1 Trigonometric Identities

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1 MTH 120 Spring 2008 Essex County College Division of Mathematics Handout Version 6 1 January 29, Trigonometric Identities 1.1 Review of The Circular Functions At this point in your mathematical development, you should be well aware of the equation of a circle that has radius r, and center (h, k. The form you most likely have seen before is: (x h 2 + (y k = r 2. For our introduction to trigonometry, we will take the center to be the origin, 2 hence the equation becomes: x 2 + y 2 = r 2. The trigonometric (circular functions are related to the coordinates along this circle as we rotate the radius. Of course the radius does not change, but the x and y do as we rotate the radius. This rotation will be with respect to the positive x axis: as we rotate couter-clockwise we say that the radius forms an positive angle with respect to the positive x-axis; and as we rotate clockwise we say that the radius forms a negative angle with respect to the positive x-axis. Here I will refer to this angle as theta, θ. Here s how they are defined. 1. The sine function. 2. The cosine function. 3. The tangent function. 4. The cotangent function. 5. The secant function. 6. The cosecant function. sin θ = y r cos θ = x r tan θ = y x cot θ = x y sec θ = r x csc θ = r y 1 This document was prepared by Ron Bannon using L A TEX 2ε. 2 Both h and k become zero. 1

2 1.2 Note An important note on potentially confusing notation. (sin θ 2 = (sin θ (sin θ = sin θ sin θ = sin 2 θ Get used to seeing the exponent on the function s name. 1.3 Fundamental Trigonometric Identities Reciprocal Identities Each will be explained using the definitions that were initially presented. These should all become second nature. csc θ = 1 sin θ sec θ = 1 cos θ cot θ = 1 tan θ tan θ = sin θ cos θ cot θ = cos θ sin θ Pythagorean Identities Each will be explained using the definitions that were initially presented, and their relationship to x 2 + y 2 = r 2. These should all become second nature. sin 2 θ + cos 2 θ = 1 tan 2 θ + 1 = sec 2 θ 1 + cot 2 θ = csc 2 θ Even-Odd Identities From the graphs you should be able to figure out which functions are even (symmetric with respect to the y-axis, and which are odd (symmetric with respect to the origin. Sine is odd, sin ( θ = sin θ Cosine is even, cos ( θ = cos θ Tangent is odd, tan ( θ = tan θ Cotangent is odd, cot ( θ = cot θ Secant is even, sec ( θ = sec θ Cosecant is odd, csc ( θ = csc θ Cofunction Identities You should be able to verify these by using simple graphs. Yes, you know how to graph each of these functions and you should verify that each of the following identities is true. sin 2 θ = cos θ cos 2 θ = sin θ tan 2 θ = cot θ 2

3 cot 2 θ = tan θ sec 2 θ = csc θ csc 2 θ = sec θ 1.4 Simplifying Trigonometric Expressions To simplify trigonometric expressions, we use the fundamental identities to rewrite as a simpler expression. 1. Simplify the trigonometric expression. 3 cos θ + tan θ sin θ 2. Simplify the trigonometric expression. sin θ cos θ + cos θ 1 + sin θ 3. Simplify the trigonometric expression. 4 cos 3 β + sin 2 β cos β 3 θ is the Greek letter theta. 4 β is the Greek letter beta. 3

4 4. Simplify the trigonometric expression. 5 sec α cos α tan α 5. Simplify the trigonometric expression. sin z csc z + cos z sec z 6. Simplify the trigonometric expression. cos y sec y + tan y 5 α is the Greek letter alpha. 4

5 7. Simplify the trigonometric expression sin γ cos γ + cos γ 1 + sin γ 1.5 Proving/Verifying Trigonometric Identities To verify that a trigonometric equation is an identity, we transform one side of the equation into the other side by a series of steps, each of which is itself an identity. Initially we will work with the fundamental identities only. Here s the general technique. Start with one side. It is usually best to choose the side that appears more complicated. Use known identities only, and try to use simple obvious steps that bring us closer to the other side. Converting to sines and cosines can in fact be very helpful in most circumstances. Examples follow. However, you need to do many more problems in order to become proficient at verifying trigonometric identities. 1. Verify the trigonometric identity. cos θ (sec θ cos θ = sin 2 θ 6 γ is the Greek letter gamma. 5

6 2. Verify the trigonometric identity. 2 tan x sec x = 1 1 sin x sin x 3. Verify the trigonometric identity. cos u = sec u + tan u 1 sin u 4. Verify the trigonometric identity cos δ cos δ = tan2 δ sec δ 1 7 δ is the Greek letter delta. 6

7 5. Verify the trigonometric identity. cos x cos y sin x + sin y + sin x sin y cos x + cos y = 0 6. Verify the trigonometric identity. sin 5 x = sin x ( 1 2 cos 2 x + cos 4 x 7. Verify the trigonometric identity. 8 sin ɛ 1 cos ɛ = 1 + cos ɛ sin ɛ 8 ɛ is the Greek letter epsilon. 7

You found trigonometric values using the unit circle. (Lesson 4-3)

You found trigonometric values using the unit circle. (Lesson 4-3) LEQ: How do we identify and use basic trigonometric identities to find trigonometric values & use basic trigonometric identities to simplify