2 sin x= 0 OR sin x – 1=0 2 sin x= 1 sin x = 1 Sin x = ½ X= π/6, 5π/6 or x= π/2 + Practice Links n http://www.regentsprep.org/Regents/math/algtrig/ATT10/ trigequations2.htm n http://www.regentsprep.org/Regents/math/algtrig/ATT9/pythpractice.htm n http://www.pc.maricopa.edu/ctlt/titleV/Math/RelatingTrigFunctions/ RelatingTrigFunctions_1_63.html